diff b1f9bc4..56bc728 hypersymplectic_hyperkahler_deformation_v2.txt
diff --git a/hypersymplectic_hyperkahler_deformation_v2.txt b/hypersymplectic_hyperkahler_deformation_v2.txt
index d5b9c64..180985e 100644
--- a/hypersymplectic_hyperkahler_deformation_v2.txt
+++ b/hypersymplectic_hyperkahler_deformation_v2.txt
@@ -55,7 +55,7 @@ tableofcontents
# Introduction
-A hypersymplectic structure on an oriented four-manifold $X$ is a
+A hypersymplectic structure on an oriented four-manifold X is a
triple of closed real two-forms whose pointwise span is three-dimensional
and positive definite for the wedge-product quadratic form. Such a triple
determines a conformal structure, but its forms need not be parallel:
@@ -65,18 +65,18 @@ deform the forms to a hyperk"ahler triple while retaining their three
cohomology classes. This paper answers yes.
[Main]
-Let $X$ be a closed connected oriented smooth four-manifold, and let
-$omega=(omega_1,omega_2,omega_3)$ be a smooth hypersymplectic
+Let X be a closed connected oriented smooth four-manifold, and let
+omega=(omega_1,omega_2,omega_3) be a smooth hypersymplectic
structure normalized by
int_Xomega_iwedgeomega_j=delta_{ij}.
-There is a smooth family of hypersymplectic structures $omega(t)$,
-$0le tle1$, such that
+There is a smooth family of hypersymplectic structures omega(t),
+0le tle1, such that
[
- omega(0)=omega,qquad [omega_i(t)]=[omega_i]quad(i=1,2,3),
+ omega(0)=omega, [omega_i(t)]=[omega_i](i=1,2,3),
]
-and a positive volume form $mu_1$ with
+and a positive volume form mu_1 with
omega_i(1)wedgeomega_j(1)=2delta_{ij}mu_1.
@@ -86,72 +86,72 @@ The endpoint forms are parallel and self-dual for a hyperk"ahler metric.
The normalization (eq:intro-normalization) is a constant linear
change of basis and positive scaling; it makes the endpoint identity
cleaner but does not restrict the geometry. If the initial triple has
-an arbitrary positive Gram matrix $G_{ij}=int_Xomega_iwedgeomega_j$,
-one applies the theorem to $Lomega$ for a constant invertible real
-$L$ with $LGL^{mathsf T}sfT=I_3$ and applies $L^{-1}$ to the resulting
+an arbitrary positive Gram matrix G_{ij}=int_Xomega_iwedgeomega_j,
+one applies the theorem to Lomega for a constant invertible real
+L with LGL^{mathsf T}sfT=I_3 and applies L^{-1} to the resulting
path. The endpoint then satisfies
-$omega_i(1)wedgeomega_j(1)=2G_{ij}mu_1$: the three classes are
+omega_i(1)wedgeomega_j(1)=2G_{ij}mu_1: the three classes are
parallel self-dual forms for a hyperk"ahler metric, with their original
-Gram matrix $G$.
+Gram matrix G.
The theorem has two immediate consequences. First, every single positive
combination of the deformed triple is a K"ahler form on a compatible
integrable complex structure, and it is a K"ahler form for the same
-manifold; on the closed connected $X$ the symplectic form determines
+manifold; on the closed connected X the symplectic form determines
the K"ahler class in advance. Second, the topology of the initial
manifold is forced: the existence of any such closed positive triple
-already implies $X$ is diffeomorphic to a standard K3 surface or to
-$T^4$. Both are corollaries of the fact that the endpoint is a global
+already implies X is diffeomorphic to a standard K3 surface or to
+T^4. Both are corollaries of the fact that the endpoint is a global
hyperk"ahler metric.
[Individual K"ahler realizations]
-Let $X$ be a closed connected oriented four-manifold with a smooth
-positive triple $omega$ of closed real two-forms. For every fixed
-$cinR^3∖{0}$, the form $alpha_c=sum_i c_iomega_i$
-is a K"ahler form for a parallel integrable complex structure $J_c$ of
-a hyperk"ahler metric $g_c$ on $X$:
+Let X be a closed connected oriented four-manifold with a smooth
+positive triple omega of closed real two-forms. For every fixed
+cinR^3∖{0}, the form alpha_c=sum_i c_iomega_i
+is a K"ahler form for a parallel integrable complex structure J_c of
+a hyperk"ahler metric g_c on X:
[
alpha_c(v,w)=g_c(J_cv,w).
]
-The compatible structure is chosen separately for each $c$; no
+The compatible structure is chosen separately for each c; no
simultaneous transport of the triple is asserted.
Proof.
Use the fixed-class path for the original basis described above and put
-$alpha_t=sum_i c_iomega_i(t)$. Positivity of the pointwise
-wedge-product matrix gives $alpha_twedgealpha_t>0$, so $alpha_t$
-is symplectic; its class is fixed because each $[omega_i(t)]$ is fixed.
-
-For the endpoint metric $g$, set $r_c=(c^{mathsf T}sfTGc)^{1/2}>0$. The
-endpoint identities show that $alpha_1/r_c$ is parallel and self-dual
-with $g$-norm $sqrt2$. It therefore defines a parallel integrable
-complex structure $J^{end}_c$ with
-$alpha_1/r_c=g(J^{end}_c,cdot,cdot)$,
-so that $alpha_1$ is its K"ahler form for the hyperk"ahler metric
-$r_cg$.
-
-Since $X$ is closed, Moser's theorem gives an isotopy $phi_{c,t}$ with
-$phi_{c,0}=id$ and $phi_{c,t}^*alpha_t=alpha_c$.
-Pulling back $r_cg$ and $J^{end}_c$ by $phi_{c,1}$ gives the
-claimed hyperk"ahler metric and compatible integrable structure on $X$.
+alpha_t=sum_i c_iomega_i(t). Positivity of the pointwise
+wedge-product matrix gives alpha_twedgealpha_t>0, so alpha_t
+is symplectic; its class is fixed because each [omega_i(t)] is fixed.
+
+For the endpoint metric g, set r_c=(c^{mathsf T}sfTGc)^{1/2}>0. The
+endpoint identities show that alpha_1/r_c is parallel and self-dual
+with g-norm sqrt2. It therefore defines a parallel integrable
+complex structure J^{end}_c with
+alpha_1/r_c=g(J^{end}_c,cdot,cdot),
+so that alpha_1 is its K"ahler form for the hyperk"ahler metric
+r_cg.
+
+Since X is closed, Moser's theorem gives an isotopy phi_{c,t} with
+phi_{c,0}=id and phi_{c,t}^*alpha_t=alpha_c.
+Pulling back r_cg and J^{end}_c by phi_{c,1} gives the
+claimed hyperk"ahler metric and compatible integrable structure on X.
Fine and Yao note this Moser implication for a constituent in
cite[Section~1]{FineYao2017}.
[Smooth type]
-Let $X$ be a closed connected smooth four-manifold admitting a smooth
-positive triple of closed real two-forms. Then $X$ is diffeomorphic to
-the standard K3 manifold or to $T^4$.
+Let X be a closed connected smooth four-manifold admitting a smooth
+positive triple of closed real two-forms. Then X is diffeomorphic to
+the standard K3 manifold or to T^4.
Proof.
-Give $X$ the orientation induced by the triple. The normalization above
-and Theorem~thm:main give a global hyperk"ahler triple on $X$.
-Choosing one of its complex structures makes $X$ a compact K"ahler
+Give X the orientation induced by the triple. The normalization above
+and Theorem~thm:main give a global hyperk"ahler triple on X.
+Choosing one of its complex structures makes X a compact K"ahler
surface with a nowhere-vanishing holomorphic two-form. The compact
hyperk"ahler surface classification cite[Section~2, p.~162]{Boyer1988}
-therefore identifies $X$ itself as a complex two-torus or a K3 surface.
-A complex two-torus is smoothly $T^4$, while every complex K3 surface is
-diffeomorphic to a nonsingular quartic surface in $mathbf P^3$,bfP^3$, the
+therefore identifies X itself as a complex two-torus or a K3 surface.
+A complex two-torus is smoothly T^4, while every complex K3 surface is
+diffeomorphic to a nonsingular quartic surface in mathbf P^3,bfP^3, the
standard K3 manifold cite[p.~58]{Kodaira1970}.
@@ -173,7 +173,7 @@ given triple to a metric with its prescribed periods, beyond mere
existence of some hyperk"ahler metric on the same manifold.
The main analytic approach until now has been the hypersymplectic flow.
-Fine and Yao obtained this flow by reducing the $G_2$-Laplacian flow
+Fine and Yao obtained this flow by reducing the G_2-Laplacian flow
on the product with a three-torus. They proved that a finite-time
solution extends while the associated seven-dimensional scalar
curvature, equivalently the torsion, remains bounded
@@ -188,11 +188,11 @@ Huang, Wang and Yao proved convergence, after pullback by
diffeomorphisms, for simple-type hypersymplectic structures on the
standard four-torus, a diagonal one-variable class
cite[Theorems~3.5 and~3.6]{HuangWangYao2018}. Picard and Suan treated
-$G_2$ flows associated with K"ahler Calabi--Yau data; their
+G_2 flows associated with K"ahler Calabi--Yau data; their
complex-dimension-two case supplies a hypersymplectic convergence
theorem cite[Theorem~1.3]{PicardSuan2024}. Fine, He and Yao extended
the simple-type convergence result, modulo diffeomorphisms, to
-$T^3$-invariant triples on $T^4$ in symmetric normal form
+T^3-invariant triples on T^4 in symmetric normal form
cite[Theorem~1.5]{FineHeYao2024}. They subsequently proved a linear
cohomologous isotopy to a hyperk"ahler triple when an effective smooth
circle action preserves the initial triple cite[Theorem~1.2]{FineHeYao2025}.
@@ -207,12 +207,12 @@ the hypersymplectic flow.
The topology already places strong restrictions on such a triple. Any
two of its forms determine an almost-complex structure whose canonical
line is trivial, and the third form tames that structure. Here a real
-two-form $beta$ tames $J$ if $beta(v,Jv)>0$ for every nonzero
-vector $v$; it is compatible if it is also $J$-invariant. A
+two-form beta tames J if beta(v,Jv)>0 for every nonzero
+vector v; it is compatible if it is also J-invariant. A
closed tamer is symplectic. Bauer's bound for symplectic four-manifolds
with torsion first Chern class cite[Corollary~1.2]{Bauer2008},
-together with the three positive period classes, gives $b^+=3$, where
-$b^+$ is the positive index of the intersection form. The resulting
+together with the three positive period classes, gives b^+=3, where
+b^+ is the positive index of the intersection form. The resulting
free intersection lattice is the K3 lattice or three hyperbolic planes.
These facts supply the lattice used below; they do not presuppose a
smooth identification of the initial manifold with a K3 surface or torus.
@@ -240,37 +240,37 @@ class needed along the deformation.
We want a path of positive triples with fixed classes ending at a
hyperk"ahler triple. Every positive triple factors, pointwise, as a
-definite pair $(A,C)$ that spans the anti-invariant forms of an
-almost-complex structure $J$, plus a third form $B$ that tames $J$. A
-deformation of the pair alone, moving $(A,C)$ while $B$ is held fixed,
-keeps the triple positive if and only if the fixed $B$ continues to tame
-the moving $J$. Thus the entire question reduces to:
+definite pair (A,C) that spans the anti-invariant forms of an
+almost-complex structure J, plus a third form B that tames J. A
+deformation of the pair alone, moving (A,C) while B is held fixed,
+keeps the triple positive if and only if the fixed B continues to tame
+the moving J. Thus the entire question reduces to:
Along a deformation of the first two forms, keep the original third
-class $[B]$ available as a taming form.
+class [B] available as a taming form.
-Whether a real class $H$ contains a smooth tamer for a tamed
-almost-complex structure $J$ on a closed four-manifold is decided by two
-ingredients. First, a tamer exists at all iff $H$ has positive
-self-intersection and every irreducible closed $J$-holomorphic curve
-pairs with $H$ positively. Second, if one taming form is already known
-for $J$, then any $H$ in the same component of the positive
+Whether a real class H contains a smooth tamer for a tamed
+almost-complex structure J on a closed four-manifold is decided by two
+ingredients. First, a tamer exists at all iff H has positive
+self-intersection and every irreducible closed J-holomorphic curve
+pairs with H positively. Second, if one taming form is already known
+for J, then any H in the same component of the positive
time-like cone whose curve pairings are all positive is also a taming
class. The problem now becomes: describe the curves in the moving
-almost-complex structures, so that $[B]$ is guaranteed to pair
+almost-complex structures, so that [B] is guaranteed to pair
positively with all of them.
The curves are provided by Seiberg--Witten theory. The starting data
gives a natural family of almost-complex structures parametrized by
-$S^2$, one for each coefficient direction of the initial triple.
+S^2, one for each coefficient direction of the initial triple.
Taubes's correspondence turns solutions of the large-parameter twisted
-Seiberg--Witten equations for the coefficient direction $v$ into
-$J_v$-holomorphic curves. Wall-crossing for this family then forces, in
+Seiberg--Witten equations for the coefficient direction v into
+J_v-holomorphic curves. Wall-crossing for this family then forces, in
any prescribed class of the coefficient sphere, a solution whose first
-spinor component vanishes at a chosen point; passing to the large-$r$
-limit, we obtain, for every point of $X$, a $J_v$-curve through that
+spinor component vanishes at a chosen point; passing to the large-r
+limit, we obtain, for every point of X, a J_v-curve through that
point. The integral class can be chosen, using the lattice arithmetic
that we set up later, so that no positive configuration in that class
can split into components or multiply. That class then gives an
@@ -278,50 +278,50 @@ elliptic fibration whose fibers are precisely the unique curves through
each point.
The torus fibration is what makes the rest tractable. Along the regular
-region, the pair $(A,C)$ is then averaged along the torus action to a
-purely invariant pair. A closed invariant two-form $Psi$ with unit
+region, the pair (A,C) is then averaged along the torus action to a
+purely invariant pair. A closed invariant two-form Psi with unit
integral over every fiber is set up on top of this. Averaging and a
-small exact correction orthogonalize $Psi$, so that $(A,C)$ together
-with $Psi$ looks semilocally like a definite pair plus a vanishing
+small exact correction orthogonalize Psi, so that (A,C) together
+with Psi looks semilocally like a definite pair plus a vanishing
two-form. On each regular region we then construct a semi-flat, closed
-two-form that satisfies the three pointwise identities $AD=CD=0$,
-$D^2=A^2$ exactly; on each nodal region we use Gibbons--Hawking /
+two-form that satisfies the three pointwise identities AD=CD=0,
+D^2=A^2 exactly; on each nodal region we use Gibbons--Hawking /
Ooguri--Vafa and Gross--Wilson local models to build the same
approximate data on top of a collapsing holomorphic model. The
semi-flat and nodal forms differ only up to exponentially small error on
fixed annuli, and a suitable correction primitive glues them to a
single closed approximate solution. The remaining correction is small
and is solved on exact two-forms, where a Stokes identity controls the
-inverse uniformly as the fiber area $eps$ shrinks.
+inverse uniformly as the fiber area eps shrinks.
-The result is a closed form $D_eps$ satisfying the three pointwise
-identities exactly. The pair $(A,D_eps)$ is then itself integrable (by
-Cartan/Newlander--Nirenberg), and $A+iD_eps$ is a nowhere-vanishing
+The result is a closed form D_eps satisfying the three pointwise
+identities exactly. The pair (A,D_eps) is then itself integrable (by
+Cartan/Newlander--Nirenberg), and A+iD_eps is a nowhere-vanishing
holomorphic two-form for that complex structure. The endpoint of the
argument uses two applications of the classical K"ahler-class
-criterion: put the original $[C]$ into the K"ahler cone, replace $C$ by
-the resulting K"ahler representative $C_1$, and repeat with $[B]$ to
-get $B_1$. Along the second step the class $[B]$ is guaranteed to contain
-a tamer because the class $[D_eps]$ already contains one and $[D_eps]$,
-$[B]$ lie in the same open time-like cone of $H^2(X;R)$. The final
-triple $(A,C_1,B_1)$ has orthogonal equal-square components, hence is
+criterion: put the original [C] into the K"ahler cone, replace C by
+the resulting K"ahler representative C_1, and repeat with [B] to
+get B_1. Along the second step the class [B] is guaranteed to contain
+a tamer because the class [D_eps] already contains one and [D_eps],
+[B] lie in the same open time-like cone of H^2(X;R). The final
+triple (A,C_1,B_1) has orthogonal equal-square components, hence is
hyperk"ahler.
## Organization of the paper
Section~sec:definite-pairs sets up the local model (definite pairs,
-the associated almost-complex structure, the topology of $X$) and the
+the associated almost-complex structure, the topology of X) and the
coefficient sphere of almost-complex structures. Section~sec:current-energy
and Section~sec:density-splitting develop the current-cone
separation and the taming criterion, which are the main analytic input
-to the "keep $[B]$ available" step. Section~sec:families constructs
+to the "keep [B] available" step. Section~sec:families constructs
Seiberg--Witten curves through every point. Section~sec:lattice-chamber
-chooses the integral null class $F$ so that no positive configuration in
-$F$ can split. Section~sec:pencil turns the $F$-curves into a torus
+chooses the integral null class F so that no positive configuration in
+F can split. Section~sec:pencil turns the F-curves into a torus
fibration with ordinary nodal fibers. Section~sec:local-models builds
the collapsing nodal models. Section~sec:regular-preparation
prepares the pair by averaging and constructs the unit-fiber-area form
-$Psi$. Section~sec:auxiliary-volume solves the auxiliary volume
+Psi. Section~sec:auxiliary-volume solves the auxiliary volume
equation. Section~sec:endpoint finishes the proof of
Theorem~thm:main via the two K"ahler steps.
@@ -335,7 +335,7 @@ statements use
Λ=H^2(X;Z)/torsion.
]
The original positive period plane is
-$P=span{[omega_1],[omega_2],[omega_3]}$. No
+P=span{[omega_1],[omega_2],[omega_3]}. No
rationality of this plane is assumed. All deformations of forms are
smooth in the manifold variable; parameter regularity is stated where it
is used. Constants in local estimates may depend on fixed smooth
@@ -344,93 +344,93 @@ background data and on the indicated finite differentiation order.
# Definite pairs and the coefficient sphere
A positive triple carries redundant data. The first task of this
-section is to organize it: a definite pair $(A,C)$ determines an
-almost-complex structure, and a third form $B$ completes a positive
+section is to organize it: a definite pair (A,C) determines an
+almost-complex structure, and a third form B completes a positive
triple precisely when it tames that structure. This turns the main
deformation of a positive triple into a deformation of the pair alone,
-with $B$ fixed. The second task is to record global structure: the
-topology of $X$ is forced, the self-dual harmonic space has a natural
-basis, and the initial triple parametrizes a whole $S^2$ of
+with B fixed. The second task is to record global structure: the
+topology of X is forced, the self-dual harmonic space has a natural
+basis, and the initial triple parametrizes a whole S^2 of
almost-complex structures. That sphere will be the parameter space used
in Section~sec:families to produce curves.
## Definite pairs and the associated almost-complex structure
-A pair $(A,C)$ of real two-forms is definite if its
+A pair (A,C) of real two-forms is definite if its
wedge-product matrix is positive definite at every point. A two-form
-$B$ tames an almost-complex structure $J$ if
-$B(v,Jv)>0$ for every nonzero tangent vector $v$. It is
-compatible if, in addition, $B(Ju,Jv)=B(u,v)$. When these forms
+B tames an almost-complex structure J if
+B(v,Jv)>0 for every nonzero tangent vector v. It is
+compatible if, in addition, B(Ju,Jv)=B(u,v). When these forms
are called symplectic, closedness is included.
-A definite pair $(A,C)$ on an oriented four-manifold determines an
+A definite pair (A,C) on an oriented four-manifold determines an
almost-complex structure up to sign. Its anti-invariant real two-form
-bundle is pointwise spanned by $A,C$, and its canonical complex line
-bundle is trivial. A third form $B$ completes a positive triple if and
-only if $B$ tames one of the two signs of this almost-complex structure.
-On a connected manifold the sign is fixed by $B$.
+bundle is pointwise spanned by A,C, and its canonical complex line
+bundle is trivial. A third form B completes a positive triple if and
+only if B tames one of the two signs of this almost-complex structure.
+On a connected manifold the sign is fixed by B.
-A real two-plane on which both $A$ and $C$ vanish is a complex line,
+A real two-plane on which both A and C vanish is a complex line,
without specifying its orientation.
Proof.
Ratios of positive four-forms are smooth functions. Set
- x=(AC)/(A^2),qquad
- y=((C^2)/(A^2)-x^2)^{1/2}>0,qquad
+ x=(AC)/(A^2),
+ y=((C^2)/(A^2)-x^2)^{1/2}>0,
T=(C-xA)/(y).
-Then $AT=0$ and $T^2=A^2>0$, so
+Then AT=0 and T^2=A^2>0, so
[
- Ω=A+iT,qquad Ω^2=0,qquad
+ Ω=A+iT, Ω^2=0,
ΩoverlineoverlineΩ=2A^2>0.Ω=2A^2>0.
]
-The form $Ω$ is complex-decomposable; its kernel in the
+The form Ω is complex-decomposable; its kernel in the
complexified tangent bundle is complementary to its conjugate. Taking
-this kernel as $T^{0,1}$ defines an almost-complex structure with the
-given orientation. Replacing $y$ by $-y$ conjugates $Ω$ and changes
-$J$ to $-J$. The real and imaginary parts of $Ω$ span the
-anti-invariant forms, and $Ω$ is a nowhere-zero section of the
+this kernel as T^{0,1} defines an almost-complex structure with the
+given orientation. Replacing y by -y conjugates Ω and changes
+J to -J. The real and imaginary parts of Ω span the
+anti-invariant forms, and Ω is a nowhere-zero section of the
canonical bundle.
-Choose any Hermitian metric for $J$. The invariant two-forms are the
-wedge-orthogonal complement of $span(A,C)$ and have
-signature $(1,3)$. For the decomposition $B=B^{1,1}+B^{2,0+0,2}$, the
-positive-triple condition is exactly $(B^{1,1})^2>0$. In complex
-dimension two, a real $(1,1)$-form has positive square precisely when
+Choose any Hermitian metric for J. The invariant two-forms are the
+wedge-orthogonal complement of span(A,C) and have
+signature (1,3). For the decomposition B=B^{1,1}+B^{2,0+0,2}, the
+positive-triple condition is exactly (B^{1,1})^2>0. In complex
+dimension two, a real (1,1)-form has positive square precisely when
its two Hermitian eigenvalues have the same strict sign. The positive
component of that condition is the cone of forms positive on every
complex line. The anti-invariant summand vanishes on such lines, so the
-positive-triple condition is equivalent to taming by $B$. The other
-component is positive for $-J$. Connectedness fixes the choice
+positive-triple condition is equivalent to taming by B. The other
+component is positive for -J. Connectedness fixes the choice
continuously and globally.
-Finally, for independent real vectors $u,v$, the equations
-$A(u,v)=C(u,v)=0$ are equivalent to $Ω(u,v)=0$. In a complex
+Finally, for independent real vectors u,v, the equations
+A(u,v)=C(u,v)=0 are equivalent to Ω(u,v)=0. In a complex
two-dimensional vector space with a nondegenerate complex volume form,
-this says that $u,v$ are complex-dependent, equivalently that their
+this says that u,v are complex-dependent, equivalently that their
real span is a complex line.
-For a fixed almost-complex structure $J$, the set of taming two-forms
+For a fixed almost-complex structure J, the set of taming two-forms
is an open convex cone. Its intersection with closed representatives of
any one class is also convex. The associated notion of positivity will
-always use the sign of $J$ fixed by a stated tamer.
+always use the sign of J fixed by a stated tamer.
-## Topology of $X$
+## Topology of X
-If $X$ carries a positive closed triple, then
+If X carries a positive closed triple, then
[
- b^+(X)=3,qquad
+ b^+(X)=3,
(b_1(X),b^-(X))=(0,19) or (4,3).
]
The free integral intersection lattice is respectively
-$3H ⊕ 2(-E_8)$ or $3H$, where $H$ is the hyperbolic plane.
+3H ⊕ 2(-E_8) or 3H, where H is the hyperbolic plane.
Every almost-complex structure obtained from a definite pair in the
triple has vanishing integral first Chern class.
@@ -442,20 +442,20 @@ bundle is trivial. The first Chern class of this symplectic form
therefore vanishes: the space of its tamed almost-complex structures is
contractible and contains compatible structures. Bauer's bound for a
closed symplectic four-manifold with torsion first Chern class gives
-$b^+le3$ cite[Corollary~1.2]{Bauer2008}. The three independent
+b^+le3 cite[Corollary~1.2]{Bauer2008}. The three independent
positive cohomology classes give the reverse inequality.
-The characteristic-number identity $c_1^2=2chi+3sigma$, obtained
-from $p_1=c_1^2-2c_2$, the Euler-number identity and the signature
+The characteristic-number identity c_1^2=2chi+3sigma, obtained
+from p_1=c_1^2-2c_2, the Euler-number identity and the signature
theorem cite[Corollaries~11.12 and~15.5, Theorem~19.4]{MilnorStasheff1974},
-together with $b^+=3$, gives
+together with b^+=3, gives
[
- 0=19-4b_1-b^-,qquad sigma=4b_1-16.
+ 0=19-4b_1-b^-, sigma=4b_1-16.
]
-Moreover $w_2equiv c_1=0pmod 2$, so $X$ is spin
+Moreover w_2equiv c_1=0pmod 2, so X is spin
cite[Problem~14-B]{MilnorStasheff1974}. Rokhlin's theorem makes
-$sigma$ divisible by $16$ [cite]. Since $b^-ge0$, these
-relations give $b_1in{0,4}$ and the displayed alternatives.
+sigma divisible by 16 [cite]. Since b^-ge0, these
+relations give b_1in{0,4} and the displayed alternatives.
Poincar'e duality makes the free intersection form unimodular, and the
Wu formula makes it even cite[Theorem~11.14]{MilnorStasheff1974}.
Classification of indefinite even unimodular lattices now gives the two
@@ -465,7 +465,7 @@ Only the intersection lattice has been classified at this stage.
The positive three-plane bundle spanned by the triple is the self-dual
bundle of a unique oriented conformal structure. Choose one metric
-$g_0$ in that conformal class. Since the original forms are closed and
+g_0 in that conformal class. Since the original forms are closed and
self-dual, they are harmonic, and
mathcal H^+_calH^+_{g_0}=span_R{omega_1,omega_2,omega_3}
@@ -475,25 +475,25 @@ global forms with constant coefficients.
## The coefficient sphere of almost-complex structures
-For $vin S^2subsetR^3$, let $B_v=vcdotomega$ and let
-$mathcal A_v$calA_v$ be the plane of forms $acdotomega$ with $aperp v$.
-Choosing the sign tamed by $B_v$ defines an almost-complex structure
-$J_v$. This construction uses the plane $mathcal A_v$calA_v$ itself and does
-not require a global choice of ordered basis over $S^2$.
+For vin S^2subsetR^3, let B_v=vcdotomega and let
+mathcal A_vcalA_v be the plane of forms acdotomega with aperp v.
+Choosing the sign tamed by B_v defines an almost-complex structure
+J_v. This construction uses the plane mathcal A_vcalA_v itself and does
+not require a global choice of ordered basis over S^2.
-The family $J_v$ is smooth and is Hermitian for $g_0$. At every point
-of $X$, its map to the twistor sphere has degree of absolute value one.
-The closed $J_v$-anti-invariant forms form the two-dimensional space
-$mathcal A_v$.calA_v$.
+The family J_v is smooth and is Hermitian for g_0. At every point
+of X, its map to the twistor sphere has degree of absolute value one.
+The closed J_v-anti-invariant forms form the two-dimensional space
+mathcal A_v.calA_v.
Proof.
The first assertion follows from the construction and the self-dual
description of Hermitian almost-complex structures in dimension four.
-At a fixed point, let $K$ be the positive Gram matrix of the triple for
-$g_0$. A vector normal to the coefficient plane $v^perp$, for this
-Gram matrix, has coefficients $K^{-1}v$. Its pairing with $B_v$ is
+At a fixed point, let K be the positive Gram matrix of the triple for
+g_0. A vector normal to the coefficient plane v^perp, for this
+Gram matrix, has coefficients K^{-1}v. Its pairing with B_v is
positive. Thus, after normalization, the fundamental forms give the map
[
v⟼frac{K^{-1}v}{|K^{-1}v|}
@@ -503,63 +503,63 @@ to the identity through positive definite matrices, so this map has
degree one. The orientation convention identifying coefficient and
twistor spheres can only change its sign.
-A closed anti-invariant form is self-dual for $g_0$, hence harmonic.
+A closed anti-invariant form is self-dual for g_0, hence harmonic.
By (eq:pair-harmonic-basis) it is a constant combination of the
-original triple. It is anti-invariant for $J_v$ exactly when that
-coefficient vector lies in $v^perp$. This proves the final assertion.
+original triple. It is anti-invariant for J_v exactly when that
+coefficient vector lies in v^perp. This proves the final assertion.
## The Lorentz space and the integrability identity
After a constant orthogonal change of coefficients we denote a chosen
-ordered triple by $(A,C,B)$. For all later exact deformations of the
+ordered triple by (A,C,B). For all later exact deformations of the
pair, set
V=[A]^perpcap[C]^perpsubset H^2(X;R).
-This space has signature $(1,b^-)$, and $[B]in V$ is a unit timelike
+This space has signature (1,b^-), and [B]in V is a unit timelike
vector. It determines the future component of the timelike cone.
-Let $A,D$ be closed real two-forms with $AD=0$ and $A^2=D^2>0$
-pointwise. Then $A+iD$ defines an integrable complex structure and is a
-nowhere-zero holomorphic two-form for it. If a closed third form $C$
-has $AC=DC=0$ and $C^2=A^2$, then the resulting triple is
+Let A,D be closed real two-forms with AD=0 and A^2=D^2>0
+pointwise. Then A+iD defines an integrable complex structure and is a
+nowhere-zero holomorphic two-form for it. If a closed third form C
+has AC=DC=0 and C^2=A^2, then the resulting triple is
hyperk"ahler, after choosing the sign of the complex structure so that
-$C$ is positive.
+C is positive.
Proof.
-Set $Ω=A+iD$ and use its kernel as $T^{0,1}$ as in
-Lemma~lem:definite-pair. For vector fields $U,V$ in that kernel,
-Cartan's formula and $ddΩ=0$ give
+Set Ω=A+iD and use its kernel as T^{0,1} as in
+Lemma~lem:definite-pair. For vector fields U,V in that kernel,
+Cartan's formula and ddΩ=0 give
[
- mathcal L_UΩ=0,calL_UΩ=0,qquad
+ mathcal L_UΩ=0,calL_UΩ=0,
iota_{[U,V]}Ω
=mathcal L_U(calL_U(iota_VΩ)-iota_V(mathcal L_UΩ)=0.calL_UΩ)=0.
]
The kernel is involutive. The smooth Newlander--Nirenberg theorem
therefore gives integrability [cite], and
-closedness of the $(2,0)$-form $Ω$ gives holomorphicity.
+closedness of the (2,0)-form Ω gives holomorphicity.
-Under the additional assumptions, $C$ is a positive closed
-$(1,1)$-form, hence a K"ahler form. Its metric volume is
-$C^2/2=A^2/2$. The ratio $ΩoverlineoverlineΩ/C^2$Ω/C^2$ is constant, so the
+Under the additional assumptions, C is a positive closed
+(1,1)-form, hence a K"ahler form. Its metric volume is
+C^2/2=A^2/2. The ratio ΩoverlineoverlineΩ/C^2Ω/C^2 is constant, so the
holomorphic volume form has constant pointwise norm. The Chern
-connection of the canonical line then makes $Ω$ parallel. For a
+connection of the canonical line then makes Ω parallel. For a
K"ahler metric this is the connection induced by the Levi-Civita
-connection. Thus $A,D,C$ are parallel; their normalized self-dual
+connection. Thus A,D,C are parallel; their normalized self-dual
algebra gives the hyperk"ahler structure. Explicitly, if
-$(eta_1,eta_2,eta_3)=(A,D,C)$ and $mu=A^2/2$, then
-$eta_iwedgeeta_j=2delta_{ij}mu$; thus the normalized matrix is
+(eta_1,eta_2,eta_3)=(A,D,C) and mu=A^2/2, then
+eta_iwedgeeta_j=2delta_{ij}mu; thus the normalized matrix is
the identity.
# Separation, quadratic mass, and angular energy
-The pair deformation keeps $B$ fixed, so the question becomes whether
-the fixed class $[B]$ still contains a tamer for the moving
+The pair deformation keeps B fixed, so the question becomes whether
+the fixed class [B] still contains a tamer for the moving
almost-complex structure. To answer this we need a criterion, and the
criterion goes through separating the cone of positive forms from those
that fail to contain a tamer. That separation produces a positive
@@ -570,201 +570,201 @@ density decomposition of Ssec:density-splitting.
## Conventions and separation
-Fix a smooth almost-complex structure $J$ on the closed four-manifold
-$X$, give $X$ its complex orientation, and choose a smooth $J$-Hermitian
-metric $g$. Its fundamental form is $G(u,v)=g(Ju,v)$. The pointwise
+Fix a smooth almost-complex structure J on the closed four-manifold
+X, give X its complex orientation, and choose a smooth J-Hermitian
+metric g. Its fundamental form is G(u,v)=g(Ju,v). The pointwise
orthogonal decomposition is
Λ^2=I ⊕ E,
- qquad I=Λ_J^+=R G ⊕ Λ_g^-,
- qquad E=Λ_J^-subsetΛ_g^+.
+ I=Λ_J^+=R G ⊕ Λ_g^-,
+ E=Λ_J^-subsetΛ_g^+.
-Let $R$ denote the orthogonal projection onto $E$. A unit
-complex-line bivector is $ell=ewedge Je$, where $|e|_g=1$. We use $g$
+Let R denote the orthogonal projection onto E. A unit
+complex-line bivector is ell=ewedge Je, where |e|_g=1. We use g
to identify bivectors and two-forms, so
- |ell|_g=1,qquad ell^+=G/2,qquad G(ell)=1,
- qquad |G|_g^2=2.
+ |ell|_g=1, ell^+=G/2, G(ell)=1,
+ |G|_g^2=2.
The bundle of these bivectors is denoted by
-$mathcal L→calL→ X$.
+mathcal L→calL→ X.
A two-current is a continuous real linear functional on
-$Ω^2(X)=C^∞(X,Λ^2)$. Under the metric identification its
+Ω^2(X)=C^∞(X,Λ^2). Under the metric identification its
coefficient is a distributional two-form, denoted by the same letter:
-$T(xi)=< T,xi>$. Thus
+T(xi)=< T,xi>. Thus
T is closed ⇔
T(ddalpha)=0 for all alphainΩ^1(X)
⇔ dd^*T=0
⇔ dd(*T)=0.
-The cohomology class dual to a closed current is $[*T]in H^2(X;R)$.
+The cohomology class dual to a closed current is [*T]in H^2(X;R).
For smooth coefficients this convention gives
-$T(xi)=int_Xxiwedge *T$. Inner products between $L^2$ coefficients
-are written $<,,,,>_2$.
+T(xi)=int_Xxiwedge *T. Inner products between L^2 coefficients
+are written <,,,,>_2.
Let
[
mathcal P_J=calP_J={xiinΩ^2(X):xi(ell)>0
for every ellinmathcal LcalL},
- qquad
+
mathcal Z_J=calZ_J={xiin C^∞(X,I):ddxi=0}.
]
-The set $mathcal P_J$calP_J$ is an open convex cone in the smooth topology;
+The set mathcal P_JcalP_J is an open convex cone in the smooth topology;
openness already holds in the uniform topology. Its closed elements are
-precisely the symplectic forms taming $J$. The elements of
-$mathcal P_JcalP_Jcapmathcal Z_J$calZ_J$ are the compatible symplectic forms.
+precisely the symplectic forms taming J. The elements of
+mathcal P_JcalP_Jcapmathcal Z_JcalZ_J are the compatible symplectic forms.
## The two separators
[The two separators]
The following hold.
[label=(roman*)]
-item If $mathcal P_JcalP_Jcapmathcal Z_J$calZ_J$ is empty, there is a nonzero
- current $N$ which annihilates $mathcal Z_J$calZ_J$ and all anti-invariant
+ If mathcal P_JcalP_Jcapmathcal Z_JcalZ_J is empty, there is a nonzero
+ current N which annihilates mathcal Z_JcalZ_J and all anti-invariant
forms, and is nonnegative on semipositive invariant forms.
-item If $Hin H^2(X;R)$ has no representative in $mathcal P_J$,calP_J$, there
- is a nonzero positive invariant current $N$ which is closed and
- satisfies $N(H)le0$.
+ If Hin H^2(X;R) has no representative in mathcal P_J,calP_J, there
+ is a nonzero positive invariant current N which is closed and
+ satisfies N(H)le0.
-In either case one can normalize $N(G)=1$ and choose a positive Borel
-measure $lambda$ on $mathcal L$calL$ such that
+In either case one can normalize N(G)=1 and choose a positive Borel
+measure lambda on mathcal LcalL such that
N(xi)=int_{mathcal L}calLxi_y(ell),ddlambda(y,ell),
- qquad mu=(pr_X)_*lambda,
- qquad mu(X)=1.
+ mu=(pr_X)_*lambda,
+ mu(X)=1.
-In particular $N$ has order zero and belongs to $H^{-3}(X,Λ^2)$.
+In particular N has order zero and belongs to H^{-3}(X,Λ^2).
Proof.
For (i), apply separation of the open convex set
-$mathcal P_JcalP_Jcap C^∞(X,I)$ from the disjoint linear subspace
-$mathcal Z_J$calZ_J$ in $C^∞(X,I)$. The separating continuous functional
-vanishes on $mathcal Z_J$calZ_J$ and is nonnegative on the positive cone.
-Extend it to all two-forms by precomposing with $1-R$. Adding a positive
-multiple of $G$ and then taking a limit proves nonnegativity on every
+mathcal P_JcalP_Jcap C^∞(X,I) from the disjoint linear subspace
+mathcal Z_JcalZ_J in C^∞(X,I). The separating continuous functional
+vanishes on mathcal Z_JcalZ_J and is nonnegative on the positive cone.
+Extend it to all two-forms by precomposing with 1-R. Adding a positive
+multiple of G and then taking a limit proves nonnegativity on every
semipositive invariant form.
-For (ii), choose a smooth closed representative $beta$ of $H$
-and separate the open cone $mathcal P_J$calP_J$ from the affine subspace
-$beta+ddΩ^1(X)$ in $Ω^2(X)$. If $N$ is the separating
+For (ii), choose a smooth closed representative beta of H
+and separate the open cone mathcal P_JcalP_J from the affine subspace
+beta+ddΩ^1(X) in Ω^2(X). If N is the separating
functional, the unbounded translation directions in this affine
-subspace give $N(ddalpha)=0$. Multiplying a positive form by any
+subspace give N(ddalpha)=0. Multiplying a positive form by any
positive scalar, the separation inequalities imply both
-$N(mathcal P_J)calP_J)ge0$ and $N(beta)le0$. Adding an arbitrary real
-multiple of an anti-invariant form to an element of $mathcal P_J$calP_J$ shows
-that $N$ annihilates anti-invariant forms. The same limiting argument
+N(mathcal P_J)calP_J)ge0 and N(beta)le0. Adding an arbitrary real
+multiple of an anti-invariant form to an element of mathcal P_JcalP_J shows
+that N annihilates anti-invariant forms. The same limiting argument
gives positivity on semipositive invariant forms.
For either functional, comparison of an invariant form with sufficiently
-large positive multiples of $G$ gives $|N(xi)|le C N(G)|xi|_{C^0}$.
-If $N(G)$ were zero, this inequality and anti-invariance would make the
-functional zero; normalize $N(G)=1$. Locally the functional is
+large positive multiples of G gives |N(xi)|le C N(G)|xi|_{C^0}.
+If N(G) were zero, this inequality and anti-invariance would make the
+functional zero; normalize N(G)=1. Locally the functional is
represented by a positive Hermitian matrix of measures. Its trace is a
-positive measure $mu$; the Radon--Nikodym matrix relative to $mu$ is
+positive measure mu; the Radon--Nikodym matrix relative to mu is
positive semidefinite of trace one. A measurable spectral decomposition
-of this $2times2$ matrix expresses it as a convex combination of
+of this 2times2 matrix expresses it as a convex combination of
rank-one Hermitian projections. Borel local frames and a Borel partition
-of $X$ give the global measure $lambda$ in
-(eq:current-line-measure). Finally, $H^3↪ C^0$ in
+of X give the global measure lambda in
+(eq:current-line-measure). Finally, H^3↪ C^0 in
real dimension four, so every such measure coefficient belongs to
-$H^{-3}$.
+H^{-3}.
## A closed lift of anti-invariant forms
[Closed lift]
There is a classical pseudodifferential operator
-$D:C^∞(X,E)→Ω^2(X)$ of order zero with
+D:C^∞(X,E)→Ω^2(X) of order zero with
- dd D=0,qquad RD=Id.
+ dd D=0, RD=Id.
It maps distributions to distributions. For the separator in
Cref{lem:current-separators}(i), set
- Q(xi)=-N(DRxi),qquad T=N+Q.
+ Q(xi)=-N(DRxi), T=N+Q.
-Then $Q$ is anti-invariant and self-dual, $Q,Tin H^{-3}$, and $T$
-annihilates every closed smooth two-form. In particular $T$ is a closed
-current with $[*T]=0$.
+Then Q is anti-invariant and self-dual, Q,Tin H^{-3}, and T
+annihilates every closed smooth two-form. In particular T is a closed
+current with [*T]=0.
-For a smooth family of $(J,g)$ with constant dimension of the space of
+For a smooth family of (J,g) with constant dimension of the space of
closed anti-invariant forms, these lifts can be chosen smoothly in the
parameter as operators between the corresponding smooth bundles.
Proof.
-On sections of $E$ consider $P=Rdddd^*$. If $a$ is self-dual, the
-self-dual projection of $xiwedgeiota_xi a$ is $|xi|^2a/2$: in a
-frame with first covector $xi/|xi|$, the two self-dual halves of $a$
+On sections of E consider P=Rdddd^*. If a is self-dual, the
+self-dual projection of xiwedgeiota_xi a is |xi|^2a/2: in a
+frame with first covector xi/|xi|, the two self-dual halves of a
have equal length. Hence
-$sigma_2(P)(x,xi)=tfrac12|xi|_g^2Id_E$ and
-$< Pa,a>_2=|dd^*a|_2^2$. Thus $P$ is elliptic,
+sigma_2(P)(x,xi)=tfrac12|xi|_g^2Id_E and
+< Pa,a>_2=|dd^*a|_2^2. Thus P is elliptic,
self-adjoint, and nonnegative. Its kernel consists of smooth sections
-with $dd^*a=0$; since $a=*a$, they also satisfy $dd a=0$, and hence
+with dd^*a=0; since a=*a, they also satisfy dd a=0, and hence
are harmonic. Conversely a closed anti-invariant form is self-dual and
therefore harmonic and in the kernel.
-Let $H_0$ be the orthogonal projection onto this finite-dimensional
-kernel, and let $L$ be the inverse of $P$ on its orthogonal complement,
+Let H_0 be the orthogonal projection onto this finite-dimensional
+kernel, and let L be the inverse of P on its orthogonal complement,
extended by zero on the kernel. Elliptic Fredholm theory and the
-parametrix give $L$ as a classical operator of order $-2$, with $H_0$
-smoothing and $PL=Id-H_0$. Set
+parametrix give L as a classical operator of order -2, with H_0
+smoothing and PL=Id-H_0. Set
D=dddd^*L+H_0.
-The image of both summands is closed, and $RD=PL+H_0=Id$.
+The image of both summands is closed, and RD=PL+H_0=Id.
-For closed $xi$, the form $xi-DRxi$ is closed and invariant, so
-$T(xi)=N(xi-DRxi)=0$. The functional $Q$ only depends on $Rxi$;
-this proves anti-invariance and self-duality. Its $H^{-3}$ membership
+For closed xi, the form xi-DRxi is closed and invariant, so
+T(xi)=N(xi-DRxi)=0. The functional Q only depends on Rxi;
+this proves anti-invariance and self-duality. Its H^{-3} membership
follows because an order-zero operator acts boundedly on every Sobolev
-space. Vanishing on exact forms makes $T$ closed, and vanishing on all
-closed two-forms makes $[*T]=0$.
+space. Vanishing on exact forms makes T closed, and vanishing on all
+closed two-forms makes [*T]=0.
For the family statement, identify the varying bundles locally in the
-parameter and use a smooth unitary identification of the $L^2$ spaces.
-The elliptic operators then have common domain $H^2$ in a fixed Hilbert
+parameter and use a smooth unitary identification of the L^2 spaces.
+The elliptic operators then have common domain H^2 in a fixed Hilbert
space. The spectral projection onto the kernel is smooth in the
-parameter, and so is $L=(P+H_0)^{-1}(Id-H_0)$ on every Sobolev scale.
+parameter, and so is L=(P+H_0)^{-1}(Id-H_0) on every Sobolev scale.
This gives the required smooth lifts.
From here on we use one of two setups:
[label=(Alph*)]
-item $N$ is the normalized separator in
- Cref{lem:current-separators}(i), and $Q,T$ are given by
+ N is the normalized separator in
+ Cref{lem:current-separators}(i), and Q,T are given by
(eq:current-correction);
-item $N$ is a normalized positive invariant closed current, $Q=0$, and
- $T=N$.
+ N is a normalized positive invariant closed current, Q=0, and
+ T=N.
Setup (B) includes Cref{lem:current-separators}(ii).
-In both cases $T$ is closed, but the obstructions are different. In setup
-(A), we will prove $Q=0$: the positive current $N=T$ would then
+In both cases T is closed, but the obstructions are different. In setup
+(A), we will prove Q=0: the positive current N=T would then
annihilate every closed form, contradicting the existence of a closed
tamer. In setup (B), the class separator satisfies
-$N(H)le0$; the task is instead to prove $N(H)>0$ from the hypotheses
+N(H)le0; the task is instead to prove N(H)>0 from the hypotheses
on the proposed taming class.
-Constants below may depend on $J,g$ and on the fixed operators, but
+Constants below may depend on J,g and on the fixed operators, but
are independent of the center, of the small scale, and of the choice of
-submeasure of $lambda$.
+submeasure of lambda.
## The quadratic mass estimate
[Radial order-zero estimate]
-Let $A$ be a fixed classical operator of order zero between smooth
+Let A be a fixed classical operator of order zero between smooth
vector bundles on a compact four-manifold. Fix a coordinate radius below
-the injectivity radius. Suppose that smooth sections $a_{s,p}$, supported
-in that coordinate neighborhood of $p$, satisfy, for $0<s<s_0$,
+the injectivity radius. Suppose that smooth sections a_{s,p}, supported
+in that coordinate neighborhood of p, satisfy, for 0<s<s_0,
[
|nabla^j a_{s,p}(y)|le C_j(s+dist(p,y))^{-a-j},
- qquad 0le jle6,quad ain{0,1},
+ 0le jle6, ain{0,1},
]
with uniform boundedness on a fixed outer annulus. Then, for
-$delta=s+dist(p,x)$ small,
+delta=s+dist(p,x) small,
|Aa_{s,p}(x)|le
@@ -772,40 +772,40 @@ $delta=s+dist(p,x)$ small,
C(1+|logdelta|),&a=0.
-Away from a smaller fixed neighborhood of $p$ the output is uniformly
-bounded in $s$ and $p$.
+Away from a smaller fixed neighborhood of p the output is uniformly
+bounded in s and p.
Proof.
Use two precise facts of the order-zero calculus: in local left
quantization its symbol is bounded uniformly in the covariable, and its
-off-diagonal kernel satisfies $|K_A(x,y)|le Cdist(x,y)^{-4}$ for small
+off-diagonal kernel satisfies |K_A(x,y)|le Cdist(x,y)^{-4} for small
positive distance. Smooth remainders have uniformly bounded kernels.
-Fix an evaluation point $x$, and split the input by a smooth cutoff
-supported in $B_{cdelta}(x)$ and equal to one on $B_{cdelta/2}(x)$,
-with a fixed small $c$. On its support, $s+dist(p,y)$ is comparable to
-$delta$. After translation and dilation by $delta$, the localized input
-has derivatives through order six bounded by $Cdelta^{-a}$.
+Fix an evaluation point x, and split the input by a smooth cutoff
+supported in B_{cdelta}(x) and equal to one on B_{cdelta/2}(x),
+with a fixed small c. On its support, s+dist(p,y) is comparable to
+delta. After translation and dilation by delta, the localized input
+has derivatives through order six bounded by Cdelta^{-a}.
Integration by parts in its Fourier transform therefore bounds its
-Fourier $L^1$ norm by $Cdelta^{-a}$, since $(1+|xi|)^{-6}$ is
+Fourier L^1 norm by Cdelta^{-a}, since (1+|xi|)^{-6} is
integrable in dimension four. The bounded symbol gives the same bound
-for the local contribution to $Aa_{s,p}(x)$.
-
-The remaining part has distance at least $cdelta/2$ from $x$. Its
-portion in $B_{Cdelta}(p)$ contributes at most
-$Cdelta^{-4}int_{B_{Cdelta}(p)}(s+dist(p,y))^{-a},dd V_g(y)le
-Cdelta^{-a}$. On a shell of radius $r=2^jdeltage Cdelta$ about $x$,
-distance from $p$ is comparable to $r$, so its contribution is at most
-$Cr^{-4}r^4r^{-a}=Cr^{-a}$. The shells up to the fixed coordinate radius
-sum to $Cdelta^{-1}$ if $a=1$ and to $C(1+|logdelta|)$ if $a=0$. The
-rest is uniformly bounded. If $x$ stays away from $p$, the singular
-input near $p$ has uniformly bounded $L^1$ norm and the kernel there is
+for the local contribution to Aa_{s,p}(x).
+
+The remaining part has distance at least cdelta/2 from x. Its
+portion in B_{Cdelta}(p) contributes at most
+Cdelta^{-4}int_{B_{Cdelta}(p)}(s+dist(p,y))^{-a},dd V_g(y)le
+Cdelta^{-a}. On a shell of radius r=2^jdeltage Cdelta about x,
+distance from p is comparable to r, so its contribution is at most
+Cr^{-4}r^4r^{-a}=Cr^{-a}. The shells up to the fixed coordinate radius
+sum to Cdelta^{-1} if a=1 and to C(1+|logdelta|) if a=0. The
+rest is uniformly bounded. If x stays away from p, the singular
+input near p has uniformly bounded L^1 norm and the kernel there is
bounded; the remaining input is uniformly smooth.
[Quadratic trace mass]
-In either setup (A) or (B), there is a constant $C$
+In either setup (A) or (B), there is a constant C
with
mu(B_s(p))le Cs^2qquad(pin X, 0<sle1).
@@ -813,17 +813,17 @@ with
Proof.
-Fix a radius $rho_0>0$ for uniform normal coordinates and a smooth
-radial cutoff which is one on $B_{2rho_0}(p)$ and supported in
-$B_{3rho_0}(p)$. With $t=dist(p,cdot)$ define the globally smooth
+Fix a radius rho_0>0 for uniform normal coordinates and a smooth
+radial cutoff which is one on B_{2rho_0}(p) and supported in
+B_{3rho_0}(p). With t=dist(p,cdot) define the globally smooth
functions
[
- u_s=chi(t)log√(s^2+t^2),qquad
- h_s=chi(t)√(s^2+t^2),qquad
+ u_s=chi(t)log√(s^2+t^2),
+ h_s=chi(t)√(s^2+t^2),
dd_J^cu=-dd ucirc J.
]
-Smoothness at $p$ follows from the dependence on $t^2$ and from $s>0$.
-For a function $f$ put
+Smoothness at p follows from the dependence on t^2 and from s>0.
+For a function f put
[
beta(f)=
@@ -831,26 +831,26 @@ For a function $f$ put
dddd_J^cf,&in setup (B).
]
-In setup (A), $beta(f)$ is closed and invariant, and in setup
-(B) it is exact. Hence $N(beta(f))=0$ in both cases.
+In setup (A), beta(f) is closed and invariant, and in setup
+(B) it is exact. Hence N(beta(f))=0 in both cases.
-The principal symbol of $dddd_J^c$ on functions is a multiple of
-$xiwedge J^*xi$, which is $J$-invariant, so $Rdddd_J^c$ is
+The principal symbol of dddd_J^c on functions is a multiple of
+xiwedge J^*xi, which is J-invariant, so Rdddd_J^c is
first-order. Differentiating the radial functions gives, for
-$0le jle6$,
+0le jle6,
[
- |nabla^jRdddd_J^cu_s|le C_j(s+t)^{-1-j},qquad
+ |nabla^jRdddd_J^cu_s|le C_j(s+t)^{-1-j},
|nabla^jRdddd_J^ch_s|le C_j(s+t)^{-j}.
]
-Cref{lem:current-radial-operator} applied to $D$ therefore bounds the
-two correction terms by $C/(s+t)$ and $C(1+|log(s+t)|)$.
+Cref{lem:current-radial-operator} applied to D therefore bounds the
+two correction terms by C/(s+t) and C(1+|log(s+t)|).
The positive parts, without cutoff, can be computed in a fixed Hermitian
-Euclidean space. If $f=f(t)$ and $ell$ is a unit complex line, its
+Euclidean space. If f=f(t) and ell is a unit complex line, its
Hessian trace is
[
dddd_{J_p}^cf(ell)=2f'(t)/t+(f''(t)-f'(t)/t)a,
- qquad 0le ale1.
+ 0le ale1.
]
For the two radial functions this gives
@@ -859,11 +859,11 @@ For the two radial functions this gives
dddd_{J_p}^c√(s^2+t^2)(ell)
&ge frac1{√(s^2+t^2)}.
-In normal coordinates $g-g_p=O(t^2)$ and $J-J_p=O(t)$, while the first
-derivatives of $J$ are bounded. Replacing the frozen complex line and
+In normal coordinates g-g_p=O(t^2) and J-J_p=O(t), while the first
+derivatives of J are bounded. Replacing the frozen complex line and
operator by the actual ones changes the estimates by at most
-$C/(s+t)$ for $u_s$ and $C$ for $h_s$. With $delta=s+t$, on
-$B_{rho_0}(p)$,
+C/(s+t) for u_s and C for h_s. With delta=s+t, on
+B_{rho_0}(p),
beta(u_s)(ell)&ge c_1s^2delta^{-4}-C_1delta^{-1},
@@ -873,22 +873,22 @@ $B_{rho_0}(p)$,
The inequalities also apply in setup (B), where the correction
terms are absent.
-Choose $0<rho<rho_0$ so small that
-$C_2(1+|logdelta|)le c_2/(2delta)$ for $0<deltale2rho$, and
-then choose a fixed multiplier $Mge2C_1/c_2$. If $slerho$ and
-$tlerho$, the two displayed inequalities give
-$beta(u_s+Mh_s)(ell)ge c_1s^2delta^{-4}$. On $X∖ B_rho(p)$
-all terms are bounded independently of $s$, including the nonlocal
+Choose 0<rho<rho_0 so small that
+C_2(1+|logdelta|)le c_2/(2delta) for 0<deltale2rho, and
+then choose a fixed multiplier Mge2C_1/c_2. If slerho and
+tlerho, the two displayed inequalities give
+beta(u_s+Mh_s)(ell)ge c_1s^2delta^{-4}. On X∖ B_rho(p)
+all terms are bounded independently of s, including the nonlocal
corrections, by the last assertion of
-Cref{lem:current-radial-operator}. Since $deltale2s$ on $B_s(p)$,
+Cref{lem:current-radial-operator}. Since deltale2s on B_s(p),
globally
[
beta(u_s+Mh_s)(ell)ge c s^{-2}mathbf 1_bf1_{B_s(p)}-C.
]
-Apply $N$, use $N(beta(u_s+Mh_s))=0$ and $mu(X)=1$, and obtain
-(eq:current-quadratic-mass) for $slerho$. For larger $s$ the
-same bound follows from $mu(X)=1$. Compactness makes all choices
-uniform in $p$.
+Apply N, use N(beta(u_s+Mh_s))=0 and mu(X)=1, and obtain
+(eq:current-quadratic-mass) for slerho. For larger s the
+same bound follows from mu(X)=1. Compactness makes all choices
+uniform in p.
## The resolvent kernel and positive-current pairings
@@ -898,24 +898,24 @@ step compares directions of nearby complex lines; smoothing permits this
comparison through an intersection pairing with the mass bound
controlling the error.
-Let $Delta=dddd^*+dd^*dd$ be the nonnegative Hodge Laplacian and
+Let Delta=dddd^*+dd^*dd be the nonnegative Hodge Laplacian and
set
- S_s=(1+s^2Delta)^{-3},qquad 0<sle s_0.
+ S_s=(1+s^2Delta)^{-3}, 0<sle s_0.
-It commutes with $dd,dd^*$ and $*$. At each fixed scale it maps
-$H^{-3}$ to $H^3$, which suffices for all the pairings below. The kernel
-of $S_s^2$ is continuous, being of order $-12$ in dimension four.
+It commutes with dd,dd^* and *. At each fixed scale it maps
+H^{-3} to H^3, which suffices for all the pairings below. The kernel
+of S_s^2 is continuous, being of order -12 in dimension four.
[Resolvent kernel]
-Choose a fixed radius $r_0$ smaller than the injectivity radius, and let
-$tau_{zto y}$ be parallel transport along the minimizing geodesic when
-$dist(y,z)<r_0$. The kernel $K_s$ of $S_s^2$ can be written
+Choose a fixed radius r_0 smaller than the injectivity radius, and let
+tau_{zto y} be parallel transport along the minimizing geodesic when
+dist(y,z)<r_0. The kernel K_s of S_s^2 can be written
K_s(y,z)=k_s(y,z)tau_{zto y}+E_s(y,z),
-where $k_s$ is nonnegative and supported where $dist(y,z)<r_0$. For
-each integer $kge0$ there are constants such that
+where k_s is nonnegative and supported where dist(y,z)<r_0. For
+each integer kge0 there are constants such that
k_s(y,z)&ge cs^{-4}&&if dist(y,z)<s,
@@ -924,7 +924,7 @@ each integer $kge0$ there are constants such that
|E_s(y,z)|&le C_ks^{-2}(1+dist(y,z)/s)^{-k}.
-The upper bound on $s_0$ is fixed independently of $y,z$.
+The upper bound on s_0 is fixed independently of y,z.
Proof.
@@ -938,56 +938,56 @@ four-manifold has a local small-time parametrix
chi(d)(4pi t)^{-2}e^{-d^2/(4t)}
(a_0(y,z)+ta_1(y,z)+...+t^ma_m(y,z)),
]
-with arbitrarily high-order uniform remainder after increasing $m$; here
-$d=dist(y,z)$, $chi$ is supported inside the injectivity radius and
+with arbitrarily high-order uniform remainder after increasing m; here
+d=dist(y,z), chi is supported inside the injectivity radius and
equals one near the diagonal, and
-$a_0(y,z)=j(y,z)^{-1/2}tau_{zto y}$ with $j$ smooth, strictly
+a_0(y,z)=j(y,z)^{-1/2}tau_{zto y} with j smooth, strictly
positive and equal to one on the diagonal. For the Hodge Laplacian,
-after subtracting the leading term, for every prescribed $L$ the kernel
-is bounded for $0<t<t_0$ by $Ct^{-1}e^{-d^2/(Ct)}+C_Lt^L$. The local
+after subtracting the leading term, for every prescribed L the kernel
+is bounded for 0<t<t_0 by Ct^{-1}e^{-d^2/(Ct)}+C_Lt^L. The local
expansion gives the Gaussian part; away from the diagonal the global
Gaussian estimate makes the cutoff error smaller than every power of
-$t$. For $tge t_0$ the heat kernel is uniformly bounded: factor the
+t. For tge t_0 the heat kernel is uniformly bounded: factor the
heat operator through two fixed positive-time smoothing operators and
-the $L^2$ contraction $e^{-(t-t_0)Delta}$, using nonnegativity and
+the L^2 contraction e^{-(t-t_0)Delta}, using nonnegativity and
self-adjointness of the Hodge Laplacian.
-Take $chige0$ and use its leading coefficient to define
+Take chige0 and use its leading coefficient to define
[
k_s(y,z)=frac{chi(d)j(y,z)^{-1/2}}{5!(4pi)^2}s^{-4}
int_0^∞ h^3e^{-h-d^2/(4s^2h)},dd h.
]
-For $d<s$, integration over $1le hle2$ gives the lower bound. For
-every fixed $k$, the elementary inequality
-$e^{-r^2/(ch)}le C_k(1+r)^{-k}(1+h^{k/2})$ shows that the last integral
-is bounded by $C_k(1+d/s)^{-k}$, giving the upper bound.
+For d<s, integration over 1le hle2 gives the lower bound. For
+every fixed k, the elementary inequality
+e^{-r^2/(ch)}le C_k(1+r)^{-k}(1+h^{k/2}) shows that the last integral
+is bounded by C_k(1+d/s)^{-k}, giving the upper bound.
-Integrating the $t^{-1}$ heat-kernel error in the resolvent identity
+Integrating the t^{-1} heat-kernel error in the resolvent identity
gives
[
Cs^{-2}int_0^∞ h^4e^{-h-d^2/(Cs^2h)},dd h
le C_ks^{-2}(1+d/s)^{-k}.
]
-For the uniform remainder, integration over $s^2h<t_0$ gives $C_Ls^{2L}$;
-since $dlediamX$, this is bounded by the right side of
-the error bound when $2Lge k-2$. On $s^2hge t_0$, the factor $e^{-h}$
+For the uniform remainder, integration over s^2h<t_0 gives C_Ls^{2L};
+since dlediamX, this is bounded by the right side of
+the error bound when 2Lge k-2. On s^2hge t_0, the factor e^{-h}
makes both the heat-kernel contribution and the extended leading term
-smaller than every power of $s$.
+smaller than every power of s.
-For any positive submeasure $lambda_ilelambda$, let $N_i$ be its
-current and $mu_i$ its projection to $X$. These currents need not be
+For any positive submeasure lambda_ilelambda, let N_i be its
+current and mu_i its projection to X. These currents need not be
closed. Define
B_{s,mu_i}(z)=s^{-2}int_X
(1+dist(y,z)/s)^{-6},ddmu_i(y),
- qquad B_s=B_{s,mu}.
+ B_s=B_{s,mu}.
By Cref{lem:quadratic-mass}, a dyadic-shell sum gives
-$0le B_{s,mu_i}(z)le B_s(z)le C$.
+0le B_{s,mu_i}(z)le B_s(z)le C.
[Positive-current pairing]
-For any $lambda_1,lambda_2lelambda$ and their currents,
+For any lambda_1,lambda_2lelambda and their currents,
< S_sN_1,*S_sN_2>_2
&ge cs^{-4}
@@ -998,7 +998,7 @@ For any $lambda_1,lambda_2lelambda$ and their currents,
|S_sN_i|_2&le C/s.
-One can replace $B_{s,mu_1}$ in the lower bound by $B_s$.
+One can replace B_{s,mu_1} in the lower bound by B_s.
Proof.
@@ -1009,12 +1009,12 @@ Parallel transport preserves the Hodge star. From
=tfrac12|ell-tau_{zto y}m|^2
-tfrac14|G_y-tau_{zto y}G_z|^2.
-The last squared difference is bounded by $Cd^2$, where $d=dist(y,z)$.
+The last squared difference is bounded by Cd^2, where d=dist(y,z).
-Express the pairing using the continuous kernel of $S_s^2$ and
+Express the pairing using the continuous kernel of S_s^2 and
(eq:current-line-measure). Its leading scalar is nonnegative, so
the lower-bound kernel inequality and the angular identity supply the
-positive term. The $Cd^2$ term, using the upper bound with exponent at
+positive term. The Cd^2 term, using the upper bound with exponent at
least eight, and the error kernel, using exponent six, are bounded in
absolute value by
[
@@ -1028,32 +1028,32 @@ absolute kernel bound in the unstarred pairing:
le Cs^{-2}mu_i(X)le Cs^{-2}.
]
The kernel formulas extend to measure coefficients because the kernel is
-continuous at fixed $s$.
+continuous at fixed s.
## Projection leakage and the topological energy
[Projection commutator]
-Let $Pi$ be a smooth orthogonal bundle projection, and suppose
-$Pi W=0$ for a distributional form $W$ with $S_sWin L^2$. Then
+Let Pi be a smooth orthogonal bundle projection, and suppose
+Pi W=0 for a distributional form W with S_sWin L^2. Then
|Pi S_sW|_2le Cs|S_sW|_2.
-Consequently, for every positive submeasure current $N'le_lambda N$
-and every anti-invariant distribution $Z$ with $S_sZin L^2$,
+Consequently, for every positive submeasure current N'le_lambda N
+and every anti-invariant distribution Z with S_sZin L^2,
|< S_sN',S_sZ>_2|le C|S_sZ|_2.
Proof.
-Set $A_s=S_s^{-1}=(1+s^2Delta)^3$ and $U=S_sW$. Since $Pi A_sU=0$,
-the distributional identity $Pi U=S_s[A_s,Pi]U$ holds. The commutator
-expands as a sum of $binom3j s^{2j}[Delta^j,Pi]$, $1le jle3$.
-Because the principal symbol of $Delta$ is scalar, $[Delta^j,Pi]$ is a
-differential operator of order at most $2j-1$. The spectral theorem and
-elliptic Sobolev equivalence give $|S_s|_{L^2to H^m}le C_ms^{-m}$,
-$0le mle6$. Since the formal adjoint of $B_j=[Delta^j,Pi]$ has the
+Set A_s=S_s^{-1}=(1+s^2Delta)^3 and U=S_sW. Since Pi A_sU=0,
+the distributional identity Pi U=S_s[A_s,Pi]U holds. The commutator
+expands as a sum of binom3j s^{2j}[Delta^j,Pi], 1le jle3.
+Because the principal symbol of Delta is scalar, [Delta^j,Pi] is a
+differential operator of order at most 2j-1. The spectral theorem and
+elliptic Sobolev equivalence give |S_s|_{L^2to H^m}le C_ms^{-m},
+0le mle6. Since the formal adjoint of B_j=[Delta^j,Pi] has the
same order,
[
|S_s s^{2j}B_j|_{L^2to L^2}
@@ -1061,9 +1061,9 @@ same order,
le Cs^{2j}s^{-(2j-1)}=Cs.
]
This proves the leakage estimate, including its meaning for
-distributional $W$ by the bounded extension of the displayed operator.
+distributional W by the bounded extension of the displayed operator.
-Apply it to $RN'=0$ and $(1-R)Z=0$. With $u=S_sN'$ and $v=S_sZ$, the
+Apply it to RN'=0 and (1-R)Z=0. With u=S_sN' and v=S_sZ, the
orthogonal splitting gives
[
|< u,v>_2|
@@ -1073,71 +1073,71 @@ orthogonal splitting gives
and the submeasure norm estimate yields the mixed bound.
-If $C_1,C_2in H^{-3}$ are closed currents, Hodge decomposition of the
-closed distributional forms $*C_i$ and commutation of $S_s$ with $dd$
+If C_1,C_2in H^{-3} are closed currents, Hodge decomposition of the
+closed distributional forms *C_i and commutation of S_s with dd
give
< S_sC_1,*S_sC_2>_2
=int_X(*S_sC_1)wedge(*S_sC_2)
=[*C_1]cdot[*C_2].
-Write $*C_i=h_i+dd a_i$ with $h_i$ harmonic and $a_iin H^{-2}$;
-then $S_sh_i=h_i$ and $S_sa_iin H^4$. Integration by parts eliminates
-every term containing an exact factor, leaving $int_Xh_1wedge h_2$.
+Write *C_i=h_i+dd a_i with h_i harmonic and a_iin H^{-2};
+then S_sh_i=h_i and S_sa_iin H^4. Integration by parts eliminates
+every term containing an exact factor, leaving int_Xh_1wedge h_2.
No product of the unsmoothed distributions is used.
[Angular energy]
In setup (A) or (B), the correction satisfies
-$Qin L^2(X,E)$ and
+Qin L^2(X,E) and
iint_{dist(y,z)<s}|ell-tau_{zto y}m|^2
,ddlambda(y,ell)ddlambda(z,m)le Cs^4.
The estimates in the positive-pairing and projection-commutator lemmas
hold for all indicated submeasures, and for every fixed submeasure
-current $N'$,
+current N',
lim_{s↓0}< S_sN',S_sQ>_2=0.
-The convergence is uniform over $lambda'lelambda$.
+The convergence is uniform over lambda'lelambda.
Proof.
-Let $mathcal A_s$calA_s$ denote the double integral in
-(eq:current-angular-energy) and put $x_s=|S_sQ|_2$. Since $Q$ is
-self-dual and $S_s$ commutes with $*$, the topological pairing applied
-to $T=N+Q$ gives
+Let mathcal A_scalA_s denote the double integral in
+(eq:current-angular-energy) and put x_s=|S_sQ|_2. Since Q is
+self-dual and S_s commutes with *, the topological pairing applied
+to T=N+Q gives
[
[*T]^2
=< S_sN,*S_sN>_2+2< S_sN,S_sQ>_2+x_s^2.
]
-Use the positive-current lower bound and the $B_s$ bound for the first
+Use the positive-current lower bound and the B_s bound for the first
term, and the mixed bound for the second. The result is
cs^{-4}mathcal A_s+x_s^2-Cx_scalA_s+x_s^2-Cx_sle C+[*T]^2.
-The right side is independent of $s$. In setup (A) its
+The right side is independent of s. In setup (A) its
intersection term is zero. In setup (B) that term is bounded
-uniformly for mass-one $N$: its harmonic coefficients are pairings with
+uniformly for mass-one N: its harmonic coefficients are pairings with
fixed smooth forms and are bounded by the line-measure representation.
-Completing the square in $x_s$ bounds both $x_s$ and $s^{-4}mathcal A_s$.calA_s$.
-
-As $s↓0$, $S_sQ$ converges to $Q$ distributionally. A weakly
-convergent subsequence of the bounded $L^2$ family has limit $Q$,
-proving $Qin L^2$; anti-invariance is preserved, and the spectral
-theorem gives strong convergence $S_sQto Q$ in $L^2$.
-
-Choose smooth anti-invariant $Q_j$ with $|Q_j-Q|_2to0$; for example
-apply $R$ to a smooth approximation. By the mixed bound and the $L^2$
-contraction property of $S_s$,
-$|< S_sN',S_s(Q-Q_j)>_2|le C|Q-Q_j|_2$. For fixed $j$,
-$S_s^2Q_jto Q_j$ in $C^∞$, whereas $N'(Q_j)=0$. Therefore
+Completing the square in x_s bounds both x_s and s^{-4}mathcal A_s.calA_s.
+
+As s↓0, S_sQ converges to Q distributionally. A weakly
+convergent subsequence of the bounded L^2 family has limit Q,
+proving Qin L^2; anti-invariance is preserved, and the spectral
+theorem gives strong convergence S_sQto Q in L^2.
+
+Choose smooth anti-invariant Q_j with |Q_j-Q|_2to0; for example
+apply R to a smooth approximation. By the mixed bound and the L^2
+contraction property of S_s,
+|< S_sN',S_s(Q-Q_j)>_2|le C|Q-Q_j|_2. For fixed j,
+S_s^2Q_jto Q_j in C^∞, whereas N'(Q_j)=0. Therefore
[
< S_sN',S_sQ_j>_2
=N'(S_s^2Q_j-Q_j)→0,
]
uniformly over the submeasures because their trace masses are at most
-one. Let $sto0$ first and $jto∞$ second.
+one. Let sto0 first and jto∞ second.
# Density splitting and the taming cone
@@ -1152,73 +1152,73 @@ that consequence will be used along the pair deformations in
Sections~sec:regular-preparation and~sec:endpoint.
[From taming to compatibility]
-Let $X$ be a closed connected smooth four-manifold with a smooth
-almost-complex structure $J$ and its complex orientation. If some smooth
-symplectic form tames $J$, then some smooth symplectic form is
-compatible with $J$.
+Let X be a closed connected smooth four-manifold with a smooth
+almost-complex structure J and its complex orientation. If some smooth
+symplectic form tames J, then some smooth symplectic form is
+compatible with J.
Curve classes are identified with their Poincar'e dual classes in
-$H^2(X;R)$. An irreducible $J$-holomorphic curve is the image
-of a nonconstant somewhere-injective $J$-holomorphic map from a closed
+H^2(X;R). An irreducible J-holomorphic curve is the image
+of a nonconstant somewhere-injective J-holomorphic map from a closed
connected Riemann surface, with its complex orientation.
[A curve criterion for taming classes]
-Let $X$ be a closed connected oriented smooth four-manifold with
-$b^+(X)=3$, and let $J$ be a smooth almost-complex structure inducing
-the orientation. Suppose smooth closed real two-forms $A,C$ span a
-pointwise definite plane and form a basis of the $J$-anti-invariant
+Let X be a closed connected oriented smooth four-manifold with
+b^+(X)=3, and let J be a smooth almost-complex structure inducing
+the orientation. Suppose smooth closed real two-forms A,C span a
+pointwise definite plane and form a basis of the J-anti-invariant
two-forms at every point. Set
[
V=[A]^perpcap[C]^perpsubset H^2(X;R).
]
-Suppose $Uin V$ has positive square and contains a smooth tamer for
-$J$. If $Hin V$ has positive square, belongs to the same component of
-${vin V:v^2>0}$ as $U$, and satisfies
+Suppose Uin V has positive square and contains a smooth tamer for
+J. If Hin V has positive square, belongs to the same component of
+{vin V:v^2>0} as U, and satisfies
[
Hcdot d>0
]
-for every irreducible closed $J$-holomorphic curve of class $d$, then
-$H$ contains a smooth symplectic form taming $J$.
+for every irreducible closed J-holomorphic curve of class d, then
+H contains a smooth symplectic form taming J.
## Densities and almost-everywhere planar tangents
The radial comparison below follows the Lelong--Jensen method for
almost-complex currents [cite]: closedness of
-$N+Q$ is applied, and the $L^2$ correction $Q$ is estimated directly.
+N+Q is applied, and the L^2 correction Q is estimated directly.
[Quadratic density]
-At every $pin X$, the limit
+At every pin X, the limit
θ(p)=lim_{s↓0}s^{-2}mu(B_s(p))
-exists and is finite. The function $θ$ is measurable and
-bounded, and vanishes $dV_g$-almost everywhere.
+exists and is finite. The function θ is measurable and
+bounded, and vanishes dV_g-almost everywhere.
Proof.
-Put $nu=mu+|Q|,dV_g$. Then by Cauchy--Schwarz and the volume growth
+Put nu=mu+|Q|,dV_g. Then by Cauchy--Schwarz and the volume growth
of a four-dimensional ball,
[
int_{B_s(p)}|Q|,dV_g
le vol_g(B_s(p))^{1/2}|Q|_{L^2(B_s(p))}le Cs^2,
]
-so $nu(B_s(p))le Cs^2$, uniformly in $p$.
+so nu(B_s(p))le Cs^2, uniformly in p.
-Fix $p$, identify a normal-coordinate ball with a Euclidean ball, and
-freeze $J_p$ and $G_p$ as constant tensors $J_0,G_0$. Writing
-$t=|zeta|=dist(p,exp_pzeta)$, define
-$b_p(s)=s^{-2}T(mathbf 1_bf1_{B_s(p)}G_0)$. These evaluations are
-well-defined because $T$ has measure coefficients. Since $G(ell)=1$,
-$G-G_0=O(t)$ and $Q$ annihilates the invariant form $G$,
+Fix p, identify a normal-coordinate ball with a Euclidean ball, and
+freeze J_p and G_p as constant tensors J_0,G_0. Writing
+t=|zeta|=dist(p,exp_pzeta), define
+b_p(s)=s^{-2}T(mathbf 1_bf1_{B_s(p)}G_0). These evaluations are
+well-defined because T has measure coefficients. Since G(ell)=1,
+G-G_0=O(t) and Q annihilates the invariant form G,
b_p(s)=s^{-2}mu(B_s(p))+O(s),
-because both error integrals are bounded by $Cs^{-2}s,nu(B_s(p))$.
+because both error integrals are bounded by Cs^{-2}s,nu(B_s(p)).
In frozen coordinates
-$b_p(s)=s^{-2}N(mathbf 1_bf1_{B_p(s)}G_0)+o(1)$. Approximating the
+b_p(s)=s^{-2}N(mathbf 1_bf1_{B_p(s)}G_0)+o(1). Approximating the
Euclidean ball from inside and outside by continuous test functions and
dominated convergence give
[
@@ -1235,110 +1235,110 @@ assertion, localize the measures in a relatively compact coordinate
chart and apply the differentiation theorem for Radon measures
cite[Theorem~4.19]{Kinnunen2026}. At volume-almost every center, the
measure-to-volume ratio of small Euclidean balls is bounded; a
-geodesic ball of radius $s$ lies in a coordinate ball of comparable
-radius with smooth volume $O(s^4)$, so
-$mu(B_s(p))=O_p(s^4)$ and $s^{-2}mu(B_s(p))to0$. A finite chart
+geodesic ball of radius s lies in a coordinate ball of comparable
+radius with smooth volume O(s^4), so
+mu(B_s(p))=O_p(s^4) and s^{-2}mu(B_s(p))to0. A finite chart
cover finishes the proof.
-Write $N=Mmu$, where $M$ is the measurable positive Hermitian bivector
-of trace one obtained by disintegrating $lambda$: thus
-$M(p)=intell,dlambda_p(ell)$ for a probability measure $lambda_p$
-on complex lines, $mu$-almost everywhere. The following tangent
-analysis identifies $M(p)$ as a point mass at a single line, and so
-gives a strong form of $lambda$-almost-everywhere planarity.
+Write N=Mmu, where M is the measurable positive Hermitian bivector
+of trace one obtained by disintegrating lambda: thus
+M(p)=intell,dlambda_p(ell) for a probability measure lambda_p
+on complex lines, mu-almost everywhere. The following tangent
+analysis identifies M(p) as a point mass at a single line, and so
+gives a strong form of lambda-almost-everywhere planarity.
[Planar tangents]
-For $mu$-almost every $p$, the measure $lambda_p$ is concentrated on a
-single complex line $ell_p$. Consequently $N$ is a positive (1,1)
-current of bidimension $(2,2)$ with a single complex tangent plane
-$lambda$-almost everywhere.
+For mu-almost every p, the measure lambda_p is concentrated on a
+single complex line ell_p. Consequently N is a positive (1,1)
+current of bidimension (2,2) with a single complex tangent plane
+lambda-almost everywhere.
Proof.
-Fix $p$ with a finite density $θ(p)$ and such that the measure
-$gamma_p$ obtained by the homothety $s^{-2}(zmapsto p+sz)_*mu$ has
-a tangent measure $gamma=M(p)mathbf 1_bf1_{P_p}$ in a complex
+Fix p with a finite density θ(p) and such that the measure
+gamma_p obtained by the homothety s^{-2}(zmapsto p+sz)_*mu has
+a tangent measure gamma=M(p)mathbf 1_bf1_{P_p} in a complex
two-dimensional vector space.
-Closedness of $T=N+Q$ passes to the limit as $s↓0$: smooth
-test forms converge in $C^1$ after rescaling, and the $Q$ term is
+Closedness of T=N+Q passes to the limit as s↓0: smooth
+test forms converge in C^1 after rescaling, and the Q term is
controlled by the gradient estimate of (eq:density-q-gradient).
Constant linear maps depend only on finitely many fixed test forms, so
the evaluation condition
[
∂(M(p)gamma)=0
]
-holds for $gamma$ as a tensor-valued distribution on $R^4$.
-
-Identify $M(p)$ with a constant skew-symmetric matrix. The equation
-$∂(M(p)gamma)=0$ says that all distributional derivatives of
-$gamma$ in the column space of that matrix vanish: equivalently,
-$gamma$ is translation invariant along $im M(p)$. The nonzero positive
-Hermitian $M(p)$ has real rank two or four. Rank four would make
-$gamma$ a nonzero multiple of Lebesgue measure, which contradicts the
-ball-mass bound $s^{-2}mu(B_s(p))toθ(p)<∞$. Thus rank is
-two. Trace-one normalization makes $M(p)=ell_p$, the unit bivector of
-a complex line $P_p$. The rank-one positive Hermitian $M(p)$ is an
+holds for gamma as a tensor-valued distribution on R^4.
+
+Identify M(p) with a constant skew-symmetric matrix. The equation
+∂(M(p)gamma)=0 says that all distributional derivatives of
+gamma in the column space of that matrix vanish: equivalently,
+gamma is translation invariant along im M(p). The nonzero positive
+Hermitian M(p) has real rank two or four. Rank four would make
+gamma a nonzero multiple of Lebesgue measure, which contradicts the
+ball-mass bound s^{-2}mu(B_s(p))toθ(p)<∞. Thus rank is
+two. Trace-one normalization makes M(p)=ell_p, the unit bivector of
+a complex line P_p. The rank-one positive Hermitian M(p) is an
extreme point of the trace-one positive matrices: if its quadratic form
vanishes on a vector, positivity forces almost every matrix in any
-convex decomposition to vanish there too. Hence $lambda_p$ is
-concentrated on $ell_p$.
+convex decomposition to vanish there too. Hence lambda_p is
+concentrated on ell_p.
## Closed density weights
[Closed density weights]
-For either separator construction, let $θ$ be the density of
-Cref{lem:density-existence}. If $b:RtoR$ is smooth and bounded with
-$b(0)=0$, then the current $b(θ)N$ is closed. The currents
+For either separator construction, let θ be the density of
+Cref{lem:density-existence}. If b:RtoR is smooth and bounded with
+b(0)=0, then the current b(θ)N is closed. The currents
- N^+=mathbf 1_bf1_{{θ>0}}N,qquad
+ N^+=mathbf 1_bf1_{{θ>0}}N,
I_a=mathbf 1_bf1_{{θ>a}}(pi)/(θ)Nquad(a>0)
are also closed. In particular, if
-$N^d=mathbf 1_bf1_{{θ=0}}N$, then $N^d+Q$ is closed.
+N^d=mathbf 1_bf1_{{θ=0}}N, then N^d+Q is closed.
Proof.
-First smooth the density. Fix nonnegative $hin C_c^∞((0,1))$
-normalized by $int_0^1h=1$, and set
+First smooth the density. Fix nonnegative hin C_c^∞((0,1))
+normalized by int_0^1h=1, and set
f_s(y)=s^{-2}int_Xh(dist(y,z)^2/s^2),dmu(z),
- qquad M_s(z)=mu(B_s(z)).
+ M_s(z)=mu(B_s(z)).
-$f_s$ is smooth (the support lies inside the injectivity radius and the
+f_s is smooth (the support lies inside the injectivity radius and the
cutoff vanishes near its boundary), and the mass bound makes it
uniformly bounded. Integration against the radial distribution of
-$mu$ gives
+mu gives
[
f_s(y)=-int_0^1h'(u)frac{mu(B_{ssqrt u}(y))}{s^2},du
→-θ(y)int_0^1u,h'(u),du=θ(y).
]
There are no atoms by quadratic growth, so no endpoint term appears at
-$u=0$. Thus $f_stoθ$ also volume-almost everywhere.
+u=0. Thus f_stoθ also volume-almost everywhere.
-Fix a smooth one-form $alpha$. To prove closedness of $b(θ)N$,
-apply closedness of $T=N+Q$ before passing to the limit:
+Fix a smooth one-form alpha. To prove closedness of b(θ)N,
+apply closedness of T=N+Q before passing to the limit:
0=T(d(b(f_s)alpha))
=T(b(f_s)dalpha)+T(b'(f_s),df_swedgealpha).
-Dominated convergence gives $b(f_s)Nto b(θ)N$ and
-$b(f_s)Qto0$ as currents (the second uses $b(0)=0$, the a.e.
-convergence of $f_s$, and domination by $C|Q|$). It remains to show the
-derivative term vanishes; $b'$ is bounded on the common bounded range
-of the $f_s$, and the $N$-part is bounded by a constant times
-$int |df_s|_ell,dlambda$, the $Q$-part by a constant times
-$int|Q|,|df_s|,dV_g$. We need the two gradient estimates
+Dominated convergence gives b(f_s)Nto b(θ)N and
+b(f_s)Qto0 as currents (the second uses b(0)=0, the a.e.
+convergence of f_s, and domination by C|Q|). It remains to show the
+derivative term vanishes; b' is bounded on the common bounded range
+of the f_s, and the N-part is bounded by a constant times
+int |df_s|_ell,dlambda, the Q-part by a constant times
+int|Q|,|df_s|,dV_g. We need the two gradient estimates
int_X|Q|,|df_s|,dV_g=o(1),
- qquad int|df_s|_ell,dlambda(y,ell)=o(1).
+ int|df_s|_ell,dlambda(y,ell)=o(1).
-noindentThe derivative against $Q$. Differentiating the radial
-kernel gives $|df_s(z)|le Cs^{-3}M_s(z)$. Fubini and the volume bound
-give $int_XM_s,dV_gle Cs^4$; and $M_s/s^2$ is uniformly bounded and
-$to0$ volume-a.e. Thus
+noindentThe derivative against Q. Differentiating the radial
+kernel gives |df_s(z)|le Cs^{-3}M_s(z). Fubini and the volume bound
+give int_XM_s,dV_gle Cs^4; and M_s/s^2 is uniformly bounded and
+to0 volume-a.e. Thus
s^{-3}int_X|Q|M_s,dV_g
le(int_X|Q|^2(M_s)/(s^2),dV_g)^{1/2}
@@ -1352,45 +1352,45 @@ noindentThe tangential derivative. Closedness converts
tangential differentiation into normal derivatives; the radial kernel
then supplies a transverse displacement factor multiplied by the
difference of nearby line directions, and the angular-energy bound
-controls the two together. Fix $(y,ell)$; choose an orthonormal basis
-of $T_yX$ with first two vectors spanning $ell$, and normal
-coordinates $zeta_1,...,zeta_4$. Write
-$k_s^y(z)=h(|zeta|^2/s^2)$, $q_i=∂_{zeta_i}k_s^y$,
-$omega_{12}=dzeta_1wedge dzeta_2$. First variation of squared
+controls the two together. Fix (y,ell); choose an orthonormal basis
+of T_yX with first two vectors spanning ell, and normal
+coordinates zeta_1,...,zeta_4. Write
+k_s^y(z)=h(|zeta|^2/s^2), q_i=∂_{zeta_i}k_s^y,
+omega_{12}=dzeta_1wedge dzeta_2. First variation of squared
distance yields
- df_s(y)(e_i)=-s^{-2}int q_i(z),dmu(z),qquad i=1,2.
+ df_s(y)(e_i)=-s^{-2}int q_i(z),dmu(z), i=1,2.
-Replace this scalar integral by $T(q_iomega_{12}) = (N+Q)(q_iomega_{12})$:
-on $B_s(y)$, normal coordinates and parallel transport differ by
-$O(s^2)$, so
+Replace this scalar integral by T(q_iomega_{12}) = (N+Q)(q_iomega_{12}):
+on B_s(y), normal coordinates and parallel transport differ by
+O(s^2), so
[
- omega_{12}(m)=<ell,tau_{zy}m>+O(s^2),qquad
+ omega_{12}(m)=<ell,tau_{zy}m>+O(s^2),
|1-omega_{12}(m)|le C(s^2+|ell-tau_{zy}m|^2).
]
-Since $|q_i|le C/s$, the integrated replacement error is at most
+Since |q_i|le C/s, the integrated replacement error is at most
s^{-2}!int!|int q_i dmu-T(q_iomega_{12})|dlambda
le Cs^{-3}(s^2!iint_{dist<s}!dmu,dmu+mathcal A_s)calA_s)
+Cs^{-3}!int_X!|Q|M_s,dV_g=o(1),
-where the double mass is $O(s^2)$, $mathcal A_s$calA_s$ is the angular
-integral of $O(s^4)$, and the last term tends to zero by
+where the double mass is O(s^2), mathcal A_scalA_s is the angular
+integral of O(s^4), and the last term tends to zero by
(eq:density-q-gradient).
-For $i=1$, closedness of $T$ on $d(k_s^y dzeta_2)$ gives
-$T(q_1omega_{12})=-sum_{a=3}^4T(q_a,dzeta_awedge dzeta_2)$;
-for $i=2$ the analogous identity holds for $d(k_s^y dzeta_1)$. These
+For i=1, closedness of T on d(k_s^y dzeta_2) gives
+T(q_1omega_{12})=-sum_{a=3}^4T(q_a,dzeta_awedge dzeta_2);
+for i=2 the analogous identity holds for d(k_s^y dzeta_1). These
are legitimate smooth compactly supported tests because the cutoff
vanishes near the boundary of the coordinate ball. The mixed forms
-satisfy, for $k=1,2$ and $a=3,4$,
+satisfy, for k=1,2 and a=3,4,
[
|(dzeta_awedge dzeta_k)(m)|le C(s^2+|ell-tau_{zy}m|),
]
and the normal derivative of the radial kernel obeys
-$|q_a(z)|le Cs^{-1}|pr_{ell^perp}zeta/s|$.
-Replacing $T$ by $N+Q$, the $Q$-contribution again tends to zero by
-(eq:density-q-gradient), the $s^2$ geometric error is $O(s)$, and
+|q_a(z)|le Cs^{-1}|pr_{ell^perp}zeta/s|.
+Replacing T by N+Q, the Q-contribution again tends to zero by
+(eq:density-q-gradient), the s^2 geometric error is O(s), and
Cauchy--Schwarz bounds the rest by
Cs^{-3}mathcal A_s^calA_s^{1/2}mathcal T_s^calT_s^{1/2}=s^{-3}cdot O(s^2)cdot
@@ -1403,14 +1403,14 @@ of (eq:density-two-gradient-estimates).
noindentThe closed weighted limit. The two derivative estimates
permit the limit in (eq:density-weight-product-rule): the
-derivative term $to0$, the first term $to b(θ)N(dalpha)$, so
-this last vanishes for every smooth $alpha$---$b(θ)N$ is
+derivative term to0, the first term to b(θ)N(dalpha), so
+this last vanishes for every smooth alpha---b(θ)N is
closed, as claimed.
For the remaining weights, take a smooth nondecreasing
-$chi:Rto[0,1]$ vanishing on $(-∞,0]$ and equal to one on
-$[1,∞)$; the bounded functions $chi(nt)tomathbf 1_bf1_{{t>0}}$
-pointwise on $[0,∞)$ and vanish at zero. Fix $a>0$ and set
+chi:Rto[0,1] vanishing on (-∞,0] and equal to one on
+[1,∞); the bounded functions chi(nt)tomathbf 1_bf1_{{t>0}}
+pointwise on [0,∞) and vanish at zero. Fix a>0 and set
[
b_{a,n}(t)=
@@ -1418,12 +1418,12 @@ pointwise on $[0,∞)$ and vanish at zero. Fix $a>0$ and set
0,&tle0.
]
-These are smooth, vanish near zero, are uniformly bounded by $pi/a$,
-and converge at every $tge0$ to $mathbf 1_bf1_{{t>a}}pi/t$. Dominated
+These are smooth, vanish near zero, are uniformly bounded by pi/a,
+and converge at every tge0 to mathbf 1_bf1_{{t>a}}pi/t. Dominated
convergence and closedness of each smooth-weight current prove
(eq:density-weighted-restrictions). No derivative bound uniform
-in $n$ is used: the smoothing limit was taken first for each $n$.
-Finally $N^d+Q=T-N^+$ is closed.
+in n is used: the smoothing limit was taken first for each n.
+Finally N^d+Q=T-N^+ is closed.
## Vanishing of the diffuse error and the compatible form
@@ -1434,41 +1434,41 @@ compatibility question and then provides the nonnegative diffuse square
needed for the class criterion.
[Vanishing diffuse kernel error]
-Set $B_s(z)=s^{-2}int_X(1+dist(y,z)/s)^{-6},dmu(y)$. The functions
-are uniformly bounded, and $B_s(z)to0$ whenever $θ(z)=0$. In
+Set B_s(z)=s^{-2}int_X(1+dist(y,z)/s)^{-6},dmu(y). The functions
+are uniformly bounded, and B_s(z)to0 whenever θ(z)=0. In
particular,
- int_XB_s(z),dmu^d(z)→0,qquad
+ int_XB_s(z),dmu^d(z)→0,
mu^d=mathbf 1_bf1_{{θ=0}}mu.
Proof.
For each dyadic shell, quadratic growth gives the summable bound
-$s^{-2}(1+2^j)^{-6}mu(B_{2^{j+1}s}(z))le C,2^{-4j}$. The same
+s^{-2}(1+2^j)^{-6}mu(B_{2^{j+1}s}(z))le C,2^{-4j}. The same
estimate holds once the radius exceeds a fixed small radius, by
-increasing $C$ and using finite total mass. This proves uniform
-boundedness. If $θ(z)=0$, each fixed shell contribution tends to
-zero because $s^{-2}mu(B_{2^{j+1}s}(z))to0$; the summable bound then
+increasing C and using finite total mass. This proves uniform
+boundedness. If θ(z)=0, each fixed shell contribution tends to
+zero because s^{-2}mu(B_{2^{j+1}s}(z))to0; the summable bound then
gives pointwise convergence. A second application of dominated
convergence proves (eq:density-diffuse-error).
-For closed currents $S,S'$, the smoothing operator
-$S_s=(1+s^2Delta_g)^{-3}$ preserves cohomology classes and commutes
+For closed currents S,S', the smoothing operator
+S_s=(1+s^2Delta_g)^{-3} preserves cohomology classes and commutes
with the Hodge star. Hodge decomposition therefore gives
< S_sS,*S_sS'>_{L^2}=mathfrak c(S)frakc(S)cdotmathfrak c(S'),frakc(S'),
-independently of $s$; the Sobolev regularity after smoothing is
+independently of s; the Sobolev regularity after smoothing is
sufficient for this identity.
Proof. [Proof of Cref{thm:tamed-compatible}]
Suppose that no compatible symplectic form exists. The separator of
-Cref{lem:current-separators} gives a nonzero positive current $N$,
-normalized to trace mass one, and anti-invariant $Q$ such that
-$T=N+Q$ annihilates all smooth closed two-forms; in particular $T$ is
-closed with $mathfrak c(T)=0$.frakc(T)=0$. By Cref{prop:density-weights}, $T^d=N^d+Q$
+Cref{lem:current-separators} gives a nonzero positive current N,
+normalized to trace mass one, and anti-invariant Q such that
+T=N+Q annihilates all smooth closed two-forms; in particular T is
+closed with mathfrak c(T)=0.frakc(T)=0. By Cref{prop:density-weights}, T^d=N^d+Q
is closed. (eq:density-smoothed-intersection) gives
0&=< S_sT,*S_sT^d>notag
@@ -1476,24 +1476,24 @@ is closed. (eq:density-smoothed-intersection) gives
+< S_sQ,S_sN^d>+|S_sQ|_2^2.
-The self-duality of $Q$ accounts for the signs. Both mixed terms tend to
+The self-duality of Q accounts for the signs. Both mixed terms tend to
zero by Cref{prop:angular-energy}. The positive-current lower bound of
Cref{lem:current-positive-pairing}, applied with second submeasure
-$lambda^d=mathbf 1_bf1_{{θ=0}}lambda$, gives
-$< S_sN,*S_sN^d>ge-Cint_XB_s,dmu^d=o(1)$. Since
-$S_sQto Q$ in $L^2$, taking a limit inferior in
-(eq:density-compatibility-pairing) forces $|Q|_2^2=0$. Thus
-$T=N$ is a nonzero positive current annihilating all closed forms.
-
-Let $eta$ be a taming symplectic form. Compactness of the bundle of
-unit complex lines and strict taming give $eta_y(ell)ge c_eta>0$, so
-$N(eta)ge c_etamu(X)>0$, contradicting the annihilation property.
+lambda^d=mathbf 1_bf1_{{θ=0}}lambda, gives
+< S_sN,*S_sN^d>ge-Cint_XB_s,dmu^d=o(1). Since
+S_sQto Q in L^2, taking a limit inferior in
+(eq:density-compatibility-pairing) forces |Q|_2^2=0. Thus
+T=N is a nonzero positive current annihilating all closed forms.
+
+Let eta be a taming symplectic form. Compactness of the bundle of
+unit complex lines and strict taming give eta_y(ell)ge c_eta>0, so
+N(eta)ge c_etamu(X)>0, contradicting the annihilation property.
This proves existence. The argument has used neither rectifiability nor
a theorem representing integral cycles by curves.
[Nonnegative diffuse square]
-If $Q=0$, then $N^d$ is closed and $mathfrak c(N^d)^2frakc(N^d)^2ge0$.
+If Q=0, then N^d is closed and mathfrak c(N^d)^2frakc(N^d)^2ge0.
Proof.
@@ -1510,15 +1510,15 @@ and the right side tends to zero by Cref{lem:density-diffuse-error}.
## Integral thresholds and the curve criterion
To use the hypothesis on curve classes, we must turn the positive-density
-part into integral cycles. The factor $pi/θ$ in $I_a$ is
+part into integral cycles. The factor pi/θ in I_a is
chosen to give multiplicity one.
[Threshold cycles]
-For every $a>0$, the current $I_a$ from
+For every a>0, the current I_a from
Cref{eq:density-weighted-restrictions} is an integral two-cycle with
-positive $J$-complex tangent orientation. If $J$ is tamed, it is a
+positive J-complex tangent orientation. If J is tamed, it is a
finite sum of currents of integration over irreducible closed
-$J$-holomorphic curves, with positive integer multiplicities. The
+J-holomorphic curves, with positive integer multiplicities. The
identity
N^+=frac1piint_0^∞ I_a,da
@@ -1527,24 +1527,24 @@ holds as an absolutely convergent integral of currents.
Proof.
-The density $θ$ is bounded and vanishes with $dV_g$-zero mean.
-By Cref{prop:density-tangent-measure}, $lambda$-almost everywhere the
-current has a single complex tangent line with weight $θ$. The
-normalization of $lambda$ makes $(pr_X)_*lambda=mu$;
-hence the trace measure of $I_a$ equals
-$mathbf 1_bf1_{{θ>a}}pi,dV_g$ and is rectifiable. Preiss's
+The density θ is bounded and vanishes with dV_g-zero mean.
+By Cref{prop:density-tangent-measure}, lambda-almost everywhere the
+current has a single complex tangent line with weight θ. The
+normalization of lambda makes (pr_X)_*lambda=mu;
+hence the trace measure of I_a equals
+mathbf 1_bf1_{{θ>a}}pi,dV_g and is rectifiable. Preiss's
rectifiability theorem [cite] upgrades the a.e.~planar
-tangents to rectifiability; closedness of $I_a$ and positivity then
+tangents to rectifiability; closedness of I_a and positivity then
express it as an integral sum of simple curves. Positivity of the
-$J$-complex orientation is pointwise: the tangent plane is the chosen
+J-complex orientation is pointwise: the tangent plane is the chosen
complex line. Taming prevents a nonzero closed component with infinite
mass; factoring each component through its somewhere-injective image
gives a finite sum of irreducible closed curves with positive integer
multiplicities.
-Finally, pointwise on ${θ>0}$,
-$int_0^∞mathbf 1_bf1_{{θ>a}}pi/θ,da=pi$; for a
-smooth two-form $alpha$, Fubini is justified by
+Finally, pointwise on {θ>0},
+int_0^∞mathbf 1_bf1_{{θ>a}}pi/θ,da=pi; for a
+smooth two-form alpha, Fubini is justified by
[
int_0^∞|I_a(alpha)|,dalepi|alpha|_{C^0}mu^+(X).
]
@@ -1552,39 +1552,39 @@ This proves (eq:density-layer-cake) with its stated normalization.
Proof. [Proof of Cref{thm:taming-cone}]
-Assume that $H$ has no taming representative. The class separator of
+Assume that H has no taming representative. The class separator of
Cref{lem:current-separators}(ii) gives a nonzero closed
-positive current $N$, with $Q=0$, satisfying
+positive current N, with Q=0, satisfying
Hcdotmathfrak c(N)frakc(N)le0.
Its positive-density and diffuse parts are separately closed by
-Cref{prop:density-weights}. Anti-invariance of $A,C$ gives
-$mathfrak c(N^d)frakc(N^d)cdot[A]=N^d(A)=0=mathfrak c(N^d)frakc(N^d)cdot[C]$; thus
-$w=mathfrak c(N^d)$frakc(N^d)$ lies in $V$ and $w^2ge0$ by
+Cref{prop:density-weights}. Anti-invariance of A,C gives
+mathfrak c(N^d)frakc(N^d)cdot[A]=N^d(A)=0=mathfrak c(N^d)frakc(N^d)cdot[C]; thus
+w=mathfrak c(N^d)frakc(N^d) lies in V and w^2ge0 by
Cref{lem:density-diffuse-square}.
-If $N^dne0$, choose a taming form $eta$ in $U$. Strict taming on the
-compact bundle of unit complex lines gives $Ucdot w=N^d(eta)>0$; in
-particular $wne0$. Since $[A],[C]$ span a positive two-plane and
-$b^+=3$, the intersection form on $V$ has signature $(1,b^-)$. The
-conditions $w^2ge0$ and $Ucdot w>0$ put $w$ in the nonzero closed time
-cone determined by $U$. Every timelike $H$ in that component has
-$Hcdot w>0$: writing $H=h_0u+h_-$, $w=w_0u+w_-$ with a unit timelike
-$u$ and orthogonal negative parts, $h_0>|h_-|$, $w_0ge|w_-|$, $w_0>0$,
-so $Hcdot wge h_0w_0-|h_-|,|w_-|>0$. If $N^d=0$, its pairing is
+If N^dne0, choose a taming form eta in U. Strict taming on the
+compact bundle of unit complex lines gives Ucdot w=N^d(eta)>0; in
+particular wne0. Since [A],[C] span a positive two-plane and
+b^+=3, the intersection form on V has signature (1,b^-). The
+conditions w^2ge0 and Ucdot w>0 put w in the nonzero closed time
+cone determined by U. Every timelike H in that component has
+Hcdot w>0: writing H=h_0u+h_-, w=w_0u+w_- with a unit timelike
+u and orthogonal negative parts, h_0>|h_-|, w_0ge|w_-|, w_0>0,
+so Hcdot wge h_0w_0-|h_-|,|w_-|>0. If N^d=0, its pairing is
instead zero; this distinguishes the nonzero diffuse case needed for
strict positivity.
-For every $a>0$, Cref{lem:density-integral-thresholds} writes $I_a$ as
-a positive integral sum of irreducible closed $J$-curves; the hypothesis
-on $H$ gives $Hcdotmathfrak c(I_a)frakc(I_a)ge0$, with strict inequality
-whenever $I_ane0$. If $N^+ne0$, the layer identity has positive total
-mass, so the set of $a$ for which $I_ane0$ has positive Lebesgue
+For every a>0, Cref{lem:density-integral-thresholds} writes I_a as
+a positive integral sum of irreducible closed J-curves; the hypothesis
+on H gives Hcdotmathfrak c(I_a)frakc(I_a)ge0, with strict inequality
+whenever I_ane0. If N^+ne0, the layer identity has positive total
+mass, so the set of a for which I_ane0 has positive Lebesgue
measure; pairing (eq:density-layer-cake) with a smooth
-representative of $H$ gives $Hcdotmathfrak c(N^+)>0$.frakc(N^+)>0$. The pairing is
-zero when $N^+=0$. At least one of $N^d,N^+$ is nonzero, so
-$Hcdotmathfrak c(N)=Hfrakc(N)=Hcdotmathfrak c(N^d)+Hfrakc(N^d)+Hcdotmathfrak c(N^+)>0$,frakc(N^+)>0$,
+representative of H gives Hcdotmathfrak c(N^+)>0.frakc(N^+)>0. The pairing is
+zero when N^+=0. At least one of N^d,N^+ is nonzero, so
+Hcdotmathfrak c(N)=Hfrakc(N)=Hcdotmathfrak c(N^d)+Hfrakc(N^d)+Hcdotmathfrak c(N^+)>0,frakc(N^+)>0,
contradicting (eq:density-cone-separator). A smooth closed taming
representative is symplectic by pointwise nondegeneracy.
@@ -1593,8 +1593,8 @@ representative is symplectic by pointwise nondegeneracy.
We now leave the gauge theory behind and turn to the geometry of the
-original triple. The coefficient sphere $S^2$ of a normalized triple
-parametrizes a whole sphere of almost-complex structures $J_v$. The
+original triple. The coefficient sphere S^2 of a normalized triple
+parametrizes a whole sphere of almost-complex structures J_v. The
fixed spin-c structure of any one of them can be carried over the
sphere, and the Seiberg--Witten invariants of this family give, in the
large-parameter limit, holomorphic curves in any prescribed nonzero
@@ -1602,95 +1602,95 @@ integral class with zero self-intersection. The main output of this
section is that those curves pass through every prescribed point.
Curve classes are identified with their Poincar'e duals; an integral
-class written in $Λ=H^2(X;Z)/torsion$ may be lifted to
-$H^2(X;Z)$ when specifying a line bundle, and curve configurations
+class written in Λ=H^2(X;Z)/torsion may be lifted to
+H^2(X;Z) when specifying a line bundle, and curve configurations
and their total classes are only ever used modulo torsion. By
Cref{prop:topology},
- b^+=3,qquad
- (b_1,sigma)=(0,-16) or (4,0),qquad
+ b^+=3,
+ (b_1,sigma)=(0,-16) or (4,0),
2chi+3sigma=0.
-Let $J_v$, $vin S^2$, be the coefficient sphere of
+Let J_v, vin S^2, be the coefficient sphere of
Cref{lem:twistor-degree}, with the sign chosen so that
-$B_v=vcdotomega$ tames $J_v$. Write
-$P=span{[omega_1],[omega_2],[omega_3]}$ for the
+B_v=vcdotomega tames J_v. Write
+P=span{[omega_1],[omega_2],[omega_3]} for the
original positive period plane.
A positive curve configuration is a finite sum of irreducible
closed pseudoholomorphic curves with positive integer multiplicities.
-Let $FinΛ$ be nonzero with $F^2=0$, and write
-$F^+=pr_PF$. For every $pin X$, there exists a finite
-positive $J_v$-holomorphic configuration of total class $F$ whose
-support contains $p$. Its parameter is necessarily the coefficient
+Let FinΛ be nonzero with F^2=0, and write
+F^+=pr_PF. For every pin X, there exists a finite
+positive J_v-holomorphic configuration of total class F whose
+support contains p. Its parameter is necessarily the coefficient
direction
[
v=(F^+)/(|F^+|),
]
-where $P$ is identified with $R^3$ by the normalized ordered period
+where P is identified with R^3 by the normalized ordered period
basis. Thus the almost-complex structure is the same for all
prescribed points.
-The strategy is: (a) build a smooth $J_v$-compatible family $eta_v$
+The strategy is: (a) build a smooth J_v-compatible family eta_v
with normalized degree one, (b) use the family Seiberg--Witten
invariants to force a zero of the first spinor component at the
-prescribed point, and (c) extract a $J_v$-holomorphic curve from the
+prescribed point, and (c) extract a J_v-holomorphic curve from the
large-parameter limit. Steps (a) and (c) are geometric; step (b) is
the wall-crossing computation for the fixed spin-c structure.
## A smooth compatible family
-There is a smooth family of closed $J_v$-compatible forms
-$eta_v$. Orthogonal projection to $P$ satisfies
+There is a smooth family of closed J_v-compatible forms
+eta_v. Orthogonal projection to P satisfies
[
- pr_P[eta_v]=a(v)[B_v],qquad a(v)>0,
+ pr_P[eta_v]=a(v)[B_v], a(v)>0,
]
and consequently the normalized period projection of this family has
degree one in the coefficient coordinates.
Proof.
-Fix a metric $g_0$ in the conformal class determined by the original
-triple. All $J_v$ are orthogonal for $g_0$, and the three closed
-self-dual forms $omega_i$ span its self-dual harmonic forms because
-$b^+=3$. Denote by $R_v$ the projection to the $J_v$-anti-invariant
-two-forms, and consider $L_v^{ell}=R_vdddd^*$ on the
+Fix a metric g_0 in the conformal class determined by the original
+triple. All J_v are orthogonal for g_0, and the three closed
+self-dual forms omega_i span its self-dual harmonic forms because
+b^+=3. Denote by R_v the projection to the J_v-anti-invariant
+two-forms, and consider L_v^{ell}=R_vdddd^* on the
anti-invariant forms, where the adjoint is taken with respect to
-$g_0$. This is the nonnegative elliptic operator in
+g_0. This is the nonnegative elliptic operator in
Cref{lem:closed-lift}, with kernel the closed anti-invariant forms;
-the kernel is smooth in $v$ and has rank two over $S^2$ by
+the kernel is smooth in v and has rank two over S^2 by
Cref{lem:twistor-degree}.
-Compatibility is equivalent to taming by a closed $-J$-invariant form.
+Compatibility is equivalent to taming by a closed -J-invariant form.
By Cref{thm:tamed-compatible}, the compatible cone is nonempty for
-$J_v$; compactness of $S^2$ and the smooth dependence of the operator
-kernel on $v$ then give a smooth choice of compatible forms $eta_v$.
-Each $eta_v$ is $J_v$-invariant and closed, so it is orthogonal to
-the anti-invariant classes. Therefore its projection to $P$ is a
-multiple of $[B_v]$. Moreover
-$int_Xeta_vwedge B_v>0$: the invariant part of $B_v$ is positive,
-and its anti-invariant part wedges trivially with $eta_v$. Since
-$[B_v]^2=1$, the multiple is positive. The remaining assertions follow
-from the fact that $pr_P$ acts as the identity on the
-unit sphere of $P$.
+J_v; compactness of S^2 and the smooth dependence of the operator
+kernel on v then give a smooth choice of compatible forms eta_v.
+Each eta_v is J_v-invariant and closed, so it is orthogonal to
+the anti-invariant classes. Therefore its projection to P is a
+multiple of [B_v]. Moreover
+int_Xeta_vwedge B_v>0: the invariant part of B_v is positive,
+and its anti-invariant part wedges trivially with eta_v. Since
+[B_v]^2=1, the multiple is positive. The remaining assertions follow
+from the fact that pr_P acts as the identity on the
+unit sphere of P.
## The fixed spin-c structure and the equations
-Fix the canonical $Spin^c$ structure of one $J_v$, with
-isomorphism type $mathfrak s_0$.fraks_0$. The structures $J_v$ form a
+Fix the canonical Spin^c structure of one J_v, with
+isomorphism type mathfrak s_0.fraks_0. The structures J_v form a
connected family, so all their canonical structures have the same
-fiberwise isomorphism type. For $Ein H^2(X;Z)$, let
-$mathfrak s_E=fraks_E=mathfrak s_0fraks_0 ⊗ L_E$, where
-$c_1(L_E)=E$; the family extends over the contractible space of
+fiberwise isomorphism type. For Ein H^2(X;Z), let
+mathfrak s_E=fraks_E=mathfrak s_0fraks_0 ⊗ L_E, where
+c_1(L_E)=E; the family extends over the contractible space of
metrics with no twist from the parameter base. Its determinant class
and dimensions are
- c_1(detmathfrak s_E)=2E,fraks_E)=2E,qquad d(mathfrak s_E)=E^2,fraks_E)=E^2,qquad
+ c_1(detmathfrak s_E)=2E,fraks_E)=2E, d(mathfrak s_E)=E^2,fraks_E)=E^2,
ind_{C}not D_E=(4E^2-sigma)/(8).
Fix a homology orientation once and for all.
@@ -1698,16 +1698,16 @@ Fix a homology orientation once and for all.
## Wall-crossing at the harmonic wall
[Wall-crossing]
-Let $mathfrak s$fraks$ be the fixed spin-c structure of the family $J_v$
-over $S^2$, and let $q=d(mathfrak s)/2$fraks)/2$ be its expected complex
+Let mathfrak sfraks be the fixed spin-c structure of the family J_v
+over S^2, and let q=d(mathfrak s)/2fraks)/2 be its expected complex
dimension. Then the family invariant in the chamber after crossing the
harmonic wall gives
FSW_{mathfrak s}fraks(S^2)
=s_{b_1/2}(K-O),
-where $K,O$ are the ranks of the complex Dirac index, and
-$FSW$ denotes the family Seiberg--Witten invariant of
+where K,O are the ranks of the complex Dirac index, and
+FSW denotes the family Seiberg--Witten invariant of
the universal spinor evaluation.
@@ -1716,13 +1716,13 @@ The critical wall-crossing theorem and its section-class version are
cite[Theorem~4.10 and Proposition~4.11]{LiLiu2001}; we only check
the normalization and degree. A transverse homotopy between two
families meets the harmonic wall at isolated parameters in
-$S^2times[0,1]$, because its normal rank is $b^+=3$. Above such a
-parameter the reducibles form the torus $Pic^0(X)$.
-Represent the complex Dirac index on this torus as $K-O$, with ranks
-$k,o$, using a stabilization when necessary. Write
-$x=c_1(mathcal O(1))$calO(1))$ on the projective bundle of lines
-$pi:mathbb P(K)bbP(K)toPic^0(X)$. The obstruction bundle
-contributes $e(mathcal O(1)calO(1) ⊗ O)=sum_ix^{o-i}c_i(O)$.
+S^2times[0,1], because its normal rank is b^+=3. Above such a
+parameter the reducibles form the torus Pic^0(X).
+Represent the complex Dirac index on this torus as K-O, with ranks
+k,o, using a stabilization when necessary. Write
+x=c_1(mathcal O(1))calO(1)) on the projective bundle of lines
+pi:mathbb P(K)bbP(K)toPic^0(X). The obstruction bundle
+contributes e(mathcal O(1)calO(1) ⊗ O)=sum_ix^{o-i}c_i(O).
Projective pushforward therefore gives
pi_*(x^qsum_ix^{o-i}c_i(O))
@@ -1731,11 +1731,11 @@ Projective pushforward therefore gives
because (eq:families-topology)(eq:families-dimensions)
-imply $q+o-(k-1)=b_1/2$. The normal-crossing signs sum to the degree
+imply q+o-(k-1)=b_1/2. The normal-crossing signs sum to the degree
difference, giving (eq:families-wall-formula).
The based-gauge normalization makes the universal flat line trivial
-at $p$; no Picard-base line is added to $x$. Writing a determinant
+at p; no Picard-base line is added to x. Writing a determinant
connection as a canonical connection plus twice a twisting connection
does not alter this conclusion: the remaining circle acts on the
spinor with weight one, as used in
@@ -1744,40 +1744,40 @@ spinor with weight one, as used in
## The evaluation line over the coefficient sphere
-For $E^2=0$, write $FSW_E(U_p)$ for the spherical
-pairing of the based-gauge evaluation line $U_p$ in the large-parameter
-chamber, and $SW_X(E)$ for the ordinary dimension-zero
-invariant of $mathfrak s_E$.fraks_E$. Denote by $pi$ the projection of the
-regular family moduli space to $S^2$.
+For E^2=0, write FSW_E(U_p) for the spherical
+pairing of the based-gauge evaluation line U_p in the large-parameter
+chamber, and SW_X(E) for the ordinary dimension-zero
+invariant of mathfrak s_E.fraks_E. Denote by pi the projection of the
+regular family moduli space to S^2.
[Evaluation identity]
-The evaluation line has first Chern class $U_p+epi^*h$, where
-$h=c_1(mathcal O(1))$calO(1))$ is the twistor parameter. Its spherical pairing
+The evaluation line has first Chern class U_p+epi^*h, where
+h=c_1(mathcal O(1))calO(1)) is the twistor parameter. Its spherical pairing
satisfies
FSW_E(U_p)=U_p([M_E^{reg}])
=SW_X(E)+eFSW_{mathfrak s_E}fraks_E(S^2).
-For every prescribed point $p$, if $FSW_E(U_p)ne0$
-then the exact family has an evaluation zero at $p$.
+For every prescribed point p, if FSW_E(U_p)ne0
+then the exact family has an evaluation zero at p.
Proof.
The evaluation line over the coefficient sphere is a line bundle
-$mathcal O(1)calO(1) ⊗ mathcal E_E|_p$.calE_E|_p$. The evaluation transforms with
+mathcal O(1)calO(1) ⊗ mathcal E_E|_p.calE_E|_p. The evaluation transforms with
the same gauge weight as the spinor, and its sign on a reducible link
-can be checked on $elltomathcal E_E|_p$:calE_E|_p$: it lies in
-$Hom(ell,mathcal E_E|_p)=calE_E|_p)=mathcal O(1)calO(1) ⊗
-mathcal E_E|_p$.calE_E|_p$. On the quotient, the evaluation section is a
+can be checked on elltomathcal E_E|_p:calE_E|_p: it lies in
+Hom(ell,mathcal E_E|_p)=calE_E|_p)=mathcal O(1)calO(1) ⊗
+mathcal E_E|_p.calE_E|_p. On the quotient, the evaluation section is a
section of the tensor product of the based-gauge evaluation line with
-the pullback of $mathcal E_E|_calE_E|_{{p}times S^2}$; its first Chern
-class is $U_p+epi^*h$. Pairing $pi^*h$ with the family moduli cycle
-gives $SW_X(E)$; equivalently, restrict the family to a
+the pullback of mathcal E_E|_calE_E|_{{p}times S^2}; its first Chern
+class is U_p+epi^*h. Pairing pi^*h with the family moduli cycle
+gives SW_X(E); equivalently, restrict the family to a
regular base point. Formula (eq:families-evaluation-identity)
is now immediate.
For the last assertion, suppose the exact family had no evaluation
-zero. Fixed-$r$ compactness (see Cref{lem:families-taubes-compactness})
+zero. Fixed-r compactness (see Cref{lem:families-taubes-compactness})
would give the same absence for all sufficiently small perturbations,
because otherwise a sequence of their zeros would limit to an exact
zero. Applying the preceding identity to a regular such perturbation
@@ -1788,9 +1788,9 @@ gives a contradiction.
[Taubes compactness]
Fix a compact smooth parameter family of backgrounds and a compact
-smooth family of allowed choices of taming pairs, and let $rto∞$.
+smooth family of allowed choices of taming pairs, and let rto∞.
After passing to a subsequence, the solutions of the perturbed
-Seiberg--Witten equations converge to a positive $J_∞$-holomorphic
+Seiberg--Witten equations converge to a positive J_∞-holomorphic
configuration of the prescribed class; the support contains all limit
points of first-component zeros.
@@ -1803,22 +1803,22 @@ first-component estimate, the off-support decay, the flux
identification, and the disk lower bound are carried out in the
original cite[Equations~(7.1)--(7.5) and Lemma~7.1]{Taubes1996}; the
uniform estimates hold because the smooth background family is
-compact and the choices of $r$ are made after the background and the
+compact and the choices of r are made after the background and the
approximation tolerance are fixed.
We state the two standard inputs we use explicitly. First, the first
-spinor component $a_n$ of a solution has a uniform integral two-form
-$T_n$ with bounded mass and periods on closed classes equal to those
-of $E$:
+spinor component a_n of a solution has a uniform integral two-form
+T_n with bounded mass and periods on closed classes equal to those
+of E:
T_n(zeta)=(i)/(2pi)int_XF_{a_n}wedgezeta,
- qquad
+
|T_n|le C(r_n)
-for a constant $C(r_n)$ that stays bounded in $r_n$; the identification
+for a constant C(r_n) that stays bounded in r_n; the identification
follows from the topological identity for the determinant line. Second,
-off the first-component zero set, the $L^1$-to-$C^1$ distance from the
+off the first-component zero set, the L^1-to-C^1 distance from the
canonical connection is
|mathcal A-calA-mathcal A_calA_{can}|
@@ -1826,56 +1826,56 @@ canonical connection is
le Cr^{-1}+Crexp(-sqrt r,
dist(x,alpha^{-1}(0))/C)
-when the distance is at least $r^{-1/2}$
+when the distance is at least r^{-1/2}
cite[Proposition~6.1]{Taubes1999}.
Fix the disk bound. Write
-$f_n=(i)/(2pi)F_{a_n}
- =(i)/(4pi)(F_{mathcal A_n}calA_n-F_{mathcal A_calA_{can,n}})$.
+f_n=(i)/(2pi)F_{a_n}
+ =(i)/(4pi)(F_{mathcal A_n}calA_n-F_{mathcal A_calA_{can,n}}).
The sharp estimate
cite[Proposition~4.2, Equation~(4.9)]{Taubes1999}, specialized to
-$|eta_n|=sqrt2$, is
+|eta_n|=sqrt2, is
|F_{mathcal A_n}calA_n^-|le(r_n)/(2sqrt2)(1-|alpha_n|^2)+C.
Its constants depend on the background family in a fixed smooth way.
-If $ell$ is a unit, positively oriented $J_n$-complex bivector, then
-$ell^+=eta_n/2$, $|ell^-|=1/sqrt2$, and the curvature equation
+If ell is a unit, positively oriented J_n-complex bivector, then
+ell^+=eta_n/2, |ell^-|=1/sqrt2, and the curvature equation
together with the second displayed identity give the disk bound
u^*f_nge-C_0,dd A_u
-for every $J_n$-holomorphic disk $u$, with $C_0ge1$ independent of $n$
+for every J_n-holomorphic disk u, with C_0ge1 independent of n
and of the parameter.
Proof. [Proof of Cref{thm:family-curves}]
-The projection $F^+$ is nonzero: otherwise $F$ would be a nonzero
-square-zero vector in the negative-definite space $P^perp$. Lift $F$
+The projection F^+ is nonzero: otherwise F would be a nonzero
+square-zero vector in the negative-definite space P^perp. Lift F
to an integral class and use the fixed spin-c structure
-$mathfrak s_F$.fraks_F$.
+mathfrak s_F.fraks_F.
The ordinary invariant vanishes. Choose the opposite coefficient
-direction $w=-F^+/|F^+|$, so that $F[B_w]<0$. If
-$SW_X(F)ne0$, ordinary invariance and fixed-parameter
+direction w=-F^+/|F^+|, so that F[B_w]<0. If
+SW_X(F)ne0, ordinary invariance and fixed-parameter
monopole compactness would give solutions of the equations for
-arbitrarily large $r$ at this fixed background. Taubes compactness
-would give a nonempty positive $J_w$-holomorphic configuration of
-class $F$, contradicting its negative $B_w$-area.
+arbitrarily large r at this fixed background. Taubes compactness
+would give a nonempty positive J_w-holomorphic configuration of
+class F, contradicting its negative B_w-area.
-The spherical $U_p$ pairing is nonzero. For $b_1=0$ this follows
-directly from Cref{lem:families-wall-crossing} with $K-O=0$; for
-$b_1=4$, the projective pushforward (eq:families-projective-pushforward)
+The spherical U_p pairing is nonzero. For b_1=0 this follows
+directly from Cref{lem:families-wall-crossing} with K-O=0; for
+b_1=4, the projective pushforward (eq:families-projective-pushforward)
reduces the family invariant to the same computation.
Since the ordinary invariant vanishes,
Cref{lem:families-evaluation-line} forces a first-component zero over
-$p$ in the exact family for each sufficiently large $r$. Extracting a
+p in the exact family for each sufficiently large r. Extracting a
convergent parameter subsequence and applying
Cref{lem:families-taubes-compactness} gives a configuration through
-$p$ in some $J_v$, with total class $F$.
+p in some J_v, with total class F.
-The two closed forms whose coefficients are perpendicular to $v$ are
-$J_v$-anti-invariant and restrict to zero on every component of this
-configuration. Thus $F^+$ is parallel to $[B_v]$. Its $B_v$-area is
+The two closed forms whose coefficients are perpendicular to v are
+J_v-anti-invariant and restrict to zero on every component of this
+configuration. Thus F^+ is parallel to [B_v]. Its B_v-area is
positive, so the proportionality factor is positive, which proves the
asserted direction.
@@ -1886,241 +1886,241 @@ The unmarked wall calculation also excludes the square-minus-two
classes needed in the lattice argument.
-If $b_1=0$, no class $EinΛ$ with $E^2=-2$ is orthogonal to $P$.
+If b_1=0, no class EinΛ with E^2=-2 is orthogonal to P.
Proof.
-Lift $E$ to an integral line-bundle class. Its ordinary expected
-dimension is $-2$, and its spherical family has dimension zero. By
+Lift E to an integral line-bundle class. Its ordinary expected
+dimension is -2, and its spherical family has dimension zero. By
Cref{lem:families-wall-crossing}, its large-parameter unmarked
-family invariant is $pm1$. For each sufficiently large $r$, take
-small regular perturbations and then a fixed-$r$ limit to obtain an
-actual solution of the equations at some parameter $v_r$. Pass to a
-sequence with $v_rto v$, and apply
+family invariant is pm1. For each sufficiently large r, take
+small regular perturbations and then a fixed-r limit to obtain an
+actual solution of the equations at some parameter v_r. Pass to a
+sequence with v_rto v, and apply
Cref{lem:families-taubes-compactness}. The limit is a nonempty
-positive $J_v$-holomorphic configuration of class $E$. Every
-nonconstant component has positive $B_v$-area, since $B_v$ tames
-$J_v$. This contradicts $E[B_v]=0$, which follows from $Eperp P$.
+positive J_v-holomorphic configuration of class E. Every
+nonconstant component has positive B_v-area, since B_v tames
+J_v. This contradicts E[B_v]=0, which follows from Eperp P.
# An integral null class with a uniform chamber
Curve existence (Cref{thm:family-curves}) is available for every
-nonzero integral class with $F^2=0$. What is needed now is one such
+nonzero integral class with F^2=0. What is needed now is one such
class for which a positive curve configuration cannot split into
components of smaller degree. This choice uses only the original
-positive period plane $P$, so the resulting restrictions on curve
+positive period plane P, so the resulting restrictions on curve
classes survive every later pair deformation that preserves the two
periods. The argument has three parts: (a) a lattice-theoretic choice
-of a primitive null class $F$ avoiding finitely many obstructions;
+of a primitive null class F avoiding finitely many obstructions;
(b) adjunction and positivity of intersections, which together turn
the lattice properties into geometric properties of the curves; and
-(c) an orthogonal adaptation of the coefficients making $[B]$ the
-direction of $F^+$.
+(c) an orthogonal adaptation of the coefficients making [B] the
+direction of F^+.
## The lattice and the chamber class
Let
[
- Λ=H^2(X;Z)/torsion,qquad
- Λ_{Q}=Λ ⊗ _{Z}Q,qquad
+ Λ=H^2(X;Z)/torsion,
+ Λ_{Q}=Λ ⊗ _{Z}Q,
Λ_{R}=Λ ⊗ _{Z}R.
]
-A curve class is the image in $Λ$ of the integral
+A curve class is the image in Λ of the integral
Poincar'e dual of a closed oriented curve, with its complex
orientation. All equalities of integral classes in this section are
equal to torsions; this does not affect intersections or integrals of
-closed real forms. The original positive period plane is $P$, and
-$d^+$ denotes the orthogonal projection of $dinΛ_{R}$ to $P$.
+closed real forms. The original positive period plane is P, and
+d^+ denotes the orthogonal projection of dinΛ_{R} to P.
[Choice of chamber class]
-There is a primitive class $FinΛ$ with $F^2=0$ such that the
-following implication holds for every $dinΛ$ and every real
-number $c>0$:
+There is a primitive class FinΛ with F^2=0 such that the
+following implication holds for every dinΛ and every real
+number c>0:
- d^2ge-2,qquad d^+=cF^+,qquad dcdot Fle0
- quad⇒quad
- cinZ_{>0},qquad dcdot F=0.
+ d^2ge-2, d^+=cF^+, dcdot Fle0
+ ⇒
+ cinZ_{>0}, dcdot F=0.
-Moreover $F^+ne0$.
+Moreover F^+ne0.
Proof.
noindentStep~1: a rational null subspace with many integral
-directions. Because $P^perp$ is negative definite and $dim P=3$,
-the restriction of the intersection form to the $(3,b^-+1)$-dimensional
-subspace $P^perp ⊕ R F_0$ for a null $F_0inΛ$ is indefinite,
+directions. Because P^perp is negative definite and dim P=3,
+the restriction of the intersection form to the (3,b^-+1)-dimensional
+subspace P^perp ⊕ R F_0 for a null F_0inΛ is indefinite,
so it contains an integral null vector. Take a rational null subspace
-$L$ of $Λ$ of maximal dimension, namely dimension three: the
-intersection form restricted to $L_{R}$ has signature $(1,2)$ and
-intersects $Λ$ in a lattice. Let $lin L$ be a primitive vector
-with $lcdot l=0$ and such that $L_{R}=l^perpcap L_{R} ⊕ R l$.
+L of Λ of maximal dimension, namely dimension three: the
+intersection form restricted to L_{R} has signature (1,2) and
+intersects Λ in a lattice. Let lin L be a primitive vector
+with lcdot l=0 and such that L_{R}=l^perpcap L_{R} ⊕ R l.
Let
[
mathcal F=calF={Fin LcapΛ: F^2=0, Fhbox{ primitive},
Fcdot l=1}.
]
-Each $Finmathcal F$calF$ is a nonzero null vector and belongs to $L$. Since
-$l$ is null and $Fcdot l=1$, we have $F^+notin P^perp$: otherwise
-$Fcdot l=F^+cdot l+F^-cdot l=F^+cdot l=F^+cdot l$ and $F^+in P$
-would force $F^+cdot l=0$, a contradiction. In particular $F^+ne0$.
-The set $mathcal F$calF$ is infinite by completing any rational null vector
-to a hyperbolic pair with $l$.
-
-noindentStep~2: projecting and pinning $d^+$. Fix $Finmathcal F$calF$
-and suppose $dinΛ$, $c>0$ satisfy the three left-hand conditions
+Each Finmathcal FcalF is a nonzero null vector and belongs to L. Since
+l is null and Fcdot l=1, we have F^+notin P^perp: otherwise
+Fcdot l=F^+cdot l+F^-cdot l=F^+cdot l=F^+cdot l and F^+in P
+would force F^+cdot l=0, a contradiction. In particular F^+ne0.
+The set mathcal FcalF is infinite by completing any rational null vector
+to a hyperbolic pair with l.
+
+noindentStep~2: projecting and pinning d^+. Fix Finmathcal FcalF
+and suppose dinΛ, c>0 satisfy the three left-hand conditions
of (eq:lattice-chamber-implication). Then
[
- r:=d-cFin P^perp,qquad r^2=d^2-2c(dcdot F)ge-2,
+ r:=d-cFin P^perp, r^2=d^2-2c(dcdot F)ge-2,
]
-because $Fcdot F=0$, so $r$ lies in the compact set
+because Fcdot F=0, so r lies in the compact set
[
K={rin P^perp:r^2ge-2}.
]
noindentStep~3: only finitely many cosets. Let
-$q:Λ_{R}toΛ_{R}/L_{R}$ be the quotient map. Because
-$L_{R}$ is rational and $L$ is saturated, a basis of $L$ extends to an
-integral basis of $Λ$; the remaining basis vectors project to a
-lattice basis in the real quotient. Hence $q(Λ)$ is discrete and
-$ker(q|_Λ)=L$. Since $q(d)=q(r)in q(K)$, only finitely many
-cosets $d+L$ can occur in the displayed expression for $r$. This finite
-set is independent of $F$.
+q:Λ_{R}toΛ_{R}/L_{R} be the quotient map. Because
+L_{R} is rational and L is saturated, a basis of L extends to an
+integral basis of Λ; the remaining basis vectors project to a
+lattice basis in the real quotient. Hence q(Λ) is discrete and
+ker(q|_Λ)=L. Since q(d)=q(r)in q(K), only finitely many
+cosets d+L can occur in the displayed expression for r. This finite
+set is independent of F.
noindentStep~4: avoiding irrational points on a torus. Orthogonal
projection restricts to an isomorphism
[
pi:L_{R}→ P:
]
-its kernel is null and lies in the negative-definite space $P^perp$,
-and both spaces have dimension three. Hence $pi(L)$ is a lattice in
-$P$. Each of the finitely many cosets just found determines one point of
+its kernel is null and lies in the negative-definite space P^perp,
+and both spaces have dimension three. Hence pi(L) is a lattice in
+P. Each of the finitely many cosets just found determines one point of
[
mathbb T=P/bbT=P/pi(L),
]
-because adding an element of $L$ changes the projection by an element of
-$pi(L)$. For $Finmathcal F$calF$ define the circle
+because adding an element of L changes the projection by an element of
+pi(L). For Finmathcal FcalF define the circle
[
mathbb S_F=bbS_F={tpi(F)+pi(L):tinR}subsetmathbb T.bbT.
]
Two such circles with different directions intersect only at torsion
-points: if $tpi(F)-spi(F')inpi(L)$ with $F,F'$ not collinear, choose
-an integral linear functional on $L$ vanishing on $F$ and nonzero on
-$F'$; applying it to $tF-sF'in L$ forces $sinQ$, so the common point
-is torsion. In particular a non-torsion point of $mathbb T$bbT$ lies on at
-most one circle $mathbb S_F$.bbS_F$.
-
-The elements of $mathcal F$calF$ have distinct directions: two collinear
-elements with $l$-pairing one are equal. Since $mathcal F$calF$ is infinite,
-we can choose $Finmathcal F$calF$ whose circle contains none of the
+points: if tpi(F)-spi(F')inpi(L) with F,F' not collinear, choose
+an integral linear functional on L vanishing on F and nonzero on
+F'; applying it to tF-sF'in L forces sinQ, so the common point
+is torsion. In particular a non-torsion point of mathbb TbbT lies on at
+most one circle mathbb S_F.bbS_F.
+
+The elements of mathcal FcalF have distinct directions: two collinear
+elements with l-pairing one are equal. Since mathcal FcalF is infinite,
+we can choose Finmathcal FcalF whose circle contains none of the
non-torsion points among the finitely many possible coset points.
-noindentStep~5: integrality and vanishing. For this $F$, every
-$d,c$ satisfying the hypotheses gives a torsion point
+noindentStep~5: integrality and vanishing. For this F, every
+d,c satisfying the hypotheses gives a torsion point
[
d^++pi(L)=cpi(F)+pi(L).
]
-So there is an integer $k>0$ with $kcFin L$. Pairing with $l$ shows
-$kcinZ$, so $cinQ$. It follows that $r=d-cFin P^perpcapΛ_Q$,
-and the lattice property of $P^perp$ now gives
+So there is an integer k>0 with kcFin L. Pairing with l shows
+kcinZ, so cinQ. It follows that r=d-cFin P^perpcapΛ_Q,
+and the lattice property of P^perp now gives
c=lcdot dinZ_{>0}.
-If $m=dcdot F<0$, then $m$ is a negative integer and
+If m=dcdot F<0, then m is a negative integer and
[
r^2=d^2-2cmge-2+2cge0.
]
-Negative definiteness of $P^perp$ forces $r=0$, which would imply
-$m=0$. Contradiction; hence $dcdot F=0$, and the implication is proved.
+Negative definiteness of P^perp forces r=0, which would imply
+m=0. Contradiction; hence dcdot F=0, and the implication is proved.
## Curve signs
-Fix $F$ as in Cref{prop:chamber-class}. Make a constant orthogonal
+Fix F as in Cref{prop:chamber-class}. Make a constant orthogonal
change of the coefficient basis of the original triple and denote the
-result again by $(A,C,B)$, choosing
+result again by (A,C,B), choosing
- [B]=(F^+)/(lVert F^+rVert),qquad
- [A]cdot F=[C]cdot F=0,qquad
+ [B]=(F^+)/(lVert F^+rVert),
+ [A]cdot F=[C]cdot F=0,
beta=[B]cdot F=lVert F^+rVert>0.
-Here the norm is the positive intersection norm on $P$. The three
-classes remain orthonormal, and $V=[A]^perpcap[C]^perp$ has
-signature $(1,b^-)$. All later statements are made for arbitrary
-definite pairs with these two periods $[A],[C]$; the choice is fixed by
-$F$ and the original period plane $P$, independent of any later pair.
+Here the norm is the positive intersection norm on P. The three
+classes remain orthonormal, and V=[A]^perpcap[C]^perp has
+signature (1,b^-). All later statements are made for arbitrary
+definite pairs with these two periods [A],[C]; the choice is fixed by
+F and the original period plane P, independent of any later pair.
[Curve signs]
-Let $J$ be an almost-complex structure with definite pair $(A,C)$
+Let J be an almost-complex structure with definite pair (A,C)
satisfying the adapted periods (eq:lattice-adapted-periods) and
-tamed by $B$. For every irreducible closed $J$-holomorphic curve with
-class $dinΛ$ (identified with its Poincar'e dual), let
-$m=Fcdot d$ and write $[B]cdot d=beta c$ with the constant
-$beta=[B]cdot F>0$. Then:
+tamed by B. For every irreducible closed J-holomorphic curve with
+class dinΛ (identified with its Poincar'e dual), let
+m=Fcdot d and write [B]cdot d=beta c with the constant
+beta=[B]cdot F>0. Then:
[label=(roman*)]
-item $mge0$.
-item $m=0$ if and only if $d=cF$ for some real $c$ (in which case
-$cinZ$, and $c>0$ if $dne0$).
+ mge0.
+ m=0 if and only if d=cF for some real c (in which case
+cinZ, and c>0 if dne0).
Proof.
By Cref{lem:definite-pair}, the definite pair trivializes the
-canonical bundle, so $c_1(TX,J)=0$ for either sign of $J$. The
+canonical bundle, so c_1(TX,J)=0 for either sign of J. The
standard adjunction and singularity theorem
cite[Theorem~2.8]{Wendl2020} give
- d^2=2g(Σ)-2+2delta(u),qquad delta(u)inZ_{ge0},
+ d^2=2g(Σ)-2+2delta(u), delta(u)inZ_{ge0},
-so in particular $d^2ge-2$.
+so in particular d^2ge-2.
Curve classes couple trivially with the anti-invariant forms, so
-$dcdot[A]=dcdot[C]=0$. Equivalently $din V=[A]^perpcap[C]^perp$.
-Since $Vcap P=R[B]$ (both spaces contain $[B]$ and $V$ is the
-codimension-two plane cut out by $[A],[C]$), the projection $d^+$
-is a multiple of $[B]=F^+/lVert F^+rVert$.
-
-noindent(i). Write $d^+=c'F^+$ with $c'inR$. If $c'>0$, apply
-Cref{prop:chamber-class} to $d$ with $c=c'$: if $dcdot Fle0$ then
-$c'inZ_{>0}$ and $dcdot F=0$; in particular, either $dcdot F=0$ or
-$dcdot F>0$. If $c'<0$, apply to $-d$ and $-c'$: same conclusion, and
-the sign of $dcdot F$ is unchanged. If $c'=0$, then $d^+=0$; by
-adjunction $d^2ge-2$ and by negativity of $P^perp$, $dcdot F=d^-cdot
-F^-le0$, so the implication traps $dcdot F=0$ (indeed $dcdot Fle0$
-would give $c'inZ_{>0}$, which $c'=0$ violates; a direct inspection
-shows $dcdot Fge0$). In all cases $mge0$.
-
-noindent(ii). If $d=cF$ with $cinR$ then $m=cF^2=0$ and
-$cinZ$ since both $d,F$ are integral and $F$ is primitive. Conversely
-assume $m=0$. From $din V$ and $m=dcdot F=0$, the class $d$ is
-orthogonal to the nondegenerate plane $span([B],F)subset
-V$; the orthogonal complement in $V$ is negative definite. So
-$d^2le0$, and adjunction bounds $d^2ge-2$. If $d^+ne0$, then
-$d^+=c'F^+$ with $c'ne0$; Cref{prop:chamber-class} forces $c'inZ$,
-and then $d^2le0$ combined with $0=m=d^+F^+ + d^-F^-$ yields
-$d^2=0$ (by completing the square in the Lorentzian space). In that case
-$d^pm$ are both null, and $d$ is a null vector in $V$. But in a
+dcdot[A]=dcdot[C]=0. Equivalently din V=[A]^perpcap[C]^perp.
+Since Vcap P=R[B] (both spaces contain [B] and V is the
+codimension-two plane cut out by [A],[C]), the projection d^+
+is a multiple of [B]=F^+/lVert F^+rVert.
+
+(i). Write d^+=c'F^+ with c'inR. If c'>0, apply
+Cref{prop:chamber-class} to d with c=c': if dcdot Fle0 then
+c'inZ_{>0} and dcdot F=0; in particular, either dcdot F=0 or
+dcdot F>0. If c'<0, apply to -d and -c': same conclusion, and
+the sign of dcdot F is unchanged. If c'=0, then d^+=0; by
+adjunction d^2ge-2 and by negativity of P^perp, dcdot F=d^-cdot
+F^-le0, so the implication traps dcdot F=0 (indeed dcdot Fle0
+would give c'inZ_{>0}, which c'=0 violates; a direct inspection
+shows dcdot Fge0). In all cases mge0.
+
+(ii). If d=cF with cinR then m=cF^2=0 and
+cinZ since both d,F are integral and F is primitive. Conversely
+assume m=0. From din V and m=dcdot F=0, the class d is
+orthogonal to the nondegenerate plane span([B],F)subset
+V; the orthogonal complement in V is negative definite. So
+d^2le0, and adjunction bounds d^2ge-2. If d^+ne0, then
+d^+=c'F^+ with c'ne0; Cref{prop:chamber-class} forces c'inZ,
+and then d^2le0 combined with 0=m=d^+F^+ + d^-F^- yields
+d^2=0 (by completing the square in the Lorentzian space). In that case
+d^pm are both null, and d is a null vector in V. But in a
Lorentzian space, a null vector in the same closure of the positive cone
-as $F$ (which it is since $dcdot[B]=beta c'>0$) is a positive
-multiple of $F$; hence $d=cF$ for some $cinZ_{>0}$. If $d^+=0$, then
-$din P^perp$ and $d^2le0$; adjunction gives $-2le d^2le0$, and
-$m=dcdot F=d^-F^-le0$ forces $d^-F^-=0$; together with $F^-ne0$
-this forces $d^-=0$ (the orthogonal complement of $F^-$ in $P^perp$
-being spacelike with intersection $ge-1$), so $d=0=cF$ with $c=0$.
+as F (which it is since dcdot[B]=beta c'>0) is a positive
+multiple of F; hence d=cF for some cinZ_{>0}. If d^+=0, then
+din P^perp and d^2le0; adjunction gives -2le d^2le0, and
+m=dcdot F=d^-F^-le0 forces d^-F^-=0; together with F^-ne0
+this forces d^-=0 (the orthogonal complement of F^- in P^perp
+being spacelike with intersection ge-1), so d=0=cF with c=0.
## No splitting
[Nonsplitting]
For any definite pair with the adapted periods (eq:lattice-adapted-periods),
-let $J$ be the almost-complex structure tamed by $B$. Then:
+let J be the almost-complex structure tamed by B. Then:
[label=(roman*)]
-item Every positive $J$-holomorphic configuration of total class $F$
+ Every positive J-holomorphic configuration of total class F
is a single simple image with multiplicity one.
-item A curve of class $F$ through any prescribed point exists; any two
+ A curve of class F through any prescribed point exists; any two
such curves are disjoint.
-item Each such curve is either an embedded torus or an immersed rational
+ Each such curve is either an embedded torus or an immersed rational
curve with exactly one transverse positive double point.
@@ -2128,34 +2128,34 @@ curve with exactly one transverse positive double point.
Proof.
For (i), write a positive configuration as
[
- F=sum_{i=1}^rn_id_iquadhbox{in }Λ,qquad n_iinZ_{>0},
+ F=sum_{i=1}^rn_id_iquadhbox{in }Λ, n_iinZ_{>0},
]
-where $d_i$ are the classes of its distinct irreducible images. Taming
+where d_i are the classes of its distinct irreducible images. Taming
excludes zero real component classes. By Cref{cor:curve-signs},
-$m_i=d_icdot Fge0$, whereas
+m_i=d_icdot Fge0, whereas
[
0=F^2=sum_{i=1}^rn_im_i.
]
-Every $m_i$ therefore vanishes, and again by Cref{cor:curve-signs}
-$d_i=c_iF$ with $c_iinZ_{>0}$, giving
+Every m_i therefore vanishes, and again by Cref{cor:curve-signs}
+d_i=c_iF with c_iinZ_{>0}, giving
[
sum_ic_in_i=1.
]
-There is exactly one summand, with $n_1=c_1=1$.
+There is exactly one summand, with n_1=c_1=1.
For (ii), apply Cref{thm:family-curves}. Its period plane is
-still $P$, and (eq:lattice-adapted-periods) forces the direction
+still P, and (eq:lattice-adapted-periods) forces the direction
of the configuration to be precisely the one whose anti-invariant plane
-is $span(widetilde A,widetilde C)$ and whose tamer is
-$widetilde B$; thus the theorem gives a positive $J$-curve of class
-$F$ through any prescribed point. Part (i) makes it one
+is span(widetilde A,widetilde C) and whose tamer is
+widetilde B; thus the theorem gives a positive J-curve of class
+F through any prescribed point. Part (i) makes it one
reduced component. Two such curves are disjoint: their classes both
-equal $F$, with $F^2=0$, while positivity of intersection for distinct
+equal F, with F^2=0, while positivity of intersection for distinct
simple images would contribute positively.
-For (iii), adjunction becomes $0=F^2=2g(Σ)-2+2delta(u)$,
-or $g(Σ)+delta(u)=1$. If $g(Σ)=1$ then $delta(u)=0$ and the
-curve is embedded. If $g(Σ)=0$ then $g+delta=1$ forces $delta=1$;
+For (iii), adjunction becomes 0=F^2=2g(Σ)-2+2delta(u),
+or g(Σ)+delta(u)=1. If g(Σ)=1 then delta(u)=0 and the
+curve is embedded. If g(Σ)=0 then g+delta=1 forces delta=1;
when the map is immersed, its defect is a sum of positive
intersection multiplicities between distinct local branches, and
total defect one forces exactly one pair meeting transversely at a
@@ -2165,20 +2165,20 @@ positive double point.
# A torus pencil with ordinary nodes
-We now use Cref{prop:nonsplitting} to turn the family of $F$-curves into
+We now use Cref{prop:nonsplitting} to turn the family of F-curves into
the fibers of a smooth proper map. Curve classes are identified with
-their Poincar'e duals modulo torsion; $F$ is primitive, $F^2=0$,
-$[A]F=[C]F=0$, and $[B]F>0$. The form $B$ stays fixed throughout.
+their Poincar'e duals modulo torsion; F is primitive, F^2=0,
+[A]F=[C]F=0, and [B]F>0. The form B stays fixed throughout.
[The pencil]
-There are smooth one-forms $a_t,c_t$, $0le tle1$, with $a_0=c_0=0$,
+There are smooth one-forms a_t,c_t, 0le tle1, with a_0=c_0=0,
such that
[
- A_t=A+dd a_t,qquad C_t=C+dd c_t,
+ A_t=A+dd a_t, C_t=C+dd c_t,
]
-and $(A_t,C_t,B)$ is a positive closed triple for every $t$. For the
-almost-complex structure $J$ determined by $(A_1,C_1)$ and tamed by
-$B$, the irreducible curves of class $F$ are precisely the fibers of a
+and (A_t,C_t,B) is a positive closed triple for every t. For the
+almost-complex structure J determined by (A_1,C_1) and tamed by
+B, the irreducible curves of class F are precisely the fibers of a
smooth proper map
[
p_X:X→ S
@@ -2188,13 +2188,13 @@ embedded torus. The singular fibers, if any, are finitely many immersed
rational curves, each with one transverse positive double point.
-## Curves of class $F$
+## Curves of class F
Fix any triple used below. Cref{prop:nonsplitting} gives a curve of
-class $F$ through every point, and every positive configuration of
-total class $F$ is one simple image with multiplicity one. Two
+class F through every point, and every positive configuration of
+total class F is one simple image with multiplicity one. Two
distinct such images are disjoint: any intersection would contribute
-positively to $F^2=0$. A simple parametrization $u:Lto X$ has by
+positively to F^2=0. A simple parametrization u:Lto X has by
adjunction
g(L)+delta(u)=1.
@@ -2209,53 +2209,53 @@ double point, and the two branches meet transversely.
## Removing critical points of sphere parametrizations
-To make the $F$-curves into a torus foliation, we first remove the
+To make the F-curves into a torus foliation, we first remove the
critical points of the sphere parametrizations by exact perturbations
-of the pair with $B$ held fixed. The next lemma is the restricted
+of the pair with B held fixed. The next lemma is the restricted
first-jet transversality statement.
[Restricted first-jet transversality]
-An arbitrarily small smooth exact perturbation of $(A,C)$, with $B$
-fixed and the triple positive, makes every simple sphere in class $F$
+An arbitrarily small smooth exact perturbation of (A,C), with B
+fixed and the triple positive, makes every simple sphere in class F
immersed. The same assertion holds for a path of such pairs, relative
to immersed endpoints.
These perturbations may preserve any finite set of specified immersed
-$F$-spheres that are common Lagrangians throughout the path. At an
+F-spheres that are common Lagrangians throughout the path. At an
endpoint they may also leave fixed closed model subdisks already
-foliated by $F$-curves.
+foliated by F-curves.
The proof occupies the remainder of this subsection. It is a
finite-dimensional transversality within the space of exact one-form
-variations of $(A,C)$, with $B$ fixed; its dimension is large, and the
-only constraints come from the invertibility of the pairing with $B$.
+variations of (A,C), with B fixed; its dimension is large, and the
+only constraints come from the invertibility of the pairing with B.
We do not need to state the full machinery; the key points are the
following.
-First, the moduli space of parametrized simple spheres of class $F$ in
+First, the moduli space of parametrized simple spheres of class F in
a fixed pair is a smooth finite-dimensional manifold of the expected
-dimension. Adjunction gives $c_1(u^*TX)=F^2=0$, and the Riemann--Roch
+dimension. Adjunction gives c_1(u^*TX)=F^2=0, and the Riemann--Roch
computation for a parametrized sphere (with a mark) gives real
-dimension $4$.
+dimension 4.
Second, the linearization of the Cauchy--Riemann operator combined
with the jet evaluation at a critical point is surjective onto the
appropriate finite-dimensional target, and remains surjective after
-restricting to exact one-form variations of $A,C$ with $B$ fixed.
+restricting to exact one-form variations of A,C with B fixed.
Testing the derivative of the universal section gives the cokernel:
-the exact variations freely adjust the pullback $u^*A_v$ and $u^*C_v$
-by adjustable pullback forms with $B$ fixed. Combining this with the
+the exact variations freely adjust the pullback u^*A_v and u^*C_v
+by adjustable pullback forms with B fixed. Combining this with the
parametrized moduli calculation yields a Fredholm projection of
-index $4$ (or $5$ for a path), which by Sard--Smale is regular for a
+index 4 (or 5 for a path), which by Sard--Smale is regular for a
residual set of exact perturbations.
-Third, the group $PSL(2,C)$ of M"obius
+Third, the group PSL(2,C) of M"obius
transformations acts freely on the simple parametrized spheres, with
the marked point transformed contragrediently, and preserves the
fiber of the jet evaluation. Freeness follows because a
reparametrization fixing a simple map fixes its injective locus and
-hence is the identity. A manifold of dimension $4$ or $5$ cannot
+hence is the identity. A manifold of dimension 4 or 5 cannot
contain this six-dimensional orbit, so the fiber is empty. This
proves the lemma. The exact preservation of specified common
Lagrangians at the endpoint uses the same argument restricted to
@@ -2266,34 +2266,34 @@ qed
## The normal equation and foliation by tori
[The normal operator]
-Let $u:Lto X$ be a closed connected common Lagrangian immersion for a
-definite closed pair $(A,C)$, with the complex orientation chosen by
-$B$. Nearby common Lagrangian immersions represented as normal graphs
-over $u$ form a smooth manifold of real dimension $b_1(L)$. Its
-linearized kernel is naturally isomorphic to $H^1(L;R)$. If $u$ is
+Let u:Lto X be a closed connected common Lagrangian immersion for a
+definite closed pair (A,C), with the complex orientation chosen by
+B. Nearby common Lagrangian immersions represented as normal graphs
+over u form a smooth manifold of real dimension b_1(L). Its
+linearized kernel is naturally isomorphic to H^1(L;R). If u is
simple, this also describes the local unparametrized moduli. These
descriptions hold with smoothly varying pairs whose periods on
-$[u(L)]$ remain zero.
+[u(L)] remain zero.
-An immersed $F$-sphere is isolated and continues uniquely under small
-pair variations. Near an embedded $F$-torus the nearby $F$-curves
+An immersed F-sphere is isolated and continues uniquely under small
+pair variations. Near an embedded F-torus the nearby F-curves
form a smooth foliation of an open neighborhood.
Proof.
-Represent nearby immersions by normal graphs over $u$, removing
-infinitesimal reparametrizations. If $u$ is simple, its stabilizer is
+Represent nearby immersions by normal graphs over u, removing
+infinitesimal reparametrizations. If u is simple, its stabilizer is
trivial, so the normal-graph slice parametrizes the local
-unparametrized moduli. The symplectic form $A$ identifies the real
-normal bundle with $T^*L$ by $wmapstoalpha=u^*iota_wA$. Write,
+unparametrized moduli. The symplectic form A identifies the real
+normal bundle with T^*L by wmapstoalpha=u^*iota_wA. Write,
along the source,
- H=(C-xA)/(y),quad Ω=A+iH,quad
- x=(Awedge C)/(A^2),quad y^2=(C^2)/(A^2)-x^2,
+ H=(C-xA)/(y), Ω=A+iH,
+ x=(Awedge C)/(A^2), y^2=(C^2)/(A^2)-x^2,
-with the sign of $y$ giving $J$ tamed by $B$. Let $*_L$ be the star on
+with the sign of y giving J tamed by B. Let *_L be the star on
one-forms for the induced conformal structure and orientation. The
-complex volume identity gives $u^*iota_wC=xalpha+y*_Lalpha$; Cartan's
+complex volume identity gives u^*iota_wC=xalpha+y*_Lalpha; Cartan's
formula then gives the normal derivative
mathcal DcalDalpha=(ddalpha,dd(xalpha+y*_Lalpha)).
@@ -2301,69 +2301,69 @@ formula then gives the normal derivative
The nonlinear pullbacks have values in pairs of exact two-forms: their
integrals are the two fixed zero periods.
-Given exact two-forms $zeta_1,zeta_2$, choose $alpha_0$ with
-$ddalpha_0=zeta_1$. Adding $dd f$ changes only the second equation,
+Given exact two-forms zeta_1,zeta_2, choose alpha_0 with
+ddalpha_0=zeta_1. Adding dd f changes only the second equation,
by
mathcal PcalP f=dd(xdd f+y*_Ldd f)
=dd xwedgedd f+dd(y*_Ldd f).
After division by a positive area form its principal part is the
-nonzero signed multiple $yDelta f$ of the scalar Laplacian. The sign
-of $y$ is constant on connected $L$, the operator has no zeroth-order
+nonzero signed multiple yDelta f of the scalar Laplacian. The sign
+of y is constant on connected L, the operator has no zeroth-order
term, and the strong maximum principle shows that its kernel consists
exactly of constants
cite[Chapter~1, Theorem~1.6]{KiselevRoquejoffreRyzhik2012}. It has
index zero, by homotopy through scalar elliptic operators to the
-signed Laplacian with the sign of $y$ held fixed; the Fredholm index
+signed Laplacian with the sign of y held fixed; the Fredholm index
is constant along this homotopy
cite[Section~7, Propositions~7.3--7.4]{TaylorFunctionalNotes}. Its
image has integral zero by Stokes' theorem, so its cokernel has
dimension one and its range is exactly the integral-zero two-forms.
-We can therefore solve $mathcal PcalP f=zeta_2-dd(xalpha_0+y*_Lalpha_0)$,
+We can therefore solve mathcal PcalP f=zeta_2-dd(xalpha_0+y*_Lalpha_0),
proving surjectivity onto the stated target with bounded elliptic
right inverses in Sobolev or H"older spaces.
-The kernel of $mathcal D$calD$ is thus ${alpha:ddalpha=0}$ together
+The kernel of mathcal DcalD is thus {alpha:ddalpha=0} together
with the solutions of the scalar equation, which are the constant
-functions; the holonomy from $L=L_0$ to nearby $L_t$ identifies the
-two kernels. The kernel of $mathcal D$calD$ is therefore isomorphic to
-$H^1(L;R)$, of dimension $b_1(L)$. The linearization is Fredholm of
-index $b_1(L)$, and the graph map from the normal graph is a
+functions; the holonomy from L=L_0 to nearby L_t identifies the
+two kernels. The kernel of mathcal DcalD is therefore isomorphic to
+H^1(L;R), of dimension b_1(L). The linearization is Fredholm of
+index b_1(L), and the graph map from the normal graph is a
diffeomorphism onto an open set of common Lagrangian immersions.
-For an immersed $F$-sphere the source is $S^2$, so $H^1(S^2;R)=0$ and
+For an immersed F-sphere the source is S^2, so H^1(S^2;R)=0 and
the kernel is trivial: such a sphere is isolated. Since the moduli
space is finite-dimensional and compact (Lemma~lem:pencil-compactness
below), it is finite. Along a smooth pair path, the parameter version
of the normal operator is a local diffeomorphism, so every sphere has
a unique smooth continuation.
-For an embedded $F$-torus, $b_1(T^2)=2$, and the kernel $H^1(T^2;R)$
+For an embedded F-torus, b_1(T^2)=2, and the kernel H^1(T^2;R)
has real dimension two. By the implicit function theorem, the nearby
-$F$-curves in the normal directions foliate a neighborhood of $u(L)$;
+F-curves in the normal directions foliate a neighborhood of u(L);
the fibers of this foliation are parametrized by the two cohomology
-classes in $H^1(T^2;R)$ and are disjoint. Smoothness of the foliation
+classes in H^1(T^2;R) and are disjoint. Smoothness of the foliation
is by the parameter version of the same elliptic solving.
[Compactness]
-The space of unparametrized $F$-spheres is compact.
+The space of unparametrized F-spheres is compact.
Proof.
Standard compactness for genus-zero curves with uniformly bounded
energy under a fixed tamed almost-complex structure: the energy is
-$F[B]=q>0$, uniform in the parameter; the moduli space is compact by
+F[B]=q>0, uniform in the parameter; the moduli space is compact by
cite[Theorems~5.3.1 and~5.5.5]{McDuffSalamon2012}. Any limit could
contain multiple components, but Cref{prop:nonsplitting} rules out
splitting. This gives compactness.
[The number of nodes along a path]
-If all $F$-spheres for a triple are immersed, there are finitely many
+If all F-spheres for a triple are immersed, there are finitely many
of them. Along a compact smooth path of triples with the original
-periods and with all $F$-spheres immersed, their number is constant.
+periods and with all F-spheres immersed, their number is constant.
Proof.
@@ -2378,10 +2378,10 @@ charts at the endpoints. Hence a finite covering of the interval.
[Insertion while retaining the sphere]
-Let $K$ be the image of an immersed $F$-sphere with one transverse
+Let K be the image of an immersed F-sphere with one transverse
positive double point. Inside an arbitrarily small neighborhood of
-$K$, an exact path of the pair, with $B$ fixed, makes the pair equal
-near $K$ to (eq:regular-input-model) for a model satisfying
+K, an exact path of the pair, with B fixed, makes the pair equal
+near K to (eq:regular-input-model) for a model satisfying
(eq:nodal-exact-identities). The triple stays positive and the
same immersed sphere is a common Lagrangian throughout. The
construction may be performed in disjoint neighborhoods of finitely
@@ -2389,67 +2389,67 @@ many such spheres.
Proof. [Proof sketch]
-At the node $p$, with preimages $p_+,p_-$ on the normalization, the
+At the node p, with preimages p_+,p_- on the normalization, the
argument first matches the local holomorphic model of
-Cref{thm:nodal-model} at $p$: a single complex-linear $T$ matching
-the two tangent maps at $p_pm$, and a complex volume matching
-$u_Y^*Σ_Y=Ω(p)$. Along the smooth part of the source, the
+Cref{thm:nodal-model} at p: a single complex-linear T matching
+the two tangent maps at p_pm, and a complex volume matching
+u_Y^*Σ_Y=Ω(p). Along the smooth part of the source, the
tangent map and volume determine the map on the normal quotient. A
partition-of-unity argument gives a local diffeomorphism whose
-derivative is $T$ at the node, and which is a complex-linear
+derivative is T at the node, and which is a complex-linear
diffeomorphism along the two branches; the normal direction is
adjusted by a shear in the flat Hopf metric with smooth bounded
-coefficients. The rest of $K$ is matched by a logarithmic transverse
+coefficients. The rest of K is matched by a logarithmic transverse
cutoff that shrinks the correction.
For the exact correction, one applies closedness and a radial
homotopy primitive at the crossing, and a normal-tube contraction
along the smooth part; both corrections are concentrated near the
-pre-specified neighborhoods of $K$ and are exact changes of the
+pre-specified neighborhoods of K and are exact changes of the
pair. The positivity is preserved by making each cutoff support small
enough and the multipliers bounded. The common Lagrangian sphere is
-preserved by choosing the primitives with zero pullback on $K$.
+preserved by choosing the primitives with zero pullback on K.
## The pencil
Proof. [Proof of Cref{thm:pencil}]
-Start from a generic state where all $F$-spheres are immersed (by
+Start from a generic state where all F-spheres are immersed (by
Cref{lem:pencil-jets}) and the number of these spheres is finite (by
-Cref{lem:pencil-compactness}). Let $K_1,...,K_N$ be the images.
+Cref{lem:pencil-compactness}). Let K_1,...,K_N be the images.
-Use Cref{lem:pencil-insertion} at each $K_i$; choose smaller closed
+Use Cref{lem:pencil-insertion} at each K_i; choose smaller closed
model subdisks at the endpoint. Make the endpoint generic by an
arbitrarily small exact perturbation outside the model subdisks
-(Cref{lem:pencil-jets} applies because any competing $F$-curve is
+(Cref{lem:pencil-jets} applies because any competing F-curve is
disjoint from every model fiber). The endpoint now has no critical
spheres, and the smaller model neighborhoods remain unchanged. Extend
this endpoint perturbation a short distance into the path using a
-parameter cutoff; all retained $K_i$ are still common Lagrangians.
+parameter cutoff; all retained K_i are still common Lagrangians.
Apply the path version of Cref{lem:pencil-jets}, keeping its
-endpoints and every $K_i$ fixed, to remove all critical spheres from
-the interior. The resulting smooth path stays positive with $B$
+endpoints and every K_i fixed, to remove all critical spheres from
+the interior. The resulting smooth path stays positive with B
fixed. By Cref{cor:pencil-node-count}, its endpoints have the same
-number of nodal spheres; all the original $N$ spheres were retained,
+number of nodal spheres; all the original N spheres were retained,
so they exhaust the endpoint spheres.
-Define $p_X:Xto S$ as follows. On the regular region
-$X^circ=X∖cup K_i$, the $F$-curves form (by
+Define p_X:Xto S as follows. On the regular region
+X^circ=X∖cup K_i, the F-curves form (by
Cref{lem:pencil-normal}) a foliation with local transverse disks.
Each leaf is an embedded torus, on which the orientation is chosen by
-the taming form $B$. On each model disk a holomorphic nodal
-projection $DeltatoDelta$ gives the base coordinate. The disk
+the taming form B. On each model disk a holomorphic nodal
+projection DeltatoDelta gives the base coordinate. The disk
charts have smooth transition maps: at a smooth point of the fiber,
both local projections are smooth submersions with the same fibers,
so a local section expresses one base coordinate smoothly in terms of
-the other. This makes $S$ a smooth surface without boundary and $p_X$
-a smooth proper surjection. Compactness and connectedness of $X$ give
-compactness and connectedness of $S$. The orientation of $S$ is
+the other. This makes S a smooth surface without boundary and p_X
+a smooth proper surjection. Compactness and connectedness of X give
+compactness and connectedness of S. The orientation of S is
determined by the complex orientation of the fibers and the normal
plane; the model gives the same orientation on its regular part.
The singular fibers are precisely the ordinary holomorphic nodes of
-the models (locally $s=z_1z_2$ after complex coordinate changes); the
+the models (locally s=z_1z_2 after complex coordinate changes); the
pair vanishes on every fiber by construction, and the prescribed
model identities hold on the unchanged smaller neighborhoods.
@@ -2463,22 +2463,22 @@ K"ahler model whose forms satisfy the pointwise equal-square
identities that the auxiliary volume equation will need, and whose
semi-flat approximation is exponentially close on fixed annuli. The
model is the Gibbons--Hawking / Ooguri--Vafa multi-instanton family: a
-collapsing $A_1$-type ALE space over a disk, with exactly one singular
+collapsing A_1-type ALE space over a disk, with exactly one singular
fiber.
## The Gibbons--Hawking ansatz
-Fix a positive real parameter $rho>0$ and a positive smooth function
-$V_rho$ on a ball $B_R(0)subsetR^3$ with a pole of order one at the
+Fix a positive real parameter rho>0 and a positive smooth function
+V_rho on a ball B_R(0)subsetR^3 with a pole of order one at the
origin. The Gibbons--Hawking construction [cite]
produces a positive definite hyperk"ahler metric on the total space of
-a circle bundle over $B_R(0)∖{0}$, and extends smoothly over
+a circle bundle over B_R(0)∖{0}, and extends smoothly over
the origin after adding a point. Concretely, choose a connection
-one-form $theta$ with
+one-form theta with
ddtheta=-ast_{R^3}dd V_rho,
-and set, for cyclic $(i,j,k)$ with $(x_1,x_2,x_3)=X$ and $t$ the
+and set, for cyclic (i,j,k) with (x_1,x_2,x_3)=X and t the
circle coordinate,
@@ -2487,118 +2487,117 @@ circle coordinate,
The signs are chosen from
-$dd(dx_iwedgetheta)=-∂_iV_rho,dx_1wedge dx_2wedge dx_3$
-and $dd(V_rho,dx_jwedge dx_k)=∂_iV_rho,dx_1wedge dx_2wedge dx_3$.
+dd(dx_iwedgetheta)=-∂_iV_rho,dx_1wedge dx_2wedge dx_3
+and dd(V_rho,dx_jwedge dx_k)=∂_iV_rho,dx_1wedge dx_2wedge dx_3.
Thus all three forms are closed. In the coframe
-$V_rho^{1/2}dx_i,V_rho^{-1/2}theta$ they form the standard
+V_rho^{1/2}dx_i,V_rho^{-1/2}theta they form the standard
orthogonal equal-square self-dual frame, with the induced
four-dimensional orientation.
To see the forms are parallel, write the connection on this frame in
-terms of three real one-forms $a_i$. After a consistent choice of
+terms of three real one-forms a_i. After a consistent choice of
frame signs, the equations for closedness are
[
- I_2a_3-I_3a_2=0,qquad
- I_3a_1-I_1a_3=0,qquad
+ I_2a_3-I_3a_2=0,
+ I_3a_1-I_1a_3=0,
I_1a_2-I_2a_1=0,
]
-with $I_ialpha=ast(alphawedgegamma_i)$ and signs chosen so that
-$I_1I_2=I_3$. The first two equations give $a_3=-I_1a_2$ and
-$a_1=-I_2a_3=-I_3a_2$; the third then gives $2I_1a_2=0$, so all
-$a_i$ vanish. The frame is parallel. In particular
+with I_ialpha=ast(alphawedgegamma_i) and signs chosen so that
+I_1I_2=I_3. The first two equations give a_3=-I_1a_2 and
+a_1=-I_2a_3=-I_3a_2; the third then gives 2I_1a_2=0, so all
+a_i vanish. The frame is parallel. In particular
Ω_rho=gamma_1+igamma_2=dswedge(theta-iV_rho,dd t),
which is decomposable with positive product with its conjugate. Its
-kernel defines the complex structure for which $gamma_3$ is
+kernel defines the complex structure for which gamma_3 is
K"ahler. This kernel is involutive: closedness and Cartan's formula
make the kernel closed under bracket.
-The metric $g_rho$ is therefore hyperk"ahler with hyperk"ahler
-triple $(gamma_1,gamma_2,gamma_3)$. The specific choice
-$V_rho(s,t)=V_0(s)+1/(4pi|X|)$ with $V_0$ a small positive constant
-gives the charge-one $A_1$ manifold over the quotient
-$B_R(0)/circle$, and the completion near $X=0$ is the
+The metric g_rho is therefore hyperk"ahler with hyperk"ahler
+triple (gamma_1,gamma_2,gamma_3). The specific choice
+V_rho(s,t)=V_0(s)+1/(4pi|X|) with V_0 a small positive constant
+gives the charge-one A_1 manifold over the quotient
+B_R(0)/circle, and the completion near X=0 is the
Gibbons--Hawking / Ooguri--Vafa multi-instanton [cite].
-We use its explicit form on the punctured disk $Delta_R∖{0}$,
-where it is a smooth elliptic fibration over $Delta_R∖{0}$
-with torus fibers of area $rho$ in the $dd t$ direction.
+We use its explicit form on the punctured disk Delta_R∖{0},
+where it is a smooth elliptic fibration over Delta_R∖{0}
+with torus fibers of area rho in the dd t direction.
-The complex surface $Y$ obtained by adding the point at $X=0$ is
+The complex surface Y obtained by adding the point at X=0 is
smooth because the Hopf chart of the charge-one instanton completes
-$W_m,dmathbf X^2+W_m^bfX^2+W_m^{-1}theta_m^2$ to a flat smooth ball. In that
-chart $mathbf X$bfX$ and $r$ are smooth quadratic expressions, and
-$rtheta_m$ is smooth. The coefficient $1/(4pi)$, with connection
+W_m,dmathbf X^2+W_m^bfX^2+W_m^{-1}theta_m^2 to a flat smooth ball. In that
+chart mathbf XbfX and r are smooth quadratic expressions, and
+rtheta_m is smooth. The coefficient 1/(4pi), with connection
period one, is the smooth Hopf normalization. For instance, near the
pole,
- W^{-1}=(4pi r)/(1+4pi rH_rho),qquad
+ W^{-1}=(4pi r)/(1+4pi rH_rho),
frac{W^{-1}-W_m^{-1}}{r^2}
=-((4pi)^2H_rho)/(1+4pi rH_rho),
-so $g_rho/rho=W,dmathbf X^2+W^bfX^2+W^{-1}(theta_m+beta)^2$ is smooth
+so g_rho/rho=W,dmathbf X^2+W^bfX^2+W^{-1}(theta_m+beta)^2 is smooth
at the added point; the coefficient of the apparently singular
squared-connection difference is smooth by the second identity.
Subtracting the flat Hopf triple in
-$gamma_i/rho=dX_iwedge(theta_m+beta)+W,dX_jwedge dX_k$ shows
+gamma_i/rho=dX_iwedge(theta_m+beta)+W,dX_jwedge dX_k shows
that the forms also extend smoothly. At the added point their values
are the flat triple, so they remain nondegenerate. We obtain a smooth
-hyperk"ahler surface $Y_rho$, and the holomorphic function $s$
+hyperk"ahler surface Y_rho, and the holomorphic function s
extends over the added point.
-subsection{The curvature form $D_eps^{loc}$}
+subsection{The curvature form D_eps^{loc}}
-Fix $hinC∖{0}$, put $Σ=h,dd swedgedd z$, and fix
-$0<r<R$. Write $s=x_1+ix_2$ and $z=a+tau(s)b$ modulo the period
-lattice. For every sufficiently small $ε>0$, the form
-$D_eps^{loc}=|h|,gamma_3$ on $Y_rho$ with $rho=eps/|h|$
-has a metric $g_eps=|h|,g_rho$, and the four identities
+Fix hinC∖{0}, put Σ=h,dd swedgedd z, and fix
+0<r<R. Write s=x_1+ix_2 and z=a+tau(s)b modulo the period
+lattice. For every sufficiently small ε>0, the form
+D_eps^{loc}=|h|,gamma_3 on Y_rho with rho=eps/|h|
+has a metric g_eps=|h|,g_rho, and the four identities
D_eps^{loc}wedgeReΣ
- =D_eps^{loc}wedgeImΣ=0,qquad
+ =D_eps^{loc}wedgeImΣ=0,
(D_eps^{loc})^2=(ReΣ)^2,
int_{p^{-1}(s)}D_eps^{loc}=εquad(sne0),
hold exactly. The three forms are parallel, mutually orthogonal and
-of equal square, and $vol_{g_eps}=(ReΣ)^2/2$.
-The scalar $rho=eps/|h|$ is the fiber area of $g_eps$ in the
+of equal square, and vol_{g_eps}=(ReΣ)^2/2.
+The scalar rho=eps/|h| is the fiber area of g_eps in the
direction of the original torus fiber, viewed in the standard
Gibbons--Hawking normalization; the identities are invariant under the
-scaling $|h|$ that we are using.
+scaling |h| that we are using.
## Annular comparison
-Fix $hinC∖{0}$ and $0<r<R$. For every sufficiently small
-$ε>0$ there is a K"ahler form $D_eps^{loc}$ on the
-preimage of $Delta_R$ with metric $g_eps$ satisfying
+Fix hinC∖{0} and 0<r<R. For every sufficiently small
+ε>0 there is a K"ahler form D_eps^{loc} on the
+preimage of Delta_R with metric g_eps satisfying
(eq:nodal-exact-identities). Moreover:
[label=(roman*)]
-item On each fixed closed annulus
-$K={r_-le|s|le r_+}subsetDelta_R∖{0}$, the form
+ On each fixed closed annulus
+K={r_-le|s|le r_+}subsetDelta_R∖{0}, the form
D_eps^{sf}
=ε,dd awedgedd b+
(|h|^2)/(eps)tau_2(s),dd x_1wedgedd x_2,
- qquad s=x_1+ix_2,
+ s=x_1+ix_2,
-is well-defined and closed. For every integer $mge0$, in fixed smooth
+is well-defined and closed. For every integer mge0, in fixed smooth
charts on this annular bundle,
|D_eps^{loc}-D_eps^{sf}|_{C^m}
- le C_me^{-c_m/eps},qquad C_m,c_m>0.
-
-item These forms have the same real cohomology class on
-$p^{-1}(K)$. Their periods are $eps$ on a fiber and zero on the torus
+ le C_me^{-c_m/eps}, C_m,c_m>0.
+ These forms have the same real cohomology class on
+p^{-1}(K). Their periods are eps on a fiber and zero on the torus
swept by the primitive connection cycle.
-item The following polynomial estimates hold on bounded branches:
-$g_eps/eps$ has polynomial bounds in $(1+|logeps|)$ and $eps^{-1}$
+ The following polynomial estimates hold on bounded branches:
+g_eps/eps has polynomial bounds in (1+|logeps|) and eps^{-1}
for its coefficient matrix, its inverse, and covariant derivatives
through any prescribed finite order, with constants depending only on
the fixed data.
-item The identifications of the regular elliptic families
+ The identifications of the regular elliptic families
(structure sheaf, relative holomorphic differential, Abel coordinate,
periods) extend to holomorphic symplectic isomorphisms of the completed
families, and the fixed complex surface is intrinsic to the model.
@@ -2607,38 +2606,38 @@ families, and the fixed complex surface is intrinsic to the model.
Proof.
Parts (i) and (ii) are the semi-flat comparison. On
-a fixed annulus ${r_-le|s|le r_+}$ away from the node, the
-hyperk"ahler metric $g_eps$ collapses to a flat metric on the torus
+a fixed annulus {r_-le|s|le r_+} away from the node, the
+hyperk"ahler metric g_eps collapses to a flat metric on the torus
fibration whose base is a fixed elliptic curve bundle. The
-semi-flat form $D_eps^{sf}$ is the leading term:
-$ε,dd awedgedd b$ gives the correct fiber integral, and
-the second term $(|h|^2)/(eps)tau_2(s)dd x_1wedgedd x_2$
+semi-flat form D_eps^{sf} is the leading term:
+ε,dd awedgedd b gives the correct fiber integral, and
+the second term (|h|^2)/(eps)tau_2(s)dd x_1wedgedd x_2
expresses the horizontal curvature. The difference
-$D_eps^{loc}-D_eps^{sf}$ is supported on the
+D_eps^{loc}-D_eps^{sf} is supported on the
regions where the target metric's curvature is non-negligible,
-concentrated near $r=0$; application of the standard Gibbons--Hawking
-instantonic decay (exponentially small in the parameter $sqrtrho$,
-and $rho=eps/|h|to0$) gives
+concentrated near r=0; application of the standard Gibbons--Hawking
+instantonic decay (exponentially small in the parameter sqrtrho,
+and rho=eps/|h|to0) gives
(eq:nodal-exponential-estimate). Since the semiflat form is
closed and has the same periods, part (ii) follows.
Part (iii) follows from the explicit formula
(eq:nodal-gh-triple) together with the Hopf completion
(eq:nodal-pole-inverse), and the polynomial boundedness of the
-Gibbons--Hawking potential $V_rho$ and its derivatives in the
-parameter $rho$.
+Gibbons--Hawking potential V_rho and its derivatives in the
+parameter rho.
-For (iv), the elliptic fibration $p:Y_rhotoDelta_R$
-carries a relative holomorphic differential $dd z$, and the period
-$tau_rho(s)=int V_rho,dd t$ on each regular fiber. The
-imaginary part $tau_2(s)=Imtau_rho(s)$ is a
+For (iv), the elliptic fibration p:Y_rhotoDelta_R
+carries a relative holomorphic differential dd z, and the period
+tau_rho(s)=int V_rho,dd t on each regular fiber. The
+imaginary part tau_2(s)=Imtau_rho(s) is a
holomorphic function on each branch of the regular base; adjusting
-the connection by a real multiple of $dd t/rho$ makes
-$tau_rho=tau$ equal to the model $z$-period. The resulting
+the connection by a real multiple of dd t/rho makes
+tau_rho=tau equal to the model z-period. The resulting
complex structure is then identified with a fixed holomorphic family
-over $Delta_R$, and the completions at $s=0$ agree with the added
+over Delta_R, and the completions at s=0 agree with the added
nodal fiber. Indeed, the completed family is proper and flat, its
-fiberwise structure sheaf has $h^0=3$ and $h^1=0$ for the nodal
+fiberwise structure sheaf has h^0=3 and h^1=0 for the nodal
fiber, and Grauert's base-change theorem
cite[Section~7, Satz~5]{Grauert1960} and its correction
[cite] then identifies it with a fixed family of plane
@@ -2648,11 +2647,11 @@ model.
The proof of part (iv) deserves a short additional comment
because it uses Grauert's theorem in a way that is easy to lose track
-of. Let $L=mathcal L_calL_rho$ be the relative polarization on
-$p:Y_rhotoDelta_R$. The nodal fiber $N=p^{-1}(0)$ is the image of
-a nodal plane cubic in $mathbb P^2$.bbP^2$. Its normalization
-$nu:mathbb P^1bbP^1to N$ is a rational curve, and the pullback
-$nu^*L$ is $mathcal O_calO_{mathbb P^1}bbP^1(3)$ up to a scalar. The gluing
+of. Let L=mathcal L_calL_rho be the relative polarization on
+p:Y_rhotoDelta_R. The nodal fiber N=p^{-1}(0) is the image of
+a nodal plane cubic in mathbb P^2.bbP^2. Its normalization
+nu:mathbb P^1bbP^1to N is a rational curve, and the pullback
+nu^*L is mathcal O_calO_{mathbb P^1}bbP^1(3) up to a scalar. The gluing
condition gives the exact sequence
[
0→ L→
@@ -2660,38 +2659,38 @@ condition gives the exact sequence
xrightarrow{ev_0-lambdaev_∞}
C_{node}→0,
]
-with $lambda$ the appropriate modular parameter of the nodal cubic.
+with lambda the appropriate modular parameter of the nodal cubic.
The map on global sections is surjective, and the normalization is
-finite, so the cohomology of $nu_*mathcal O_calO_{mathbb P^1}bbP^1(3)$ is
-that of $mathcal O_calO_{mathbb P^1}bbP^1(3)$. In particular $h^0(N,L)=3$ and
-$h^1(N,L)=0$.
-
-Grauert's base-change theorem now applies: $p$ is proper and
-surjective, the disk $Delta_R$ is reduced, the sheaf $mathcal L_calL_rho$
-is flat, and the fiber dimensions $h^0$ and $h^1$ are constant. So
-$E_rho=(p_rho)_*mathcal L_calL_rho$ is a holomorphic rank-three bundle,
-and its fiber at $s$ is $H^0(p^{-1}(s),L_s)$. Evaluation gives a
-relative closed embedding in $mathbb P(E_bbP(E_rho^*)$. At the nodal
+finite, so the cohomology of nu_*mathcal O_calO_{mathbb P^1}bbP^1(3) is
+that of mathcal O_calO_{mathbb P^1}bbP^1(3). In particular h^0(N,L)=3 and
+h^1(N,L)=0.
+
+Grauert's base-change theorem now applies: p is proper and
+surjective, the disk Delta_R is reduced, the sheaf mathcal L_calL_rho
+is flat, and the fiber dimensions h^0 and h^1 are constant. So
+E_rho=(p_rho)_*mathcal L_calL_rho is a holomorphic rank-three bundle,
+and its fiber at s is H^0(p^{-1}(s),L_s). Evaluation gives a
+relative closed embedding in mathbb P(E_bbP(E_rho^*). At the nodal
point, the Zariski tangent space of the fiber is the total tangent
space of the smooth surface; tangent separation there makes the whole
map an immersion of the surface, and properness gives a closed
embedding. This determines the completed family up to biholomorphism. The
-argument is uniform in $eps$ and applies verbatim at each nodal disk.
+argument is uniform in eps and applies verbatim at each nodal disk.
# Preparing the regular fibers
-The pencil of Cref{thm:pencil} gives an elliptic fibration $p:Xto S$
-with $F$-fibers. Before we can build the auxiliary form $D$, we need
-to take the pair $(A,C)$ to a special form on the regular region:
+The pencil of Cref{thm:pencil} gives an elliptic fibration p:Xto S
+with F-fibers. Before we can build the auxiliary form D, we need
+to take the pair (A,C) to a special form on the regular region:
invariant under the torus action that translations along the fibers
-induce, with a closed two-form $Psi$ orthogonal to them both and of
+induce, with a closed two-form Psi orthogonal to them both and of
unit integral on each fiber. On the nodal neighborhoods we preserve
the local holomorphic model of Cref{thm:nodal-model}.
-Write $S^circ=S∖Delta$ and $X^circ=p^{-1}(S^circ)$, where
-$Deltasubset S$ is the finite set of critical values. The class $F$
-and its $q_0=[B]F>0$ are fixed.
+Write S^circ=S∖Delta and X^circ=p^{-1}(S^circ), where
+Deltasubset S is the finite set of critical values. The class F
+and its q_0=[B]F>0 are fixed.
## Averaging along the torus action
@@ -2699,42 +2698,42 @@ The regular torus fibration is a torsor: there is no global
origin, but the natural translations act transitively on each fiber.
[Torus action]
-If $alpha_sin T_s^*S^circ$, the equation
-$iota_{Y_alpha}A_0=-p^*alpha$ defines a unique vertical vector
-field along $p^{-1}(s)$, because the fibers are Lagrangian for $A_0$.
-Locally, take $alpha=dd f$ on the base; the fields
-$Y_{dd f}$ are Hamiltonian, their brackets vanish, and their complete
+If alpha_sin T_s^*S^circ, the equation
+iota_{Y_alpha}A_0=-p^*alpha defines a unique vertical vector
+field along p^{-1}(s), because the fibers are Lagrangian for A_0.
+Locally, take alpha=dd f on the base; the fields
+Y_{dd f} are Hamiltonian, their brackets vanish, and their complete
flows on each compact fiber give a transitive
-$T_s^*S^circ$-action with a rank-two stabilizer lattice
-$mathcal L_s$.calL_s$. Locally the lattices vary smoothly, and the two
+T_s^*S^circ-action with a rank-two stabilizer lattice
+mathcal L_s.calL_s. Locally the lattices vary smoothly, and the two
periods continue smoothly.
Proof.
-The fibers are Lagrangian for $A_0$, so $A_0|_F$ has rank zero on
-$TF$ and its image under the contraction with vertical vectors is an
-isomorphism onto $T^*_sS^circ$. A local smooth section $f$ of the
-base gives a global vertical vector field $Y_f$; the identity
- $dd(iota_{Y_f}A_0)=-p^*dd f$ closes with
-$p^*dd f$ a base form, so $Y_f$ is Hamiltonian for
-$-p^*f$. Locally, two such fields commute because their Hamiltonians
+The fibers are Lagrangian for A_0, so A_0|_F has rank zero on
+TF and its image under the contraction with vertical vectors is an
+isomorphism onto T^*_sS^circ. A local smooth section f of the
+base gives a global vertical vector field Y_f; the identity
+ dd(iota_{Y_f}A_0)=-p^*dd f closes with
+p^*dd f a base form, so Y_f is Hamiltonian for
+-p^*f. Locally, two such fields commute because their Hamiltonians
are pulled back from the base and their fields are vertical. Complete
flows therefore give a local group action.
-To identify the lattice, use a local section $smapstosigma(s)$ of
-$p$; each element of the stabilizer satisfies
-$int_{F_s}alphainmathcal L_s$calL_s$ for a small piecewise-smooth disk
+To identify the lattice, use a local section smapstosigma(s) of
+p; each element of the stabilizer satisfies
+int_{F_s}alphainmathcal L_scalL_s for a small piecewise-smooth disk
whose boundary is the corresponding translation. The two periods
continue smoothly by the implicit function theorem. Equivalently,
-a local lattice basis ${alpha_1,alpha_2}$ of closed one-forms is a
+a local lattice basis {alpha_1,alpha_2} of closed one-forms is a
smooth frame, and the corresponding flows give the two
structure-preserving translations.
[Invariant smooth pairs]
-The averaging map $Av$ defined by the torus action
+The averaging map Av defined by the torus action
preserves smoothness, exactness, the cohomology class, and the class
-in the image of $C^∞(X,Λ^2)to H^2(X;R)$. If a form
+in the image of C^∞(X,Λ^2)to H^2(X;R). If a form
vanishes on the fibers, so does its average. The average of any
closed form is closed. The average of any smooth family of forms is
smooth in the parameter.
@@ -2743,12 +2742,12 @@ smooth in the parameter.
Proof.
Average over the compact torus factor. Smoothness, base-locality and
closedness are immediate. Exactness is preserved because the average
-of $ddbeta$ equals $dd$ of the average of $beta$. Vanishing on
+of ddbeta equals dd of the average of beta. Vanishing on
the fibers is preserved because fiberwise the tangent space is spanned
-by the vector fields $Y_alpha$; the average of a form that vanishes
+by the vector fields Y_alpha; the average of a form that vanishes
on these stays zero. Classes are preserved because the average of a
closed form is cohomologous to it via the standard homotopy
-$H(t,theta)$ between the identity and the average, which is closed
+H(t,theta) between the identity and the average, which is closed
when the input is closed.
@@ -2757,249 +2756,249 @@ when the input is closed.
The main statement of this section is the following.
-Let $(A_0,C_0,B_0)$ and $pcolon Xto S$ be the positive triple and
+Let (A_0,C_0,B_0) and pcolon Xto S be the positive triple and
pencil of Cref{thm:pencil}. Retain the normalized original periods
-and the class $F$ of Cref{cor:curve-signs}. On disjoint disks $U_j$
+and the class F of Cref{cor:curve-signs}. On disjoint disks U_j
about the critical values, use the model of Cref{thm:nodal-model}
with
- Σ_j=h_j,dd swedgedd z,quad
- A_0=ReΣ_j,quad
+ Σ_j=h_j,dd swedgedd z,
+ A_0=ReΣ_j,
C_0=x_jReΣ_j+y_jImΣ_j,
- quad y_jne0,
+ y_jne0,
-where $h_jne0$, $x_j$, $y_j$ are constants and
-$z=a+tau(s)b$ modulo the elliptic period lattice.
+where h_jne0, x_j, y_j are constants and
+z=a+tau(s)b modulo the elliptic period lattice.
-There are smaller disks $U'_jSubset U_j$ and a smooth path of
-positive closed triples $(A_t,C_t,B_t)$, $0le tle1$, starting at
-$(A_0,C_0,B_0)$, with the following properties.
+There are smaller disks U'_jSubset U_j and a smooth path of
+positive closed triples (A_t,C_t,B_t), 0le tle1, starting at
+(A_0,C_0,B_0), with the following properties.
[label=(roman*)]
-item Each cohomology class is constant. The restrictions of $A_t,C_t$
-to the fibers vanish, and the pair equals $(A_0,C_0)$ on $p^{-1}(U'_j)$
-for every $t$.
-item The endpoint pair $A=A_1$, $C=C_1$ is invariant under the torus
-group bundle acting on $X^circ$ by translations.
-item There is a smooth closed invariant two-form $Psi$ on $X^circ$
+ Each cohomology class is constant. The restrictions of A_t,C_t
+to the fibers vanish, and the pair equals (A_0,C_0) on p^{-1}(U'_j)
+for every t.
+ The endpoint pair A=A_1, C=C_1 is invariant under the torus
+group bundle acting on X^circ by translations.
+ There is a smooth closed invariant two-form Psi on X^circ
with
- int_{p^{-1}(s)}Psi=1,qquad
+ int_{p^{-1}(s)}Psi=1,
Psiwedge A=Psiwedge C=0quadon X^circ.
-On each punctured $p^{-1}(U'_j∖{s_j})$ it equals
-$dd awedgedd b$, so $Psi^2=0$ there.
+On each punctured p^{-1}(U'_j∖{s_j}) it equals
+dd awedgedd b, so Psi^2=0 there.
Proof.
-noindentStep 1: averaging the pair. Let $A^a=Av(A_0)$
-and $C^a=Av(C_0)$ be the averages along the torus action
+noindentStep 1: averaging the pair. Let A^a=Av(A_0)
+and C^a=Av(C_0) be the averages along the torus action
(Lemma~lem:regular-averaging). These are closed and cohomologous
-to $A_0,C_0$; by linearity of the wedge matrix, $(A^a,C^a)$ is a
+to A_0,C_0; by linearity of the wedge matrix, (A^a,C^a) is a
definite pair; the restrictions to the fibers vanish because the
input pair does.
-Near the nodal model of Cref{thm:nodal-model}, $A_0$ and $C_0$ are
+Near the nodal model of Cref{thm:nodal-model}, A_0 and C_0 are
already invariant under the natural translations of the model family,
-so on $p^{-1}(U'_j)$ we have $A^a=A_0$ and $C^a=C_0$.
+so on p^{-1}(U'_j) we have A^a=A_0 and C^a=C_0.
noindentStep 2: the positive cone along the interpolation.
-We must check that the straight-line interpolation from $(A_0,C_0)$ to
-$(A^a,C^a)$ stays positive. For a fixed $s$, consider the family
+We must check that the straight-line interpolation from (A_0,C_0) to
+(A^a,C^a) stays positive. For a fixed s, consider the family
[
- A_lambda=A_0+lambda(A^a-A_0),qquad
- C_lambda=C_0+lambda(C^a-C_0),qquad 0lelambdale1.
+ A_lambda=A_0+lambda(A^a-A_0),
+ C_lambda=C_0+lambda(C^a-C_0), 0lelambdale1.
]
-At a point of $X^circ$ with $p(x)=s$, the fibers are Lagrangian for
-$A_0$ and for $A^a$ (the latter by the torus invariance); the
-anti-invariant forms for the underlying complex structure $J$ are
+At a point of X^circ with p(x)=s, the fibers are Lagrangian for
+A_0 and for A^a (the latter by the torus invariance); the
+anti-invariant forms for the underlying complex structure J are
spanned by the horizontal piece.
-By Cref{lem:regular-mixed-cone} below, the set of forms $C$ that are
-definite-pair with $A^a$ and vanish on the fibers is a convex cone.
+By Cref{lem:regular-mixed-cone} below, the set of forms C that are
+definite-pair with A^a and vanish on the fibers is a convex cone.
Adding a form pulled back from the base contributes a semidefinite
quantity on the fiber, which is absorbed by the positive definite
contribution on the horizontal plane. This makes the straight-line
-interpolation $(A_lambda,C_lambda)$ a positive pair for every
-$lambda$ on a fixed compact set. Compactness, combined with the
+interpolation (A_lambda,C_lambda) a positive pair for every
+lambda on a fixed compact set. Compactness, combined with the
invariance of the model disks and the equality of the model
semi-flat data on them, extends this to a uniform open cover.
Since the disk models do not change and are already invariant, and
-the rest of $X$ is covered by finitely many local bundle charts
+the rest of X is covered by finitely many local bundle charts
within which the interpolation is a finite-dimensional convex family,
the interpolation is positive everywhere. Combined with the standard
smoothing of tamers in the third slot (Cref{lem:smooth-tamers}), we
obtain a smooth path of positive closed triples.
noindentStep 3: the unit fiber area form. The class
-$[B_0]/q_0$ has integral one on every regular fiber $F$, where
-$q_0=[B_0]F>0$. Define
+[B_0]/q_0 has integral one on every regular fiber F, where
+q_0=[B_0]F>0. Define
[
Psi^{(0)}=Av(B_0/q_0)quadon X^circ.
]
This is closed and invariant, and restricts to the invariant two-form
of unit integral on every fiber.
-We next adjust it to equal $dd awedgedd b$ near the punctures.
+We next adjust it to equal dd awedgedd b near the punctures.
On a punctured model disk, the two real two-homology generators are
-the fiber and the $a$-sweep at $b=0$. The smooth closed form
-$B_0/q_0$ has periods $1$ and $0$ on these generators. The same holds
-for $dd awedgedd b$. Averaging does not change the periods by
+the fiber and the a-sweep at b=0. The smooth closed form
+B_0/q_0 has periods 1 and 0 on these generators. The same holds
+for dd awedgedd b. Averaging does not change the periods by
Cref{lem:regular-averaging}. Both closed forms with the same periods
-on the mapping torus differ by an exact two-form. Since $H^2$ of the
+on the mapping torus differ by an exact two-form. Since H^2 of the
punctured mapping torus is generated by the fiber and the sweep, the
-same holds after subtracting a constant multiple of $dd awedgedd b$
-if necessary. Replacing $Psi^{(0)}$ by $Psi^{(0)}$ minus a small
-exact two-form $ddbeta$ with $beta$ a one-form supported in
-$p^{-1}(U_j∖{s_j})$, we can make
-$Psi^{(0)}=dd awedgedd b$ on $p^{-1}(U'_j∖{s_j})$
+same holds after subtracting a constant multiple of dd awedgedd b
+if necessary. Replacing Psi^{(0)} by Psi^{(0)} minus a small
+exact two-form ddbeta with beta a one-form supported in
+p^{-1}(U_j∖{s_j}), we can make
+Psi^{(0)}=dd awedgedd b on p^{-1}(U'_j∖{s_j})
without changing closedness or the cohomology class, and preserving
the invariance of the average.
-noindentStep 4: orthogonalizing against $A$ and $C$. We have
-$Psi^{(0)}$ closed, invariant, of unit fiber integral, equal to
-$dd awedgedd b$ near the punctures. The integrals
+noindentStep 4: orthogonalizing against A and C. We have
+Psi^{(0)} closed, invariant, of unit fiber integral, equal to
+dd awedgedd b near the punctures. The integrals
[
- I_A=int_{X^circ}Psi^{(0)}wedge A,quad
+ I_A=int_{X^circ}Psi^{(0)}wedge A,
I_C=int_{X^circ}Psi^{(0)}wedge C
]
are well-defined: their integrands vanish on the smaller punctured
disks. Choose closed smooth invariant forms
-$widetilde A,widetilde C$ on $X$, representing $[A]$ and $[C]$,
-supported over a compact subset of $S^circ$ and vanishing on the
+widetilde A,widetilde C on X, representing [A] and [C],
+supported over a compact subset of S^circ and vanishing on the
vertical planes. Such forms exist because on a full one-node
neighborhood the second real homology is generated by the fiber, so
-$A$ and $C$ are exact there (their fiber integrals vanish). Subtracting
-$dd$ of a local primitive times a basic cutoff gives cohomologous
+A and C are exact there (their fiber integrals vanish). Subtracting
+dd of a local primitive times a basic cutoff gives cohomologous
forms vanishing on smaller model neighborhoods. Averaging and
-extending by zero near the nodes gives $widetilde A,widetilde C$
+extending by zero near the nodes gives widetilde A,widetilde C
with the same properties.
-The residual pairing matrix between the triple $(A,C)$ and the forms
-$widetilde A,widetilde C$ is the cohomology Gram matrix, i.e. the
+The residual pairing matrix between the triple (A,C) and the forms
+widetilde A,widetilde C is the cohomology Gram matrix, i.e. the
identity under our normalization. Set
Psi=Psi^{(0)}-I_Awidetilde A-I_Cwidetilde C.
-Then $Psi$ is closed, invariant, of unit fiber area, and equal to
-$dd awedgedd b$ near the punctures, with total pairings
+Then Psi is closed, invariant, of unit fiber area, and equal to
+dd awedgedd b near the punctures, with total pairings
int_{X^circ}Psiwedge A=int_{X^circ}Psiwedge C=0.
noindentStep 5: assembling the path. Concatenate the
-interpolation from $(A_0,C_0)$ to $(A^a,C^a)$ with the constant
-family $(A^a,C^a,B_t)$ for a smooth choice of third form
-$B_tin[B_0]$, using Cref{lem:smooth-tamers} with $B_0$ at the
+interpolation from (A_0,C_0) to (A^a,C^a) with the constant
+family (A^a,C^a,B_t) for a smooth choice of third form
+B_tin[B_0], using Cref{lem:smooth-tamers} with B_0 at the
initial endpoint and any prescribed endpoint. Each step preserves the
class, is smooth in the parameter, and keeps the pair positive.
Reparametrizing to be constant at the endpoints, and concatenating
all such paths using the same construction in reverse for the
round-trip, gives a single smooth path of positive closed triples.
-The endpoint $(A,C)=(A^a,C^a)$ is invariant, and the form $Psi$
+The endpoint (A,C)=(A^a,C^a) is invariant, and the form Psi
satisfies (eq:regular-prepared-data). This completes the proof.
[Positivity in the mixed cone]
-Fix a symplectic two-form $A$ with $A^2>0$ and a Lagrangian two-plane
-$L$ in an oriented four-dimensional vector space. Among two-forms $C$
-satisfying $C|_L=0$, the condition that $(A,C)$ be a definite pair has
+Fix a symplectic two-form A with A^2>0 and a Lagrangian two-plane
+L in an oriented four-dimensional vector space. Among two-forms C
+satisfying C|_L=0, the condition that (A,C) be a definite pair has
two open convex components. Adding a two-form pulled back from the
-quotient by $L$ to either member does not change the wedge matrix.
+quotient by L to either member does not change the wedge matrix.
Proof.
-Choose a linear basis of $V$ with first two vectors spanning $L$ and
-last two spanning $L^perp$ (non-degenerate because $L$ is Lagrangian).
-On $L$, $A$ vanishes, so the wedge-product matrix depends only on the
-$L^perp$ piece. Write $A|_{L^perp}$ via its two-by-two matrix, which
+Choose a linear basis of V with first two vectors spanning L and
+last two spanning L^perp (non-degenerate because L is Lagrangian).
+On L, A vanishes, so the wedge-product matrix depends only on the
+L^perp piece. Write A|_{L^perp} via its two-by-two matrix, which
is nondegenerate. Reduce to a canonical symplectic basis of a
-two-dimensional complex vector space. In that basis, $(A,C)$ is a
-definite pair iff the $L^perp$-components of $A$ and $C$ are linearly
+two-dimensional complex vector space. In that basis, (A,C) is a
+definite pair iff the L^perp-components of A and C are linearly
independent and the wedge-sign satisfies a fixed choice. This is the
-union of two open convex cones in $C$. Adding a form that vanishes on
-$L^perp$ (i.e., is pulled back from $L^perp$) does not change the
-$L^perp$ component.
+union of two open convex cones in C. Adding a form that vanishes on
+L^perp (i.e., is pulled back from L^perp) does not change the
+L^perp component.
[Smooth tamers]
-Let $J_t$, $0le tle1$, be a smooth family of almost-complex
-structures on a closed four-manifold $X$, $Hin H^2(X;R)$ a class
+Let J_t, 0le tle1, be a smooth family of almost-complex
+structures on a closed four-manifold X, Hin H^2(X;R) a class
whose restriction to some fiberwise positive cone is nonempty, and
-$B_0$, $B_1in H$ prescribed tamer representatives at the endpoints.
-Then there is a smooth family $B_tin H$ with $B_t$ taming $J_t$ for
-every $t$ and prescribed $B_0,B_1$ at $t=0,1$.
+B_0, B_1in H prescribed tamer representatives at the endpoints.
+Then there is a smooth family B_tin H with B_t taming J_t for
+every t and prescribed B_0,B_1 at t=0,1.
Proof.
-Compatible and taming classes are open and convex over each fixed $J$.
-The standard finite-dimensional smoothing of a $C^0$ family of taming
+Compatible and taming classes are open and convex over each fixed J.
+The standard finite-dimensional smoothing of a C^0 family of taming
forms by averaging under a smooth bump on the parameter interval,
followed by convolving with a small mollifier near each endpoint,
gives the requisite smooth family with matching endpoints. Since the
-class $H$ has a tamer at each parameter (the taming cone over
-$H$ is open and nonempty on both endpoints, and the tamer along
-$J_t$ interpolates), the interpolation is nowhere zero.
+class H has a tamer at each parameter (the taming cone over
+H is open and nonempty on both endpoints, and the tamer along
+J_t interpolates), the interpolation is nowhere zero.
# The auxiliary volume equation
The prepared pair of Cref{prop:regular-preparation} satisfies
-$A^2=C^2=1$, $Awedge C=0$ on $X$, and the restrictions of $A,C$ to
-the fibers vanish, with unit fiber area form $Psi$ orthogonal to both
-and equal to $dd awedgedd b$ near the nodes. We now construct a
-closed two-form $D_eps$ with $Awedge D_eps=Cwedge D_eps=0$ and
-$D_eps^2=A^2$ pointwise. The correction to a semi-flat approximate
+A^2=C^2=1, Awedge C=0 on X, and the restrictions of A,C to
+the fibers vanish, with unit fiber area form Psi orthogonal to both
+and equal to dd awedgedd b near the nodes. We now construct a
+closed two-form D_eps with Awedge D_eps=Cwedge D_eps=0 and
+D_eps^2=A^2 pointwise. The correction to a semi-flat approximate
solution is small and is solved on exact two-forms, where a
Stokes identity controls the inverse uniformly as the fiber area
-$eps$ shrinks.
+eps shrinks.
## The semi-flat approximation
-On the regular region $X^circ$, consider the form
+On the regular region X^circ, consider the form
E_eps=eps,Psi+(1)/(eps)cdot A^2
cdot(semi-flat horizontal)/(int_X A^2).
Concretely, in local invariant action-angle coordinates
-$(s;,a,b)$ adapted to the torus action, the semi-flat form is
+(s;,a,b) adapted to the torus action, the semi-flat form is
E_eps^{sf}=eps,dd awedgedd b
+(tau_2(s))/(eps),dd x_1wedgedd x_2,
- qquad s=x_1+ix_2,
+ s=x_1+ix_2,
-with the volume form $nu=A^2/2$ and $tau_2=Imtau$.
+with the volume form nu=A^2/2 and tau_2=Imtau.
This is exactly the semi-flat form of Cref{thm:nodal-model},
-scaled by $eps|h|^{-1}$. It is closed and of class that will be
+scaled by eps|h|^{-1}. It is closed and of class that will be
determined below.
-On each nodal disk $U_j$ the model form
-$D_eps^{loc}$ of Cref{thm:nodal-model} satisfies the
-pointwise identities $D^{loc}wedgeReΣ
-=D^{loc}wedgeImΣ=0$ and
-$(D^{loc})^2=(ReΣ)^2=(ReΣ_j)^2$,
-and its fiber integral is $eps$, matching $E_eps^{sf}$
+On each nodal disk U_j the model form
+D_eps^{loc} of Cref{thm:nodal-model} satisfies the
+pointwise identities D^{loc}wedgeReΣ
+=D^{loc}wedgeImΣ=0 and
+(D^{loc})^2=(ReΣ)^2=(ReΣ_j)^2,
+and its fiber integral is eps, matching E_eps^{sf}
up to exponential error by
(eq:nodal-exponential-estimate).
[Gluing]
-There is a closed two-form $E_eps$ on $X$ and an exact two-form
-$zeta_eps$ on the annular regions (fixed annuli near the punctures)
+There is a closed two-form E_eps on X and an exact two-form
+zeta_eps on the annular regions (fixed annuli near the punctures)
such that
[label=(roman*)]
-item
-$E_eps$ equals $E_eps^{sf}$ on the regular region outside
+
+E_eps equals E_eps^{sf} on the regular region outside
the outer boundary of the annuli, and equals
-$D_eps^{loc}$ on the preimage of the inner disks $U'_j$.
-item
+D_eps^{loc} on the preimage of the inner disks U'_j.
+
On each annulus, the difference
-$E_eps-E_eps^{sf}-zeta_eps$ has
-$|E_eps-E_eps^{sf}|_{C^m}$ bounded by
-$C_me^{-c_m/eps}$ for every $mge0$ (the constants depend only on
+E_eps-E_eps^{sf}-zeta_eps has
+|E_eps-E_eps^{sf}|_{C^m} bounded by
+C_me^{-c_m/eps} for every mge0 (the constants depend only on
fixed smooth data).
-item
+
The three wedge residuals
[
mathcal R_calR_eps=
@@ -3012,49 +3011,49 @@ satisfy
|mathcal R_calR_eps|_{H^k}
le Ceps^{-s}e^{-c/eps}
-for every fixed $k$, with $C,s,c>0$ depending only on $k$ and the
+for every fixed k, with C,s,c>0 depending only on k and the
fixed construction data.
Proof.
-On the regular region, $E_eps^{sf}$ is closed and satisfies
+On the regular region, E_eps^{sf} is closed and satisfies
the pointwise identities by direct computation:
-$dd awedgedd b$ wedges trivially with $A$ and $C$ because
-$A,C$ have only base--fiber mixed components; and
-$(E_eps^{sf})^2=A^2$ because both $A$ and $C$ have
-vanishing $dd x_1wedgedd x_2$ component while $Psi$ has only
-$dd awedgedd b$.
+dd awedgedd b wedges trivially with A and C because
+A,C have only base--fiber mixed components; and
+(E_eps^{sf})^2=A^2 because both A and C have
+vanishing dd x_1wedgedd x_2 component while Psi has only
+dd awedgedd b.
-On the nodal disk, the model form $D^{loc}$ satisfies
+On the nodal disk, the model form D^{loc} satisfies
(eq:nodal-exact-identities).
On each annulus, use a fixed bounded primitive construction. Pick a
radial trivialization of the annular model torus bundle
-$Mtimes[r_0,r_1]$. For an exact closed two-form $zeta$ on the
+Mtimes[r_0,r_1]. For an exact closed two-form zeta on the
annulus, radial homotopy gives
[
zeta=pi_M^*(zeta|_{Mtimes{r_*}})+dd Kzeta,
- qquad
+
Kzeta(r)=int_{r_*}^{r}iota_{∂_r}zeta(u),dd u.
]
-The restriction at $r_*$ is exact; a Green operator on the fixed
-compact $M$ provides a fixed primitive. Adding its pullback to
-$Kzeta$ gives a global primitive on a slightly smaller annulus, with
-$C^j$ norm bounded by finitely many $C^ell$ norms of $zeta$.
-All operators and domains are independent of $eps$.
+The restriction at r_* is exact; a Green operator on the fixed
+compact M provides a fixed primitive. Adding its pullback to
+Kzeta gives a global primitive on a slightly smaller annulus, with
+C^j norm bounded by finitely many C^ell norms of zeta.
+All operators and domains are independent of eps.
Applying this to the exponential difference of
(eq:nodal-exponential-estimate) (extended by zero outside the
-annuli and blunted inside) produces a closed $E_eps$ on $X$
+annuli and blunted inside) produces a closed E_eps on X
satisfying (i) and (ii). The residual bound (iii) now follows from
the explicit bounds
- a_eps:=int_XE_epswedge A=O(e^{-c/eps}),quad
+ a_eps:=int_XE_epswedge A=O(e^{-c/eps}),
b_eps:=int_XE_epswedge C=O(e^{-c/eps}),
which hold because the exact gluing adjusted the two pairings. The
-remaining residual $mathcal R_calR_eps$ is exponentially small in the
+remaining residual mathcal R_calR_eps is exponentially small in the
fixed Sobolev norms after the standard Sobolev inequalities on the
fixed chart cover.
@@ -3069,51 +3068,51 @@ Set
D^0_eps&=r_epswidehatwidehat E_E_eps.
-The period normalization gives $int A^2=int C^2=1$ and
-$int Awedge C=0$. So
+The period normalization gives int A^2=int C^2=1 and
+int Awedge C=0. So
- int_XAwedge D^0_eps=int_XCwedge D^0_eps=0,qquad
+ int_XAwedge D^0_eps=int_XCwedge D^0_eps=0,
int_X(D^0_eps)^2=1.
The approximate equations give
-$a_eps,b_eps=O(e^{-c/eps})$ and $q_eps=1+O(e^{-c/eps})$, after
-reducing $c>0$; thus $q_eps>0$ and $r_eps=1+O(e^{-c/eps})$.
-Subtracting constant multiples of $A,C$ and multiplying by a
+a_eps,b_eps=O(e^{-c/eps}) and q_eps=1+O(e^{-c/eps}), after
+reducing c>0; thus q_eps>0 and r_eps=1+O(e^{-c/eps}).
+Subtracting constant multiples of A,C and multiplying by a
positive constant do not change the pointwise three-plane spanned by
the triple.
## The nonlinear correction on exact forms
-Let $g_eps$ be the metric whose self-dual three-plane is
-$span(A,C,D^0_eps)$ and whose volume form is
+Let g_eps be the metric whose self-dual three-plane is
+span(A,C,D^0_eps) and whose volume form is
nu=(A^2)/(2).
-Since we may replace $E_eps$ by $D^0_eps$ in this definition, we get
-$vol_{g_eps}(X)=1/2$. Sobolev norms in the rest of the section are
-intrinsic norms for $g_eps$, using its covariant derivative and
+Since we may replace E_eps by D^0_eps in this definition, we get
+vol_{g_eps}(X)=1/2. Sobolev norms in the rest of the section are
+intrinsic norms for g_eps, using its covariant derivative and
volume.
[Bounds for the approximate triple]
-For every fixed $kge0$, the following hold for sufficiently small
-$eps$:
+For every fixed kge0, the following hold for sufficiently small
+eps:
[label=(roman*)]
-item
-The metric $g_eps$ has a finite coordinate cover whose cardinality,
+
+The metric g_eps has a finite coordinate cover whose cardinality,
inverse radii, margins, metric comparisons, and coefficient
derivatives through any prescribed order are bounded by powers of
-$eps^{-1}$.
-item
-The frame $(A,C,D^0_eps)$ and the inverse of its pointwise Gram
-matrix have polynomial intrinsic derivative bounds through order $k$.
+eps^{-1}.
+
+The frame (A,C,D^0_eps) and the inverse of its pointwise Gram
+matrix have polynomial intrinsic derivative bounds through order k.
The Gram matrix itself is uniformly positive and bounded.
-item
-Sobolev embedding and multiplication at a fixed large order $kge6$
-have polynomial constants: there are $C,q>0$ with
+
+Sobolev embedding and multiplication at a fixed large order kge6
+have polynomial constants: there are C,q>0 with
|hwedgewidetilde h|_{H^k}le Ceps^{-q}|h|_{H^k}|widetilde h|_{H^k}
-for two-forms $h,widetilde h$.
+for two-forms h,widetilde h.
@@ -3121,28 +3120,28 @@ Proof.
The explicit formulas
(eq:auxiliary-semiflat-local) and (eq:nodal-gh-triple)
give the coefficients and their derivatives on the regular and nodal
-neighborhoods. The construction of $g_eps$ from the frame
-$(A,C,D^0_eps)$ proceeds by normalizing against the wedge-form
+neighborhoods. The construction of g_eps from the frame
+(A,C,D^0_eps) proceeds by normalizing against the wedge-form
identities; matrix inversion and square root preserve the polynomial
bounds.
For the coordinate cover: shrink the charts uniformly by powers of
-$eps$ so every point has a chart containing an intrinsic ball of
-radius $rho_epsge c_1eps^{a_1}$. A maximal disjoint collection of
-such balls has at most $(2c_2eps^{a_2})^{-1}$ members by the volume
-bound $vol(X)=1/2$; the enlarged balls cover $X$.
+eps so every point has a chart containing an intrinsic ball of
+radius rho_epsge c_1eps^{a_1}. A maximal disjoint collection of
+such balls has at most (2c_2eps^{a_2})^{-1} members by the volume
+bound vol(X)=1/2; the enlarged balls cover X.
A coordinate cutoff and a normalized partition of unity then have
polynomial derivative bounds.
Euclidean Sobolev estimates on this cover give polynomial intrinsic
-Sobolev constants; since $dim X=4$ and $kge6$, derivatives of order
-at most $lfloor k/2rfloor$ are $L^∞$-bounded by $H^k$. The
-product rule on each derivative of $hwedgewidetilde h$ gives
+Sobolev constants; since dim X=4 and kge6, derivatives of order
+at most lfloor k/2rfloor are L^∞-bounded by H^k. The
+product rule on each derivative of hwedgewidetilde h gives
(eq:auxiliary-product).
[Exact inversion]
-For every fixed $kge6$ and every sufficiently small $eps$, there is
+For every fixed kge6 and every sufficiently small eps, there is
a bounded linear map
[
mathcal L_calL_eps: (t_1,t_2,t_3)in H^ktimes H^ktimes H^k
@@ -3150,18 +3149,18 @@ a bounded linear map
]
such that
[label=(roman*)]
-item
-The form $h$ is exact: $int_Xhwedgevarphi=0$ for every closed
-$varphi$; equivalently its harmonic projection is zero.
-item
-$h$ solves
+
+The form h is exact: int_Xhwedgevarphi=0 for every closed
+varphi; equivalently its harmonic projection is zero.
+
+h solves
[
(< h,A>_{L^2},
< h,C>_{L^2},
int_Xhwedge h)=(t_1,t_2,t_3),
]
-in the sense that $h$ is the unique element of $mathcal E^k_calE^k_eps$
-(exact $H^k$ two-forms) with those three moments, and its $H^k$ norm
+in the sense that h is the unique element of mathcal E^k_calE^k_eps
+(exact H^k two-forms) with those three moments, and its H^k norm
satisfies
[
|h|_{H^k}le Ceps^{-q}sup_i|t_i|_{H^k}.
@@ -3170,58 +3169,58 @@ satisfies
Proof. [Proof sketch]
-The self-dual part $phi=h^+$ is uniquely specified by requiring
-$f_i=t_inu$; then
+The self-dual part phi=h^+ is uniquely specified by requiring
+f_i=t_inu; then
phi=sum_{j=1}^3(M^{-1}t)_jE_j,
- qquad E_1=A, E_2=C, E_3=D^0_eps.
+ E_1=A, E_2=C, E_3=D^0_eps.
Moreover
<phi,E_i>_{L^2}=int_Xf_i=t_i,
-and the $E_i$ are closed and self-dual, hence harmonic. Since
-$b^+=3$, the $E_i$ form a basis of the harmonic self-dual forms, and
+and the E_i are closed and self-dual, hence harmonic. Since
+b^+=3, the E_i form a basis of the harmonic self-dual forms, and
(eq:auxiliary-cokernel-moments) is exactly the solvability
-condition for $dd^+a=phi$.
+condition for dd^+a=phi.
-Existence follows from the Hodge Green operator $G$ for $g_eps$: since
-$phi$ is orthogonal to harmonic self-dual forms and the Hodge
-Laplacian commutes with the star, $Gphi$ is self-dual with
-$Delta Gphi=phi$. For a self-dual two-form $psi$,
-$2(dddd^*psi)^+=Deltapsi$, so
+Existence follows from the Hodge Green operator G for g_eps: since
+phi is orthogonal to harmonic self-dual forms and the Hodge
+Laplacian commutes with the star, Gphi is self-dual with
+Delta Gphi=phi. For a self-dual two-form psi,
+2(dddd^*psi)^+=Deltapsi, so
h=2dddd^*Gphi
-is exact and has self-dual part $phi=h^+$. This is a Green formula;
+is exact and has self-dual part phi=h^+. This is a Green formula;
we control it by the standard Sobolev bounds for the Green operator on
-$g_eps$ (polynomial in $eps^{-1}$ by Cref{lem:auxiliary-geometry}).
-The remaining moments in $h$ are matched by adding
-$dddelta$ to $h$ for $delta$ a one-form, which does not change $h^+$
+g_eps (polynomial in eps^{-1} by Cref{lem:auxiliary-geometry}).
+The remaining moments in h are matched by adding
+dddelta to h for delta a one-form, which does not change h^+
but does change the anti-self-dual part; the exactness condition and
-the third moment $t_3$ can be arranged by a partition of unity.
+the third moment t_3 can be arranged by a partition of unity.
The stated Sobolev bound follows from the elliptic estimate in the
-standard way. Uniqueness is asserted for $h$, not for its one-form
+standard way. Uniqueness is asserted for h, not for its one-form
primitive.
## The contraction
-There is a positive constant $c$ and, for every sufficiently small
-$eps>0$, a smooth closed two-form $D_eps=D^0_eps+h_eps$, exact in
-the class of $D^0_eps$, such that
+There is a positive constant c and, for every sufficiently small
+eps>0, a smooth closed two-form D_eps=D^0_eps+h_eps, exact in
+the class of D^0_eps, such that
- Awedge D_eps=Cwedge D_eps=0,qquad (D_eps)^2=A^2.
+ Awedge D_eps=Cwedge D_eps=0, (D_eps)^2=A^2.
-The associated triple $(A,C,D_eps)$ is positive; $h_eps$ is
-bounded by $Ceps^{-N}e^{-c/eps}$ in every fixed Sobolev norm.
+The associated triple (A,C,D_eps) is positive; h_eps is
+bounded by Ceps^{-N}e^{-c/eps} in every fixed Sobolev norm.
Proof.
-Fix $kge6$. Define the closed subspace
-$mathcal E^k_calE^k_epssubset H^k$ of exact two-forms. Writing
-$D_eps=D^0_eps+h$ with $hinmathcal E^k_calE^k_eps$, the desired
+Fix kge6. Define the closed subspace
+mathcal E^k_calE^k_epssubset H^k of exact two-forms. Writing
+D_eps=D^0_eps+h with hinmathcal E^k_calE^k_eps, the desired
equations become the fixed-point equation
h=mathcal T_calT_eps(h):=mathcal L_calL_eps
@@ -3229,32 +3228,32 @@ equations become the fixed-point equation
-Cwedge D^0_eps,,
(A^2-(D^0_eps)^2-hwedge h)/(2)).
-This map is defined on all of $mathcal E^k_calE^k_eps$: the first two data
+This map is defined on all of mathcal E^k_calE^k_eps: the first two data
have integral zero by (eq:auxiliary-exact-moments), as does the
free term in the third slot; for each exact iterate,
-$int_Xhwedge h=0$ by approximation and Stokes' theorem. Every
+int_Xhwedge h=0 by approximation and Stokes' theorem. Every
iteration satisfies the three mean conditions of
Cref{lem:auxiliary-inverse}.
-The map $mathcal T_calT_eps$ is a contraction on a small ball of
-$mathcal E^k_calE^k_eps$: by the exponential smallness of the free terms
+The map mathcal T_calT_eps is a contraction on a small ball of
+mathcal E^k_calE^k_eps: by the exponential smallness of the free terms
(Cref{lem:auxiliary-gluing}) and the polynomial bounds of
Cref{lem:auxiliary-geometry,lem:auxiliary-inverse}, the quadratic part
-$hwedge h$ is a contraction with a factor $le1/2$ on a ball of
-radius $Ceps^{-N}e^{-c/eps}$. The Banach fixed-point theorem gives
-a unique smooth $h_eps$, and
+hwedge h is a contraction with a factor le1/2 on a ball of
+radius Ceps^{-N}e^{-c/eps}. The Banach fixed-point theorem gives
+a unique smooth h_eps, and
(eq:auxiliary-exact-equations) holds exactly.
-To see spatial smoothness, choose a Coulomb primitive $ain H^{k+1}$
-of $h$, so $dd a=h$ and $dd^*a=0$. The three equations for
-$D^0_eps+dd a$ together with the gauge are a first-order nonlinear
+To see spatial smoothness, choose a Coulomb primitive ain H^{k+1}
+of h, so dd a=h and dd^*a=0. The three equations for
+D^0_eps+dd a together with the gauge are a first-order nonlinear
system. Its linearization has invertible principal symbol: if a
-covector-valued unknown $b$ satisfies
-$xiwedge bwedge A=xiwedge bwedge C=xiwedge bwedge D_eps=0$
-and $iota_{xi^sharp}b=0$, then $xiwedge b$ is in the negative-definite
+covector-valued unknown b satisfies
+xiwedge bwedge A=xiwedge bwedge C=xiwedge bwedge D_eps=0
+and iota_{xi^sharp}b=0, then xiwedge b is in the negative-definite
wedge-orthogonal complement of the positive three-plane. But
-$(xiwedge b)^2=0$ and this complement is negative definite, so
-$xiwedge b=0$, forcing $b$ proportional to $xi$ and then to zero
+(xiwedge b)^2=0 and this complement is negative definite, so
+xiwedge b=0, forcing b proportional to xi and then to zero
by the gauge. Elliptic bootstrapping then gives smoothness in every
finite order.
@@ -3262,79 +3261,79 @@ finite order.
## The color class and the numerical chamber property
The last step is to extract, from the prepared construction of
-$D_eps$, a uniform numerical estimate in the cohomology lattice.
-Because the pair $(A,D_eps)$ is integrable and $A+iD_eps$ is
-nowhere-vanishing, the class $[D_eps]$ has positive square.
-The estimate is that $[D_eps]$ belongs to the same timelike cone of
-$V$ as $[B]$, and pairs positively with every curve class whose
-pairing with $[D_eps]$ is positive.
+D_eps, a uniform numerical estimate in the cohomology lattice.
+Because the pair (A,D_eps) is integrable and A+iD_eps is
+nowhere-vanishing, the class [D_eps] has positive square.
+The estimate is that [D_eps] belongs to the same timelike cone of
+V as [B], and pairs positively with every curve class whose
+pairing with [D_eps] is positive.
[Uniform chamber estimate]
-Retain the prepared pair $(A,C)$, the torus fibration
-$p:Xto S$ of Cref{thm:pencil}, and the chamber class $F$ of
-Cref{prop:chamber-class}. There is a positive constant $eps_0$
+Retain the prepared pair (A,C), the torus fibration
+p:Xto S of Cref{thm:pencil}, and the chamber class F of
+Cref{prop:chamber-class}. There is a positive constant eps_0
depending only on the fixed construction data and the original period
-plane $P$ such that, for every $0<eps<eps_0$, the class $[D_eps]$
+plane P such that, for every 0<eps<eps_0, the class [D_eps]
of Cref{thm:auxiliary-solution} satisfies
- [D_eps]=lambda_eps F+eps u_eps,qquad
- |lambda_eps-1/(2eps)|le C,qquad |u_eps|le C,
+ [D_eps]=lambda_eps F+eps u_eps,
+ |lambda_eps-1/(2eps)|le C, |u_eps|le C,
-with $C$ independent of $eps$, and the following uniform
-implication: for every positive real number $eps$ and every
-$dinΛ$ with $d^2ge-2$ and $[D_eps]cdot d>0$, we have
-$[B]cdot d>0$.
+with C independent of eps, and the following uniform
+implication: for every positive real number eps and every
+dinΛ with d^2ge-2 and [D_eps]cdot d>0, we have
+[B]cdot d>0.
Proof.
-By the construction of $D_eps$, its intersection with $F$ is
-$[D_eps]F=eps+O(eps)$ (the exact gluing changed only exponentially
-small). By $([D_eps])^2=A^2=1$ and $F^2=0$, we have
-$[D_eps]=lambda_eps F+eps u_eps$ with $u_eps$ in a fixed
-complement to $F$ in $V$, and
-$lambda_eps=1/(2eps)+O(1)$ from the linear and quadratic
-identities. The exact construction data are independent of $eps$, so
-$C$ and $eps_0$ are too.
+By the construction of D_eps, its intersection with F is
+[D_eps]F=eps+O(eps) (the exact gluing changed only exponentially
+small). By ([D_eps])^2=A^2=1 and F^2=0, we have
+[D_eps]=lambda_eps F+eps u_eps with u_eps in a fixed
+complement to F in V, and
+lambda_eps=1/(2eps)+O(1) from the linear and quadratic
+identities. The exact construction data are independent of eps, so
+C and eps_0 are too.
For the implication: by Cref{cor:curve-signs}, either
-$d=cF$ for some $cinR$ (in which case $[B]cdot d =
-c[B]cdot F=cbeta>0$ whenever $c>0$), or $d$ has the sign of the
-coefficient of $F^+$ in $d^+$. In the first case, $[B]cdot d=
-beta c$ and the sign of $[D_eps]cdot d = (lambda_eps)(c)+O(eps)$
-agrees with $c$ for small enough $eps$. In the second case, adjunction
-gives $d^2ge-2$, and Cref{cor:curve-signs} gives the sign of
-$[B]cdot d$; the condition $[D_eps]cdot d>0$ together with the
+d=cF for some cinR (in which case [B]cdot d =
+c[B]cdot F=cbeta>0 whenever c>0), or d has the sign of the
+coefficient of F^+ in d^+. In the first case, [B]cdot d=
+beta c and the sign of [D_eps]cdot d = (lambda_eps)(c)+O(eps)
+agrees with c for small enough eps. In the second case, adjunction
+gives d^2ge-2, and Cref{cor:curve-signs} gives the sign of
+[B]cdot d; the condition [D_eps]cdot d>0 together with the
expansion (eq:auxiliary-class-expansion) then forces the same
-sign. In both cases $[B]cdot d>0$.
+sign. In both cases [B]cdot d>0.
# The hyperk"ahler endpoint
-We combine the auxiliary form $D_eps$ of Cref{thm:auxiliary-solution}
+We combine the auxiliary form D_eps of Cref{thm:auxiliary-solution}
with the tamability criterion of Cref{thm:taming-cone} to reach a
genuine hyperk"ahler triple, while every triple in the path stays
positive and each of its three classes stays fixed. The argument has
-two K"ahler steps: the first replaces $[C]$ by the K"ahler class
-$[C_1]$ in a fixed integrable pair; the numerical chamber estimate
-keeps $[B]$ available along that step. The second then replaces
-$[B]$ by the K"ahler representative $B_1$ of $[B]$ in the new
+two K"ahler steps: the first replaces [C] by the K"ahler class
+[C_1] in a fixed integrable pair; the numerical chamber estimate
+keeps [B] available along that step. The second then replaces
+[B] by the K"ahler representative B_1 of [B] in the new
integrable pair; the smooth-tamer lemma makes the third forms along
the path consecutive.
## The K"ahler-class criteria
We need the following standard facts about compact complex surfaces.
-Both are applied to the compact complex surface $(X,J)$ built from a
-pair $(A,D)$ with $AD=0$ and $A^2=D^2>0$ (Lemma~lem:complex-symplectic):
+Both are applied to the compact complex surface (X,J) built from a
+pair (A,D) with AD=0 and A^2=D^2>0 (Lemma~lem:complex-symplectic):
[K"ahler classes and volumes]
-Let $(X,J)$ be a compact complex surface with $b_1$ even, and let
-$Hin H^2(X;R)$ be a class such that $H^2>0$ and $H$ pairs positively
-with every irreducible closed $J$-holomorphic curve. Then $H$
-contains a K"ahler form. Moreover, if $nu$ is a positive
-$(1,1)$-volume form, there is a K"ahler representative $B_1in H$ of
-$H$ with $B_1^2=int_Xnu$.
+Let (X,J) be a compact complex surface with b_1 even, and let
+Hin H^2(X;R) be a class such that H^2>0 and H pairs positively
+with every irreducible closed J-holomorphic curve. Then H
+contains a K"ahler form. Moreover, if nu is a positive
+(1,1)-volume form, there is a K"ahler representative B_1in H of
+H with B_1^2=int_Xnu.
Proof.
@@ -3342,132 +3341,131 @@ The first assertion is the classical Buchdahl--Lamari criterion in
the form [cite], together with the
Demailly--Paun numerical characterization of the K"ahler cone
[cite]. The class criterion enters through the
-positivity of $H$ on every irreducible curve. The volume
+positivity of H on every irreducible curve. The volume
normalization is Yau's theorem [cite]: the K"ahler
representative can be deformed in its class to have prescribed total
-volume $int_Xnu$.
+volume int_Xnu.
[Hodge type at the endpoint]
-Suppose $(A,C_1)$ is an integrable equal-square closed pair with
-$AC_1=0$ and $A^2=C_1^2>0$, and let $J$ be the complex structure with
-holomorphic volume form $A+iC_1$ (for the appropriate sign). Let
-$[B]$ be a class in $V=[A]^perpcap[C_1]^perp$. Then $[B]$
-belongs to $H^{1,1}(X;J)$.
+Suppose (A,C_1) is an integrable equal-square closed pair with
+AC_1=0 and A^2=C_1^2>0, and let J be the complex structure with
+holomorphic volume form A+iC_1 (for the appropriate sign). Let
+[B] be a class in V=[A]^perpcap[C_1]^perp. Then [B]
+belongs to H^{1,1}(X;J).
Proof.
-Because the triple $(A,C_1,B)$ is an orthonormal triple with respect
-to the positive intersection form, and $A+iC_1$ is a
-nonvanishing $(2,0)$ form, the classes $[A]$ and $[C_1]$ span the
-$(1,1)$-part of the period plane. Any class in $V$ is orthogonal to
-both, and is therefore of Hodge type $(1,1)$: use the Hodge
-decomposition and the fact that $span([A],[C_1])$ is
-the $-1$-eigenspace of the complex structure lifted to $H^2$ for the
+Because the triple (A,C_1,B) is an orthonormal triple with respect
+to the positive intersection form, and A+iC_1 is a
+nonvanishing (2,0) form, the classes [A] and [C_1] span the
+(1,1)-part of the period plane. Any class in V is orthogonal to
+both, and is therefore of Hodge type (1,1): use the Hodge
+decomposition and the fact that span([A],[C_1]) is
+the -1-eigenspace of the complex structure lifted to H^2 for the
appropriate orientation.
## The prepared endpoint
Recall from the combining theorems:
-
-item Cref{thm:pencil,prop:nonsplitting} give an exact pair
+ Cref{thm:pencil,prop:nonsplitting} give an exact pair
deformation to a torus fibration with the standard nodal
-neighborhoods, keeping $B$ fixed.
-item Cref{prop:regular-preparation} gives a further path of positive
-closed triples to invariant regular data with auxiliary form $Psi$.
-item Cref{thm:auxiliary-solution,prop:auxiliary-class} then supply
-$D_eps$ exactly.
+neighborhoods, keeping B fixed.
+ Cref{prop:regular-preparation} gives a further path of positive
+closed triples to invariant regular data with auxiliary form Psi.
+ Cref{thm:auxiliary-solution,prop:auxiliary-class} then supply
+D_eps exactly.
All stages preserve the three individual cohomology classes and start
at the actual representatives supplied by the preceding stage.
-We retain the notation $(A,C,B)$ for the resulting triple. In this
+We retain the notation (A,C,B) for the resulting triple. In this
notation
- [A]^2=[C]^2=[B]^2=1,qquad
+ [A]^2=[C]^2=[B]^2=1,
[A][C]=[A][B]=[C][B]=0.
-Fix a sufficiently small $ε>0$ for
+Fix a sufficiently small ε>0 for
Cref{thm:auxiliary-solution,prop:auxiliary-class}, and set
-$D=D_ε$. The form $D$ is closed and smooth and satisfies
+D=D_ε. The form D is closed and smooth and satisfies
- AD=CD=0,qquad D^2=A^2>0.
+ AD=CD=0, D^2=A^2>0.
-The triple with forms $A,C,D$ is positive. The same choice of
-$ε$ gives the numerical conclusion of
+The triple with forms A,C,D is positive. The same choice of
+ε gives the numerical conclusion of
Cref{prop:auxiliary-class} for every later definite pair with period
-classes $[A],[C]$. These choices precede the K"ahler steps that
+classes [A],[C]. These choices precede the K"ahler steps that
follow.
## The first K"ahler step
-By Cref{lem:complex-symplectic}, the pair $(A,D)$ is integrable and
-$A+iD$ is a nowhere-vanishing holomorphic two-form for the complex
-structure $J_0$ that it defines. Choose the sign of $J_0$ so that $C$
-tames $J_0$. This is possible because $C$ is one of the two direct
-components of the positive cone spanned by the $J_0$-compatible
-forms, and $C^2>0$.
-
-Apply Cref{lem:endpoint-kahler-class} with $H=[C]$, $J=J_0$ and
-$nu=A^2/2$. The hypotheses are: $[C]^2=1>0$ and $[C]$ pairs
-positively with every irreducible $J_0$-holomorphic curve. The latter
-follows because $C$ tames $J_0$ strictly (each nonconstant curve has
-positive area) and $X$ is compact. So $[C]$ contains a K"ahler
-representative $C_1$, with $C_1^2=A^2$.
-
-The pair $(A,C_1)$ is then also integrable and of equal square (by
-Cref{lem:complex-symplectic}, since $AC_1=0$ and $C_1^2=A^2$). Let
-$J_1$ denote the complex structure it defines. This $J_1$ equals
-$J_0$ (as complex structures on the complex surface: the two
-structures share the same $(2,0)$-form $A+iD = A+iC_1$ up to
-normalization). In particular, $[B]in V=[A]^perpcap[C_1]^perp$
-Hodge type $(1,1)$ by Cref{lem:endpoint-hodge-type}.
-
-We now interpolate the pair from $(A,C)$ to $(A,C_1)$. Set
-$E_lambda=C+lambda(C_1-C)$, $lambdain[0,1]$. The pointwise
-definite-pair condition with fixed $A$ is convex in $E_lambda$ at
-every point, so the family $(A,E_lambda)$ is positive throughout.
-Its associated almost-complex structures $J_lambda$ vary smoothly
-with $lambda$. Let $eta_lambda=(1)/(2)Re(A+iE_lambda)$
-be a smooth $J_lambda$-compatible tamer (semipositive where the
-pair degenerates). Since $C$ strictly tames $J_0$ and $C_1$ strictly
-tames $J_1$, and $[B]$ contains no tamer for $J_lambda$ for some
-$lambda$ would contradict Cref{thm:taming-cone} (the class $[B]$
-pairs positively with all $J_lambda$-curves by
-Cref{prop:auxiliary-class}), $[B]$ contains a tamer along the whole
+By Cref{lem:complex-symplectic}, the pair (A,D) is integrable and
+A+iD is a nowhere-vanishing holomorphic two-form for the complex
+structure J_0 that it defines. Choose the sign of J_0 so that C
+tames J_0. This is possible because C is one of the two direct
+components of the positive cone spanned by the J_0-compatible
+forms, and C^2>0.
+
+Apply Cref{lem:endpoint-kahler-class} with H=[C], J=J_0 and
+nu=A^2/2. The hypotheses are: [C]^2=1>0 and [C] pairs
+positively with every irreducible J_0-holomorphic curve. The latter
+follows because C tames J_0 strictly (each nonconstant curve has
+positive area) and X is compact. So [C] contains a K"ahler
+representative C_1, with C_1^2=A^2.
+
+The pair (A,C_1) is then also integrable and of equal square (by
+Cref{lem:complex-symplectic}, since AC_1=0 and C_1^2=A^2). Let
+J_1 denote the complex structure it defines. This J_1 equals
+J_0 (as complex structures on the complex surface: the two
+structures share the same (2,0)-form A+iD = A+iC_1 up to
+normalization). In particular, [B]in V=[A]^perpcap[C_1]^perp
+Hodge type (1,1) by Cref{lem:endpoint-hodge-type}.
+
+We now interpolate the pair from (A,C) to (A,C_1). Set
+E_lambda=C+lambda(C_1-C), lambdain[0,1]. The pointwise
+definite-pair condition with fixed A is convex in E_lambda at
+every point, so the family (A,E_lambda) is positive throughout.
+Its associated almost-complex structures J_lambda vary smoothly
+with lambda. Let eta_lambda=(1)/(2)Re(A+iE_lambda)
+be a smooth J_lambda-compatible tamer (semipositive where the
+pair degenerates). Since C strictly tames J_0 and C_1 strictly
+tames J_1, and [B] contains no tamer for J_lambda for some
+lambda would contradict Cref{thm:taming-cone} (the class [B]
+pairs positively with all J_lambda-curves by
+Cref{prop:auxiliary-class}), [B] contains a tamer along the whole
family. We apply Cref{lem:smooth-tamers} to get a smooth family
-$B_tin[B]$ of tamers, with $B_0=B$ at endpoint~0.
+B_tin[B] of tamers, with B_0=B at endpoint~0.
-For the endpoint at $lambda=1$: the interpolation path joined to
-the constant $B_t$ gives a smooth path of triples
-$(A,E_lambda,B_lambda)$, all positive. This is the first K"ahler
+For the endpoint at lambda=1: the interpolation path joined to
+the constant B_t gives a smooth path of triples
+(A,E_lambda,B_lambda), all positive. This is the first K"ahler
step.
## The second K"ahler step
-Now use the integrable pair $(A,C_1)$ and its structure $J_1$. Since
-$[B]in H^{1,1}(X;J_1)$ (Cref{lem:endpoint-hodge-type}), we can
-apply Cref{lem:endpoint-kahler-class} with $H=[B]$, $J=J_1$, and
-$nu=A^2/2$ to obtain a K"ahler representative $B_1in[B]$ with
-$B_1^2=A^2$. This time the hypothesis that $[B]$ pairs positively
-with every irreducible $J_1$-curve follows from
-Cref{prop:auxiliary-class}: $D$ and $B$ lie in the same timelike cone
-of $V$ and $[D]$ pairs positively on every $J$-curve.
-
-The pair $(A,B_1)$ is then also integrable of equal square, with
-complex structure $J_1$ (the same as $(A,C_1)$: the pair $(A,D)$
-originates from the same $J_1$). Interpolate from $(A,B)$ to
-$(A,B_1)$ as before. The endpoint pair $(A,B_1)$ has $[A]cdot[B_1]
-=0$ and $B_1^2=A^2$ (the first is $[A][B]=0$; the second is
-$B_1^2=A^2$). Concatenate with the interpolation to $(A,C_1)$.
+Now use the integrable pair (A,C_1) and its structure J_1. Since
+[B]in H^{1,1}(X;J_1) (Cref{lem:endpoint-hodge-type}), we can
+apply Cref{lem:endpoint-kahler-class} with H=[B], J=J_1, and
+nu=A^2/2 to obtain a K"ahler representative B_1in[B] with
+B_1^2=A^2. This time the hypothesis that [B] pairs positively
+with every irreducible J_1-curve follows from
+Cref{prop:auxiliary-class}: D and B lie in the same timelike cone
+of V and [D] pairs positively on every J-curve.
+
+The pair (A,B_1) is then also integrable of equal square, with
+complex structure J_1 (the same as (A,C_1): the pair (A,D)
+originates from the same J_1). Interpolate from (A,B) to
+(A,B_1) as before. The endpoint pair (A,B_1) has [A]cdot[B_1]
+=0 and B_1^2=A^2 (the first is [A][B]=0; the second is
+B_1^2=A^2). Concatenate with the interpolation to (A,C_1).
## Finishing
Concatenate all stages, using the specific positive closed triples
provided by the construction of each stage, and reparametrize each
path to be constant at its endpoints. The result is a smooth path of
-closed positive triples on $[0,1]$:
+closed positive triples on [0,1]:
[
(A_0,C_0,B_0)
@@ -3480,10 +3478,10 @@ closed positive triples on $[0,1]$:
Each component is closed and remains in its original class. Every
parameter triple is positive. At the endpoint, we have an
-orthogonal equal-square triple $(A,C_1,B_1)$ (by the second K"ahler
+orthogonal equal-square triple (A,C_1,B_1) (by the second K"ahler
step and Cref{lem:endpoint-hodge-type,lem:endpoint-kahler-class}).
By Cref{lem:complex-symplectic}, if
-$(eta_1,eta_2,eta_3)=(A,C_1,B_1)$ and $mu_1=A^2/2$, then
+(eta_1,eta_2,eta_3)=(A,C_1,B_1) and mu_1=A^2/2, then
[
eta_iwedgeeta_j=2delta_{ij}mu_1;
]
@@ -3498,13 +3496,13 @@ centering
[node distance=1.4cm,
every node/.style={font=small},
box/.style={draw,rounded corners,fill=gray!8,inner sep=3pt}]
- node[box] (orig) at (0,0) {original $(A,C,B)$};
- node[box,right=of orig] (pencil) {pencil $(A,C,B)$};
- node[box,right=of pencil] (prep) {prepared $(A,C,B)$ + $Psi$};
- node[box,right=of prep] (aux) {auxiliary $D=D_eps$};
- node[box,below=of prep] (K1) {1st K"ahler: $C_1$};
- node[box,below=of aux] (K2) {2nd K"ahler: $B_1$};
- node[box,below=of K1] (end) {$(A,C_1,B_1)$ hyperk"ahler};
+ node[box] (orig) at (0,0) {original (A,C,B)};
+ node[box,right=of orig] (pencil) {pencil (A,C,B)};
+ node[box,right=of pencil] (prep) {prepared (A,C,B) + Psi};
+ node[box,right=of prep] (aux) {auxiliary D=D_eps};
+ node[box,below=of prep] (K1) {1st K"ahler: C_1};
+ node[box,below=of aux] (K2) {2nd K"ahler: B_1};
+ node[box,below=of K1] (end) {(A,C_1,B_1) hyperk"ahler};
draw[->] (orig) -- (pencil);
draw[->] (pencil) -- (prep);
@@ -3518,276 +3516,276 @@ centering
Caption: The final deformation. Solid arrows trace the actual triple
path; dashed arrows indicate auxiliary dependencies. The auxiliary
form supplies the first integrable pair and the numerical control that
-keeps $[B]$ available along the moving pair. After both K"ahler
-steps, the tamers $B_t$ are chosen with initial value $B$ and final
-value $B_1$.
+keeps [B] available along the moving pair. After both K"ahler
+steps, the tamers B_t are chosen with initial value B and final
+value B_1.
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