diff b34550f..b1f9bc4 hypersymplectic_hyperkahler_deformation_v2.txt
diff --git a/hypersymplectic_hyperkahler_deformation_v2.txt b/hypersymplectic_hyperkahler_deformation_v2.txt
index 6076f55..d5b9c64 100644
--- a/hypersymplectic_hyperkahler_deformation_v2.txt
+++ b/hypersymplectic_hyperkahler_deformation_v2.txt
@@ -7,7 +7,7 @@
-\usetikzlibrary{arrows.meta,positioning,calc,fit}
+usetikzlibrary{arrows.meta,positioning,calc,fit}
@@ -37,21 +37,21 @@
-\setcounter{tocdepth}{1}
-\emergencystretch=1.5em
+setcounter{tocdepth}{1}
+emergencystretch=1.5em
-\title[Deforming hypersymplectic four-manifolds]{Deforming hypersymplectic four-manifolds to hyperk\"ahler triples}
+title[Deforming hypersymplectic four-manifolds]{Deforming hypersymplectic four-manifolds to hyperk"ahler triples}
Abstract:
Every smooth normalized positive triple of closed two-forms on a closed
connected oriented four-manifold admits a smooth deformation, with each
-cohomology class fixed, to a hyperk\"ahler triple. This resolves
+cohomology class fixed, to a hyperk"ahler triple. This resolves
Donaldson's hypersymplectic deformation conjecture.
-\maketitle
-\tableofcontents
+maketitle
+tableofcontents
# Introduction
@@ -59,83 +59,83 @@ 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:
-the induced metric need not be hyperk\"ahler. After a constant
+the induced metric need not be hyperk"ahler. After a constant
normalization of the periods, the natural question is whether one can
-deform the forms to a hyperk\"ahler triple while retaining their three
+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
+$omega=(omega_1,omega_2,omega_3)$ be a smooth hypersymplectic
structure normalized by
- \int_X\omega_i\wedge\omega_j=\delta_{ij}.
+ int_Xomega_iwedgeomega_j=delta_{ij}.
-There is a smooth family of hypersymplectic structures $\omega(t)$,
-$0\le t\le1$, such that
-\[
- \omega(0)=\omega,\qquad [\omega_i(t)]=[\omega_i]\quad(i=1,2,3),
-\]
-and a positive volume form $\mu_1$ with
+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),
+]
+and a positive volume form $mu_1$ with
- \omega_i(1)\wedge\omega_j(1)=2\delta_{ij}\mu_1.
+ omega_i(1)wedgeomega_j(1)=2delta_{ij}mu_1.
-The endpoint forms are parallel and self-dual for a hyperk\"ahler metric.
+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 $G_{ij}=\int_X\omega_i\wedge\omega_j$,
-one applies the theorem to $L\omega$ for a constant invertible real
-$L$ with $LGL^{\mathsf T}=I_3$ 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
-$\omega_i(1)\wedge\omega_j(1)=2G_{ij}\mu_1$: the three classes are
-parallel self-dual forms for a hyperk\"ahler metric, with their original
+$omega_i(1)wedgeomega_j(1)=2G_{ij}mu_1$: the three classes are
+parallel self-dual forms for a hyperk"ahler metric, with their original
Gram matrix $G$.
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
+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
-the K\"ahler class in advance. Second, the topology of the initial
+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
-hyperk\"ahler metric.
+hyperk"ahler metric.
-[Individual K\"ahler realizations]
+[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
-$c\in\R^3∖\{0\}$, the form $\alpha_c=\sum_i c_i\omega_i$
-is a K\"ahler form for a parallel integrable complex structure $J_c$ of
-a hyperk\"ahler metric $g_c$ on $X$:
-\[
- \alpha_c(v,w)=g_c(J_cv,w).
-\]
+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
simultaneous transport of the triple is asserted.
Proof.
Use the fixed-class path for the original basis described above and put
-$\alpha_t=\sum_i c_i\omega_i(t)$. Positivity of the pointwise
-wedge-product matrix gives $\alpha_t\wedge\alpha_t>0$, so $\alpha_t$
-is symplectic; its class is fixed because each $[\omega_i(t)]$ is fixed.
+$alpha_t=sum_i c_iomega_i(t)$. 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 $r_c=(c^{\mathsf T}Gc)^{1/2}>0$. The
-endpoint identities show that $\alpha_1/r_c$ is parallel and self-dual
-with $g$-norm $\sqrt2$. It therefore defines a parallel integrable
+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
+$alpha_1/r_c=g(J^{end}_c,cdot,cdot)$,
+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 $X$.
+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}.
+cite[Section~1]{FineYao2017}.
[Smooth type]
@@ -145,14 +145,14 @@ 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
+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}
+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 $\mathbf P^3$, the
-standard K3 manifold \cite[p.~58]{Kodaira1970}.
+diffeomorphic to a nonsingular quartic surface in $mathbf P^3$,bfP^3$, the
+standard K3 manifold cite[p.~58]{Kodaira1970}.
## History and significance
@@ -161,41 +161,41 @@ Donaldson's study of closed two-forms on four-manifolds links their
pointwise wedge-product geometry to nonlinear elliptic equations
[cite]. His Question~3 in Section~5.3 asks whether a
compact oriented four-manifold carrying a positive closed triple admits
-a hyperk\"ahler structure. In the same section he proposes a
+a hyperk"ahler structure. In the same section he proposes a
prescribed-volume continuity argument in the simply connected case,
conditional on a priori estimates and an auxiliary involution. The
involution keeps the evolving form in a fixed cohomology class
-\cite[Section~5.3]{Donaldson2006}. Fine and Yao formulate the normalized
+cite[Section~5.3]{Donaldson2006}. Fine and Yao formulate the normalized
deformation problem, retaining all three cohomology classes, as
Conjecture~1.1 of [cite]. Theorem~thm:main resolves
this deformation conjecture positively. Keeping the classes connects the
given triple to a metric with its prescribed periods, beyond mere
-existence of some hyperk\"ahler metric on the same manifold.
+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
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
-\cite[Theorem~1.3]{FineYao2017}. Their later work gives an integral
+cite[Theorem~1.3]{FineYao2017}. Their later work gives an integral
torsion extension criterion, global existence near pointwise
orthogonality, and subsequential convergence of global solutions under
additional geometric bounds
-\cite[Theorems~4.7, 4.10, 4.15 and~4.16]{FineYao2020}.
+cite[Theorems~4.7, 4.10, 4.15 and~4.16]{FineYao2020}.
Several restricted classes admit complete convergence results.
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
+cite[Theorems~3.5 and~3.6]{HuangWangYao2018}. Picard and Suan treated
+$G_2$ 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
+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
-\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}.
+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}.
Related reductions and coflow constructions appear in work of Petcu,
Yao and Zhou, and Karigiannis, Picard and Suan
[cite]. All of these
@@ -207,10 +207,10 @@ 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
+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},
+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
free intersection lattice is the K3 lattice or three hyperbolic planes.
@@ -219,27 +219,27 @@ smooth identification of the initial manifold with a K3 surface or torus.
For controlling closed positive forms, Sullivan's structure currents
provide the cone-duality framework [cite]. Harvey and
-Lawson developed the positive-current characterization of K\"ahler
+Lawson developed the positive-current characterization of K"ahler
geometry [cite], and Li and Zhang compared the taming
and compatible cones in almost-complex geometry assuming the compatible
cone is nonempty [cite]. A separate article proves that
every tamed smooth almost-complex structure on a closed connected real
four-manifold admits a compatible symplectic form
-\cite[Theorem~1.1]{OpenAITaming2026}. Its current estimates and
+cite[Theorem~1.1]{OpenAITaming2026}. Its current estimates and
density-splitting argument are also used here, with the needed proofs
-included in \Ssec:current-energy--\Ssec:density-splitting.
+included in Ssec:current-energy--Ssec:density-splitting.
In particular, smooth density-weight closedness already appears in the
proof in Section~6 of that article. Its compatible class is unrestricted.
The additional class criterion below converts closed normalized density
thresholds into curves, using Preiss's rectifiability theorem and
-Rivi\`ere--Tian's regularity theorem for integral cycles
+Rivi`ere--Tian's regularity theorem for integral cycles
[cite]. This supplies the prescribed taming
class needed along the deformation.
## How the proof works: a plain-language sketch
We want a path of positive triples with fixed classes ending at a
-hyperk\"ahler triple. Every positive triple factors, pointwise, as 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,
@@ -279,10 +279,10 @@ 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
+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 /
@@ -292,20 +292,20 @@ 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
+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
+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.
+hyperk"ahler.
## Organization of the paper
@@ -321,21 +321,21 @@ $F$ 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.
+Theorem~thm:main via the two K"ahler steps.
## Conventions
Products of two-forms are wedge products; products of degree-two
cohomology classes are their intersection pairing. Curve classes are
-identified with their Poincar\'e duals in cohomology. Integral lattice
+identified with their Poincar'e duals in cohomology. Integral lattice
statements use
-\[
- Λ=H^2(X;\Z)/torsion.
-\]
+[
+ 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
@@ -380,15 +380,15 @@ 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),qquad
+ y=((C^2)/(A^2)-x^2)^{1/2}>0,qquad
T=(C-xA)/(y).
Then $AT=0$ and $T^2=A^2>0$, so
-\[
- Ω=A+iT,\qquad Ω^2=0,\qquad
- Ω\overlineΩ=2A^2>0.
-\]
+[
+ Ω=A+iT,qquad Ω^2=0,qquad
+ ΩoverlineoverlineΩ=2A^2>0.Ω=2A^2>0.
+]
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
@@ -425,10 +425,10 @@ always use the sign of $J$ fixed by a stated tamer.
If $X$ carries a positive closed triple, then
-\[
- b^+(X)=3,\qquad
- (b_1(X),b^-(X))=(0,19)\ or\ (4,3).
-\]
+[
+ b^+(X)=3,qquad
+ (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.
Every almost-complex structure obtained from a definite pair in the
@@ -442,24 +442,24 @@ 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 $c_1^2=2\chi+3\sigma$, obtained
+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},
+theorem cite[Corollaries~11.12 and~15.5, Theorem~19.4]{MilnorStasheff1974},
together with $b^+=3$, gives
-\[
- 0=19-4b_1-b^-,\qquad \sigma=4b_1-16.
-\]
-Moreover $w_2\equiv c_1=0\pmod 2$, so $X$ is spin
-\cite[Problem~14-B]{MilnorStasheff1974}. Rokhlin's theorem makes
-$\sigma$ divisible by $16$ [cite]. Since $b^-\ge0$, these
-relations give $b_1\in\{0,4\}$ 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}.
+[
+ 0=19-4b_1-b^-,qquad sigma=4b_1-16.
+]
+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.
+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
-stated forms \cite[Chapter~V, Section~2.2, Theorems~5--6]{Serre1973}.
+stated forms cite[Chapter~V, Section~2.2, Theorems~5--6]{Serre1973}.
Only the intersection lattice has been classified at this stage.
@@ -468,36 +468,36 @@ bundle of a unique oriented conformal structure. Choose one metric
$g_0$ in that conformal class. Since the original forms are closed and
self-dual, they are harmonic, and
- \mathcal H^+_{g_0}=span_\R\{\omega_1,\omega_2,\omega_3\}
+ mathcal H^+_calH^+_{g_0}=span_R{omega_1,omega_2,omega_3}
by Proposition~prop:topology. Here the span is the vector space of
global forms with constant coefficients.
## The coefficient sphere of almost-complex structures
-For $v\in S^2\subset\R^3$, let $B_v=v\cdot\omega$ and let
-$\mathcal A_v$ be the plane of forms $a\cdot\omega$ with $a\perp v$.
+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 $\mathcal A_v$ itself and does
+$J_v$. This construction uses the plane $mathcal A_v$calA_v$ itself and does
not require a global choice of ordered basis over $S^2$.
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
-$\mathcal A_v$.
+$mathcal A_v$.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
+$g_0$. 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
-\[
- v⟼\frac{K^{-1}v}{|K^{-1}v|}
-\]
+[
+ v⟼frac{K^{-1}v}{|K^{-1}v|}
+]
in radial coefficient coordinates. Positive definite matrices connect
to the identity through positive definite matrices, so this map has
degree one. The orientation convention identifying coefficient and
@@ -506,7 +506,7 @@ twistor spheres can only change its sign.
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.
+coefficient vector lies in $v^perp$. This proves the final assertion.
## The Lorentz space and the integrability identity
@@ -515,9 +515,9 @@ After a constant orthogonal change of coefficients we denote a chosen
ordered triple by $(A,C,B)$. For all later exact deformations of the
pair, set
- V=[A]^\perp\cap[C]^\perp\subset H^2(X;\R).
+ 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.
@@ -525,33 +525,33 @@ 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
+hyperk"ahler, after choosing the sign of the complex structure so that
$C$ 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
-\[
- \mathcal L_UΩ=0,\qquad
- \iota_{[U,V]}Ω
- =\mathcal L_U(\iota_VΩ)-\iota_V(\mathcal L_UΩ)=0.
-\]
+Cartan's formula and $ddΩ=0$ give
+[
+ mathcal L_UΩ=0,calL_UΩ=0,qquad
+ 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.
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 $Ω\overlineΩ/C^2$ is constant, so the
+$(1,1)$-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
holomorphic volume form has constant pointwise norm. The Chern
connection of the canonical line then makes $Ω$ parallel. For a
-K\"ahler metric this is the connection induced by the Levi-Civita
+K"ahler metric this is the connection induced by the Levi-Civita
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
-$\eta_i\wedge\eta_j=2\delta_{ij}\mu$; thus the normalized matrix is
+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
the identity.
@@ -566,7 +566,7 @@ that fail to contain a tamer. That separation produces a positive
current whose mass we must control at the two-dimensional scale, and
whose complex-line directions must not turn too fast at small scales.
The present section develops both estimates. They are the input to the
-density decomposition of \Ssec:density-splitting.
+density decomposition of Ssec:density-splitting.
## Conventions and separation
@@ -576,98 +576,98 @@ 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^+.
+ qquad I=Λ_J^+=R G ⊕ Λ_g^-,
+ qquad E=Λ_J^-subsetΛ_g^+.
Let $R$ denote the orthogonal projection onto $E$. A unit
-complex-line bivector is $\ell=e\wedge Je$, where $|e|_g=1$. We use $g$
+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,qquad ell^+=G/2,qquad G(ell)=1,
+ qquad |G|_g^2=2.
The bundle of these bivectors is denoted by
-$\mathcal L→ X$.
+$mathcal L→calL→ X$.
A two-current is a continuous real linear functional on
$Ω^2(X)=C^∞(X,Λ^2)$. Under the metric identification its
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(\dd\alpha)=0\ for all \alpha\inΩ^1(X)
- \ ⇔\ \dd^*T=0
- \ ⇔\ \dd(*T)=0.
+ 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
-$T(\xi)=\int_X\xi\wedge *T$. Inner products between $L^2$ coefficients
-are written $<\,\,,\,>_2$.
+$T(xi)=int_Xxiwedge *T$. Inner products between $L^2$ coefficients
+are written $<,,,,>_2$.
Let
-\[
- \mathcal P_J=\{\xi\inΩ^2(X):\xi(\ell)>0
- for every \ell\in\mathcal L\},
- \qquad
- \mathcal Z_J=\{\xi\in C^∞(X,I):\dd\xi=0\}.
-\]
-The set $\mathcal P_J$ is an open convex cone in the smooth topology;
+[
+ 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;
openness already holds in the uniform topology. Its closed elements are
precisely the symplectic forms taming $J$. The elements of
-$\mathcal P_J\cap\mathcal Z_J$ are the compatible symplectic forms.
+$mathcal P_JcalP_Jcapmathcal Z_J$calZ_J$ are the compatible symplectic forms.
## The two separators
[The two separators]
The following hold.
-[label=(\roman*)]
-\item If $\mathcal P_J\cap\mathcal Z_J$ is empty, there is a nonzero
- current $N$ which annihilates $\mathcal Z_J$ and all anti-invariant
+[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
forms, and is nonnegative on semipositive invariant forms.
-\item If $H\in H^2(X;\R)$ has no representative in $\mathcal P_J$, there
+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 $N(H)\le0$.
+ satisfies $N(H)le0$.
In either case one can normalize $N(G)=1$ and choose a positive Borel
-measure $\lambda$ on $\mathcal L$ such that
+measure $lambda$ on $mathcal L$calL$ such that
- N(\xi)=\int_{\mathcal L}\xi_y(\ell)\,\dd\lambda(y,\ell),
- \qquad \mu=(pr_X)_*\lambda,
- \qquad \mu(X)=1.
+ N(xi)=int_{mathcal L}calLxi_y(ell),ddlambda(y,ell),
+ qquad mu=(pr_X)_*lambda,
+ qquad mu(X)=1.
In particular $N$ has order zero and belongs to $H^{-3}(X,Λ^2)$.
Proof.
For (i), apply separation of the open convex set
-$\mathcal P_J\cap C^∞(X,I)$ from the disjoint linear subspace
-$\mathcal Z_J$ in $C^∞(X,I)$. The separating continuous functional
-vanishes on $\mathcal Z_J$ and is nonnegative on the positive cone.
+$mathcal P_JcalP_Jcap C^∞(X,I)$ 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
semipositive invariant form.
-For (ii), choose a smooth closed representative $\beta$ of $H$
-and separate the open cone $\mathcal P_J$ 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 $mathcal P_J$calP_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 $N(\dd\alpha)=0$. Multiplying a positive form by any
+subspace give $N(ddalpha)=0$. Multiplying a positive form by any
positive scalar, the separation inequalities imply both
-$N(\mathcal P_J)\ge0$ and $N(\beta)\le0$. Adding an arbitrary real
-multiple of an anti-invariant form to an element of $\mathcal P_J$ shows
+$N(mathcal P_J)calP_J)ge0$ 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
gives positivity on semipositive invariant forms.
For either functional, comparison of an invariant form with sufficiently
-large positive multiples of $G$ gives $|N(\xi)|\le C N(G)\|\xi\|_{C^0}$.
+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 $2\times2$ 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
+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
$H^{-3}$.
@@ -679,14 +679,14 @@ $H^{-3}$.
There is a classical pseudodifferential operator
$D:C^∞(X,E)→Ω^2(X)$ of order zero with
- \dd D=0,\qquad RD=\Id.
+ dd D=0,qquad RD=Id.
It maps distributions to distributions. For the separator in
-\Cref{lem:current-separators}(i), set
+Cref{lem:current-separators}(i), set
- Q(\xi)=-N(DR\xi),\qquad T=N+Q.
+ Q(xi)=-N(DRxi),qquad T=N+Q.
-Then $Q$ is anti-invariant and self-dual, $Q,T\in H^{-3}$, and $T$
+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 $[*T]=0$.
@@ -696,14 +696,14 @@ parameter as operators between the corresponding smooth bundles.
Proof.
-On sections of $E$ consider $P=R\dd\dd^*$. If $a$ is self-dual, the
-self-dual projection of $\xi\wedge\iota_\xi a$ 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
-$\sigma_2(P)(x,\xi)=\tfrac12|\xi|_g^2\Id_E$ 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.
@@ -711,14 +711,14 @@ 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
+smoothing and $PL=Id-H_0$. Set
- D=\dd\dd^*L+H_0.
+ 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 $\xi-DR\xi$ is closed and invariant, so
-$T(\xi)=N(\xi-DR\xi)=0$. The functional $Q$ only depends on $R\xi$;
+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
@@ -728,29 +728,29 @@ 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
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
+[label=(Alph*)]
+item $N$ 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
+item $N$ is a normalized positive invariant closed current, $Q=0$, and
$T=N$.
-Setup (B) includes \Cref{lem:current-separators}(ii).
+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
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
are independent of the center, of the small scale, and of the choice of
-submeasure of $\lambda$.
+submeasure of $lambda$.
## The quadratic mass estimate
@@ -759,17 +759,17 @@ 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 $0<s<s_0$,
-\[
- |\nabla^j a_{s,p}(y)|\le C_j(s+\dist(p,y))^{-a-j},
- \qquad 0\le j\le6,\quad a\in\{0,1\},
-\]
+[
+ |nabla^j a_{s,p}(y)|le C_j(s+dist(p,y))^{-a-j},
+ qquad 0le jle6,quad 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
+ |Aa_{s,p}(x)|le
- C\delta^{-1},&a=1,\\
- C(1+|\log\delta|),&a=0.
+ Cdelta^{-1},&a=1,
+ C(1+|logdelta|),&a=0.
Away from a smaller fixed neighborhood of $p$ the output is uniformly
@@ -779,26 +779,26 @@ 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 $|K_A(x,y)|\le C\dist(x,y)^{-4}$ 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 $B_{c\delta}(x)$ and equal to one on $B_{c\delta/2}(x)$,
-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 $C\delta^{-a}$.
+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 $C\delta^{-a}$, 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 $c\delta/2$ from $x$. Its
-portion in $B_{C\delta}(p)$ contributes at most
-$C\delta^{-4}\int_{B_{C\delta}(p)}(s+\dist(p,y))^{-a}\,\dd V_g(y)\le
-C\delta^{-a}$. On a shell of radius $r=2^j\delta\ge C\delta$ about $x$,
+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 $C\delta^{-1}$ if $a=1$ and to $C(1+|\log\delta|)$ if $a=0$. The
+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.
@@ -808,86 +808,86 @@ bounded; the remaining input is uniformly smooth.
In either setup (A) or (B), there is a constant $C$
with
- \mu(B_s(p))\le Cs^2\qquad(p\in X,\ 0<s\le1).
+ mu(B_s(p))le Cs^2qquad(pin X, 0<sle1).
Proof.
-Fix a radius $\rho_0>0$ for uniform normal coordinates and a smooth
-radial cutoff which is one on $B_{2\rho_0}(p)$ and supported in
-$B_{3\rho_0}(p)$. 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
-\[
- u_s=\chi(t)\log√(s^2+t^2),\qquad
- h_s=\chi(t)√(s^2+t^2),\qquad
- \dd_J^cu=-\dd u\circ J.
-\]
+[
+ u_s=chi(t)log√(s^2+t^2),qquad
+ h_s=chi(t)√(s^2+t^2),qquad
+ 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
-\[
- \beta(f)=
+[
+ beta(f)=
- (1-DR)\dd\dd_J^cf,&in setup (A),\\
- \dd\dd_J^cf,&in setup (B).
+ (1-DR)dddd_J^cf,&in setup (A),
+ dddd_J^cf,&in setup (B).
-\]
-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 $\dd\dd_J^c$ on functions is a multiple of
-$\xi\wedge J^*\xi$, which is $J$-invariant, so $R\dd\dd_J^c$ 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
-$0\le j\le6$,
-\[
- |\nabla^jR\dd\dd_J^cu_s|\le C_j(s+t)^{-1-j},\qquad
- |\nabla^jR\dd\dd_J^ch_s|\le C_j(s+t)^{-j}.
-\]
-\Cref{lem:current-radial-operator} applied to $D$ therefore bounds the
-two correction terms by $C/(s+t)$ and $C(1+|\log(s+t)|)$.
+$0le jle6$,
+[
+ |nabla^jRdddd_J^cu_s|le C_j(s+t)^{-1-j},qquad
+ |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 $C(1+|log(s+t)|)$.
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
-\[
- \dd\dd_{J_p}^cf(\ell)=2f'(t)/t+(f''(t)-f'(t)/t)a,
- \qquad 0\le a\le1.
-\]
+[
+ dddd_{J_p}^cf(ell)=2f'(t)/t+(f''(t)-f'(t)/t)a,
+ qquad 0le ale1.
+]
For the two radial functions this gives
- \dd\dd_{J_p}^c\log√(s^2+t^2)(\ell)
- &\ge (2s^2)/((s^2+t^2)^2),\\
- \dd\dd_{J_p}^c√(s^2+t^2)(\ell)
- &\ge \frac1{√(s^2+t^2)}.
+ dddd_{J_p}^clog√(s^2+t^2)(ell)
+ &ge (2s^2)/((s^2+t^2)^2),
+ 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
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^2\delta^{-4}-C_1\delta^{-1},
- \\
- \beta(h_s)(\ell)&\ge c_2\delta^{-1}-C_2(1+|\log\delta|).
+ beta(u_s)(ell)&ge c_1s^2delta^{-4}-C_1delta^{-1},
+
+ beta(h_s)(ell)&ge c_2delta^{-1}-C_2(1+|logdelta|).
The inequalities also apply in setup (B), where the correction
terms are absent.
-Choose $0<\rho<\rho_0$ so small that
-$C_2(1+|\log\delta|)\le c_2/(2\delta)$ for $0<\delta\le2\rho$, and
-then choose a fixed multiplier $M\ge2C_1/c_2$. If $s\le\rho$ and
-$t\le\rho$, the two displayed inequalities give
-$\beta(u_s+Mh_s)(\ell)\ge c_1s^2\delta^{-4}$. On $X∖ B_\rho(p)$
+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 $\delta\le2s$ on $B_s(p)$,
+Cref{lem:current-radial-operator}. Since $deltale2s$ on $B_s(p)$,
globally
-\[
- \beta(u_s+Mh_s)(\ell)\ge c s^{-2}\mathbf 1_{B_s(p)}-C.
-\]
-Apply $N$, use $N(\beta(u_s+Mh_s))=0$ and $\mu(X)=1$, and obtain
-(eq:current-quadratic-mass) for $s\le\rho$. For larger $s$ the
-same bound follows from $\mu(X)=1$. Compactness makes all choices
+[
+ 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 $p$.
@@ -898,30 +898,30 @@ step compares directions of nearby complex lines; smoothing permits this
comparison through an intersection pairing with the mass bound
controlling the error.
-Let $\Delta=\dd\dd^*+\dd^*\dd$ be the nonnegative Hodge Laplacian and
+Let $Delta=dddd^*+dd^*dd$ be the nonnegative Hodge Laplacian and
set
- S_s=(1+s^2\Delta)^{-3},\qquad 0<s\le s_0.
+ S_s=(1+s^2Delta)^{-3},qquad 0<sle s_0.
-It commutes with $\dd,\dd^*$ and $*$. At each fixed scale it maps
+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
-$\tau_{z\to y}$ be parallel transport along the minimizing geodesic when
-$\dist(y,z)<r_0$. The kernel $K_s$ of $S_s^2$ can be written
+$tau_{zto y}$ 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_{z\to y}+E_s(y,z),
+ 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 $k\ge0$ 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,
- \\
- k_s(y,z)&\le C_ks^{-4}(1+\dist(y,z)/s)^{-k},
- \\
- |E_s(y,z)|&\le C_ks^{-2}(1+\dist(y,z)/s)^{-k}.
+ k_s(y,z)&ge cs^{-4}&&if dist(y,z)<s,
+
+ k_s(y,z)&le C_ks^{-4}(1+dist(y,z)/s)^{-k},
+
+ |E_s(y,z)|&le C_ks^{-2}(1+dist(y,z)/s)^{-k}.
The upper bound on $s_0$ is fixed independently of $y,z$.
@@ -930,86 +930,86 @@ The upper bound on $s_0$ is fixed independently of $y,z$.
Proof.
The spectral theorem gives the operator identity
- S_s^2=\frac1{5!}\int_0^∞ h^5e^{-h}e^{-s^2h\Delta}\,\dd h.
+ S_s^2=frac1{5!}int_0^∞ h^5e^{-h}e^{-s^2hDelta},dd h.
The heat kernel of the nonnegative Hodge Laplacian on a closed
four-manifold has a local small-time parametrix
-\[
- \chi(d)(4\pi t)^{-2}e^{-d^2/(4t)}
+[
+ chi(d)(4pi t)^{-2}e^{-d^2/(4t)}
(a_0(y,z)+ta_1(y,z)+...+t^ma_m(y,z)),
-\]
+]
with arbitrarily high-order uniform remainder after increasing $m$; here
-$d=\dist(y,z)$, $\chi$ is supported inside the injectivity radius and
+$d=dist(y,z)$, $chi$ is supported inside the injectivity radius and
equals one near the diagonal, and
-$a_0(y,z)=j(y,z)^{-1/2}\tau_{z\to y}$ 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
expansion gives the Gaussian part; away from the diagonal the global
Gaussian estimate makes the cutoff error smaller than every power of
-$t$. For $t\ge t_0$ 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 $\chi\ge0$ and use its leading coefficient to define
-\[
- k_s(y,z)=\frac{\chi(d)j(y,z)^{-1/2}}{5!(4\pi)^2}s^{-4}
- \int_0^∞ h^3e^{-h-d^2/(4s^2h)}\,\dd h.
-\]
-For $d<s$, integration over $1\le h\le2$ gives the lower bound. For
+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
+$e^{-r^2/(ch)}le C_k(1+r)^{-k}(1+h^{k/2})$ 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
gives
-\[
- Cs^{-2}\int_0^∞ h^4e^{-h-d^2/(Cs^2h)}\,\dd h
- \le C_ks^{-2}(1+d/s)^{-k}.
-\]
+[
+ 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 $d\lediamX$, this is bounded by the right side of
-the error bound when $2L\ge k-2$. On $s^2h\ge t_0$, the factor $e^{-h}$
+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 $s$.
-For any positive submeasure $\lambda_i\le\lambda$, 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}\,\dd\mu_i(y),
- \qquad B_s=B_{s,\mu}.
+ B_{s,mu_i}(z)=s^{-2}int_X
+ (1+dist(y,z)/s)^{-6},ddmu_i(y),
+ qquad B_s=B_{s,mu}.
-By \Cref{lem:quadratic-mass}, a dyadic-shell sum gives
-$0\le B_{s,\mu_i}(z)\le B_s(z)\le C$.
+By Cref{lem:quadratic-mass}, a dyadic-shell sum gives
+$0le B_{s,mu_i}(z)le B_s(z)le C$.
[Positive-current pairing]
-For any $\lambda_1,\lambda_2\le\lambda$ and their currents,
+For any $lambda_1,lambda_2lelambda$ and their currents,
< S_sN_1,*S_sN_2>_2
- &\ge cs^{-4}
- \iint_{\dist(y,z)<s}|\ell-\tau_{z\to y}m|^2
- \,\dd\lambda_1(y,\ell)\dd\lambda_2(z,m)
- \notag\\[-2pt]
- &\hspace{8mm}-C\int_X B_{s,\mu_1}(z)\,\dd\mu_2(z),
- \\
- \|S_sN_i\|_2&\le C/s.
+ &ge cs^{-4}
+ iint_{dist(y,z)<s}|ell-tau_{zto y}m|^2
+ ,ddlambda_1(y,ell)ddlambda_2(z,m)
+ notag[-2pt]
+ &hspace{8mm}-Cint_X B_{s,mu_1}(z),ddmu_2(z),
+
+ |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.
Parallel transport preserves the Hodge star. From
(eq:current-line-normalization), direct expansion gives
- <\ell,*\tau_{z\to y}m>
- =\tfrac12|\ell-\tau_{z\to y}m|^2
- -\tfrac14|G_y-\tau_{z\to y}G_z|^2.
+ <ell,*tau_{zto y}m>
+ =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
(eq:current-line-measure). Its leading scalar is nonnegative, so
@@ -1017,16 +1017,16 @@ the lower-bound kernel inequality and the angular identity supply the
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
-\[
- Cs^{-2}\iint(1+d/s)^{-6}\,\dd\mu_1(y)\dd\mu_2(z),
-\]
+[
+ Cs^{-2}iint(1+d/s)^{-6},ddmu_1(y)ddmu_2(z),
+]
which is the asserted error. The norm estimate follows by using the
absolute kernel bound in the unstarred pairing:
-\[
- \|S_sN_i\|_2^2
- \le Cs^{-4}\iint(1+d/s)^{-6}\,\dd\mu_i(y)\dd\mu_i(z)
- \le Cs^{-2}\mu_i(X)\le Cs^{-2}.
-\]
+[
+ |S_sN_i|_2^2
+ le Cs^{-4}iint(1+d/s)^{-6},ddmu_i(y)ddmu_i(z)
+ le Cs^{-2}mu_i(X)le Cs^{-2}.
+]
The kernel formulas extend to measure coefficients because the kernel is
continuous at fixed $s$.
@@ -1034,110 +1034,110 @@ 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 $S_sW\in L^2$. 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\|_2\le Cs\|S_sW\|_2.
+ |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 $S_sZ\in L^2$,
+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.
+ |< S_sN',S_sZ>_2|le C|S_sZ|_2.
Proof.
-Set $A_s=S_s^{-1}=(1+s^2\Delta)^3$ 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 $\binom3j s^{2j}[\Delta^j,\Pi]$, $1\le j\le3$.
-Because the principal symbol of $\Delta$ is scalar, $[\Delta^j,\Pi]$ is a
+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 $\|S_s\|_{L^2\to H^m}\le C_ms^{-m}$,
-$0\le m\le6$. Since the formal adjoint of $B_j=[\Delta^j,\Pi]$ has the
+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,
-\[
- \|S_s s^{2j}B_j\|_{L^2\to L^2}
- =\|s^{2j}B_j^*S_s\|_{L^2\to L^2}
- \le Cs^{2j}s^{-(2j-1)}=Cs.
-\]
+[
+ |S_s s^{2j}B_j|_{L^2to L^2}
+ =|s^{2j}B_j^*S_s|_{L^2to L^2}
+ 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.
Apply it to $RN'=0$ and $(1-R)Z=0$. With $u=S_sN'$ and $v=S_sZ$, the
orthogonal splitting gives
-\[
+[
|< u,v>_2|
- \le\|Ru\|_2\|Rv\|_2+\|(1-R)u\|_2\|(1-R)v\|_2
- \le Cs\|u\|_2\|v\|_2,
-\]
+ le|Ru|_2|Rv|_2+|(1-R)u|_2|(1-R)v|_2
+ le Cs|u|_2|v|_2,
+]
and the submeasure norm estimate yields the mixed bound.
-If $C_1,C_2\in H^{-3}$ 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].
+ =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 $a_i\in H^{-2}$;
-then $S_sh_i=h_i$ and $S_sa_i\in H^4$. Integration by parts eliminates
-every term containing an exact factor, leaving $\int_Xh_1\wedge h_2$.
+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
-$Q\in L^2(X,E)$ and
+$Qin L^2(X,E)$ and
- \iint_{\dist(y,z)<s}|\ell-\tau_{z\to y}m|^2
- \,\dd\lambda(y,\ell)\dd\lambda(z,m)\le Cs^4.
+ 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 $N'$,
- \lim_{s↓0}< S_sN',S_sQ>_2=0.
+ lim_{s↓0}< S_sN',S_sQ>_2=0.
-The convergence is uniform over $\lambda'\le\lambda$.
+The convergence is uniform over $lambda'lelambda$.
Proof.
-Let $\mathcal A_s$ denote the double integral in
-(eq:current-angular-energy) and put $x_s=\|S_sQ\|_2$. Since $Q$ is
+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
-\[
+[
[*T]^2
=< S_sN,*S_sN>_2+2< S_sN,S_sQ>_2+x_s^2.
-\]
+]
Use the positive-current lower bound and the $B_s$ bound for the first
term, and the mixed bound for the second. The result is
- cs^{-4}\mathcal A_s+x_s^2-Cx_s\le C+[*T]^2.
+ 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
intersection term is zero. In setup (B) that term is bounded
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 $s^{-4}\mathcal A_s$.
+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 $Q\in L^2$; anti-invariance is preserved, and the spectral
-theorem gives strong convergence $S_sQ\to Q$ in $L^2$.
+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 $\|Q_j-Q\|_2\to0$; for example
+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 $j$,
-$S_s^2Q_j\to Q_j$ in $C^∞$, whereas $N'(Q_j)=0$. Therefore
-\[
+$|< S_sN',S_s(Q-Q_j)>_2|le C|Q-Q_j|_2$. For fixed $j$,
+$S_s^2Q_jto Q_j$ in $C^∞$, whereas $N'(Q_j)=0$. Therefore
+[
< S_sN',S_sQ_j>_2
=N'(S_s^2Q_j-Q_j)→0,
-\]
+]
uniformly over the submeasures because their trace masses are at most
-one. Let $s\to0$ first and $j\to∞$ second.
+one. Let $sto0$ first and $jto∞$ second.
# Density splitting and the taming cone
@@ -1158,8 +1158,8 @@ 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
+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
connected Riemann surface, with its complex orientation.
@@ -1169,15 +1169,15 @@ $b^+(X)=3$, 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
-\[
- V=[A]^\perp\cap[C]^\perp\subset H^2(X;\R).
-\]
-Suppose $U\in V$ has positive square and contains a smooth tamer for
-$J$. If $H\in V$ has positive square, belongs to the same component of
-$\{v\in V:v^2>0\}$ as $U$, and satisfies
-\[
- H\cdot d>0
-\]
+[
+ 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
+[
+ Hcdot d>0
+]
for every irreducible closed $J$-holomorphic curve of class $d$, then
$H$ contains a smooth symplectic form taming $J$.
@@ -1189,41 +1189,41 @@ almost-complex currents [cite]: closedness of
$N+Q$ is applied, and the $L^2$ correction $Q$ is estimated directly.
[Quadratic density]
-At every $p\in X$, the limit
+At every $pin X$, the limit
- θ(p)=\lim_{s↓0}s^{-2}\mu(B_s(p))
+ 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.
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 $p$.
+[
+ 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 $p$.
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
-$t=|\zeta|=\dist(p,\exp_p\zeta)$, define
-$b_p(s)=s^{-2}T(\mathbf 1_{B_s(p)}G_0)$. These evaluations are
-well-defined because $T$ has measure coefficients. Since $G(\ell)=1$,
+$t=|zeta|=dist(p,exp_pzeta)$, 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),
+ 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
-$b_p(s)=s^{-2}N(\mathbf 1_{B_p(s)}G_0)+o(1)$. 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
-\[
- b_p(s)→\lim_{s↓0}s^{-2}\mu(B_s(p)).
-\]
+[
+ b_p(s)→lim_{s↓0}s^{-2}mu(B_s(p)).
+]
Indeed, the measure of a manifold ball is comparable to the measure of
the corresponding Euclidean ball, the ratio of the two measures is
bounded, and by uniqueness of the differentiation limit the two
@@ -1233,167 +1233,167 @@ Measurability follows from a fixed sequence of radii tending to zero;
boundedness from (eq:density-frozen-ratio). For the last
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
+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
+$mu(B_s(p))=O_p(s^4)$ and $s^{-2}mu(B_s(p))to0$. A finite chart
cover finishes the proof.
-Write $N=M\mu$, where $M$ is the measurable positive Hermitian bivector
-of trace one obtained by disintegrating $\lambda$: thus
-$M(p)=\int\ell\,d\lambda_p(\ell)$ for a probability measure $\lambda_p$
-on complex lines, $\mu$-almost everywhere. The following tangent
+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.
+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)
+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.
+$lambda$-almost everywhere.
Proof.
Fix $p$ with a finite density $θ(p)$ and such that the measure
-$\gamma_p$ obtained by the homothety $s^{-2}(z\mapsto p+sz)_*\mu$ has
-a tangent measure $\gamma=M(p)\mathbf 1_{P_p}$ in a complex
+$gamma_p$ 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
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 $\R^4$.
+[
+ ∂(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
+$∂(M(p)gamma)=0$ 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
+$gamma$ 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 $b:\R\to\R$ is smooth and bounded with
+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_{\{θ>0\}}N,\qquad
- I_a=\mathbf 1_{\{θ>a\}}(\pi)/(θ)N\quad(a>0)
+ N^+=mathbf 1_bf1_{{θ>0}}N,qquad
+ I_a=mathbf 1_bf1_{{θ>a}}(pi)/(θ)Nquad(a>0)
are also closed. In particular, if
-$N^d=\mathbf 1_{\{θ=0\}}N$, 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 $h\in C_c^∞((0,1))$
-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)\,d\mu(z),
- \qquad M_s(z)=\mu(B_s(z)).
+ f_s(y)=s^{-2}int_Xh(dist(y,z)^2/s^2),dmu(z),
+ qquad M_s(z)=mu(B_s(z)).
$f_s$ 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
-\[
- f_s(y)=-\int_0^1h'(u)\frac{\mu(B_{s\sqrt u}(y))}{s^2}\,du
- →-θ(y)\int_0^1u\,h'(u)\,du=θ(y).
-\]
+$mu$ 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 $f_s\toθ$ also volume-almost everywhere.
+$u=0$. Thus $f_stoθ$ also volume-almost everywhere.
-Fix a smooth one-form $\alpha$. To prove closedness of $b(θ)N$,
+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)d\alpha)+T(b'(f_s)\,df_s\wedge\alpha).
+ 0=T(d(b(f_s)alpha))
+ =T(b(f_s)dalpha)+T(b'(f_s),df_swedgealpha).
-Dominated convergence gives $b(f_s)N\to b(θ)N$ and
-$b(f_s)Q\to0$ as currents (the second uses $b(0)=0$, the a.e.
+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
-$\int |df_s|_\ell\,d\lambda$, the $Q$-part by a constant times
-$\int|Q|\,|df_s|\,dV_g$. We need the two gradient estimates
+$int |df_s|_ell,dlambda$, 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\,d\lambda(y,\ell)=o(1).
+ int_X|Q|,|df_s|,dV_g=o(1),
+ qquad 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 $\int_XM_s\,dV_g\le Cs^4$; 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}
- (s^{-4}\int_XM_s\,dV_g)^{1/2}=o(1),
+ s^{-3}int_X|Q|M_s,dV_g
+ le(int_X|Q|^2(M_s)/(s^2),dV_g)^{1/2}
+ (s^{-4}int_XM_s,dV_g)^{1/2}=o(1),
the first factor by dominated convergence and the second by
boundedness. This proves the first estimate of
(eq:density-two-gradient-estimates).
-\noindentThe tangential derivative. Closedness converts
+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
-$k_s^y(z)=h(|\zeta|^2/s^2)$, $q_i=∂_{\zeta_i}k_s^y$,
-$\omega_{12}=d\zeta_1\wedge d\zeta_2$. 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)\,d\mu(z),\qquad i=1,2.
+ df_s(y)(e_i)=-s^{-2}int q_i(z),dmu(z),qquad i=1,2.
-Replace this scalar integral by $T(q_i\omega_{12}) = (N+Q)(q_i\omega_{12})$:
+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
-\[
- \omega_{12}(m)=<\ell,\tau_{zy}m>+O(s^2),\qquad
- |1-\omega_{12}(m)|\le C(s^2+|\ell-\tau_{zy}m|^2).
-\]
-Since $|q_i|\le C/s$, the integrated replacement error is at most
+[
+ omega_{12}(m)=<ell,tau_{zy}m>+O(s^2),qquad
+ |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
- s^{-2}\!\int\!|\int q_i d\mu-T(q_i\omega_{12})|d\lambda
- \le Cs^{-3}(s^2\!\iint_{\dist<s}\!d\mu\,d\mu+\mathcal A_s)
- +Cs^{-3}\!\int_X\!|Q|M_s\,dV_g=o(1),
+ 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 $O(s^2)$, $\mathcal A_s$ is the angular
+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
(eq:density-q-gradient).
-For $i=1$, closedness of $T$ on $d(k_s^y d\zeta_2)$ gives
-$T(q_1\omega_{12})=-\sum_{a=3}^4T(q_a\,d\zeta_a\wedge d\zeta_2)$;
-for $i=2$ the analogous identity holds for $d(k_s^y d\zeta_1)$. 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 $a=3,4$,
-\[
- |(d\zeta_a\wedge d\zeta_k)(m)|\le C(s^2+|\ell-\tau_{zy}m|),
-\]
+[
+ |(dzeta_awedge dzeta_k)(m)|le C(s^2+|ell-tau_{zy}m|),
+]
and the normal derivative of the radial kernel obeys
-$|q_a(z)|\le Cs^{-1}|pr_{\ell^\perp}\zeta/s|$.
+$|q_a(z)|le Cs^{-1}|pr_{ell^perp}zeta/s|$.
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^{1/2}\mathcal T_s^{1/2}=s^{-3}\cdot O(s^2)\cdot
+ Cs^{-3}mathcal A_s^calA_s^{1/2}mathcal T_s^calT_s^{1/2}=s^{-3}cdot O(s^2)cdot
o(s)=o(1).
Combining (eq:density-first-variation),
@@ -1401,25 +1401,25 @@ Combining (eq:density-first-variation),
(eq:density-angular-transverse-product) proves the second estimate
of (eq:density-two-gradient-estimates).
-\noindentThe closed weighted limit. The two derivative 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 $\to b(θ)N(d\alpha)$, 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
-$\chi:\R\to[0,1]$ vanishing on $(-∞,0]$ and equal to one on
-$[1,∞)$; the bounded functions $\chi(nt)\to\mathbf 1_{\{t>0\}}$
+$chi:Rto[0,1]$ 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
-\[
+[
b_{a,n}(t)=
- \pi\,\chi(n(t-a))/t,&t>0,\\
- 0,&t\le0.
+ pi,chi(n(t-a))/t,&t>0,
+ 0,&tle0.
-\]
-These are smooth, vanish near zero, are uniformly bounded by $\pi/a$,
-and converge at every $t\ge0$ to $\mathbf 1_{\{t>a\}}\pi/t$. 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 $n$.
@@ -1434,105 +1434,105 @@ compatibility question and then provides the nonnegative diffuse square
needed for the class criterion.
[Vanishing diffuse kernel error]
-Set $B_s(z)=s^{-2}\int_X(1+\dist(y,z)/s)^{-6}\,d\mu(y)$. 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)\,d\mu^d(z)→0,\qquad
- \mu^d=\mathbf 1_{\{θ=0\}}\mu.
+ int_XB_s(z),dmu^d(z)→0,qquad
+ 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
+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
-$S_s=(1+s^2\Delta_g)^{-3}$ preserves cohomology classes and commutes
+$S_s=(1+s^2Delta_g)^{-3}$ preserves cohomology classes and commutes
with the Hodge star. Hodge decomposition therefore gives
- < S_sS,*S_sS'>_{L^2}=\mathfrak c(S)\cdot\mathfrak c(S'),
+ < 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
sufficient for this identity.
-Proof. [Proof of \Cref{thm:tamed-compatible}]
+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 $N$,
+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 $\mathfrak c(T)=0$. By \Cref{prop:density-weights}, $T^d=N^d+Q$
+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\\
+ 0&=< S_sT,*S_sT^d>notag
&=< S_sN,*S_sN^d>+< S_sN,S_sQ>
- +< S_sQ,S_sN^d>+\|S_sQ\|_2^2.
+ +< S_sQ,S_sN^d>+|S_sQ|_2^2.
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
-$\lambda^d=\mathbf 1_{\{θ=0\}}\lambda$, gives
-$< S_sN,*S_sN^d>\ge-C\int_XB_s\,d\mu^d=o(1)$. Since
-$S_sQ\to Q$ in $L^2$, taking a limit inferior in
-(eq:density-compatibility-pairing) forces $\|Q\|_2^2=0$. Thus
+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
-$N(\eta)\ge c_\eta\mu(X)>0$, contradicting the annihilation property.
+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 $\mathfrak c(N^d)^2\ge0$.
+If $Q=0$, then $N^d$ is closed and $mathfrak c(N^d)^2frakc(N^d)^2ge0$.
Proof.
-Closedness is \Cref{prop:density-weights}. By
+Closedness is Cref{prop:density-weights}. By
(eq:density-smoothed-intersection) and
-\Cref{lem:current-positive-pairing},
-\[
- \mathfrak c(N^d)^2=< S_sN^d,*S_sN^d>
- \ge-C\int_XB_s\,d\mu^d,
-\]
-and the right side tends to zero by \Cref{lem:density-diffuse-error}.
+Cref{lem:current-positive-pairing},
+[
+ mathfrak c(N^d)^2=<frakc(N^d)^2=< S_sN^d,*S_sN^d>
+ ge-Cint_XB_s,dmu^d,
+]
+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
-\Cref{eq:density-weighted-restrictions} is an integral two-cycle with
+Cref{eq:density-weighted-restrictions} is an integral two-cycle with
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
identity
- N^+=\frac1\pi\int_0^∞ I_a\,da
+ N^+=frac1piint_0^∞ I_a,da
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
+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 $(pr_X)_*\lambda=\mu$;
+normalization of $lambda$ makes $(pr_X)_*lambda=mu$;
hence the trace measure of $I_a$ equals
-$\mathbf 1_{\{θ>a\}}\pi\,dV_g$ and is rectifiable. Preiss's
+$mathbf 1_bf1_{{θ>a}}pi,dV_g$ and is rectifiable. Preiss's
rectifiability theorem [cite] upgrades the a.e.~planar
tangents to rectifiability; closedness of $I_a$ and positivity then
express it as an integral sum of simple curves. Positivity of the
@@ -1542,49 +1542,49 @@ mass; factoring each component through its somewhere-injective image
gives a finite sum of irreducible closed curves with positive integer
multiplicities.
-Finally, pointwise on $\{θ>0\}$,
-$\int_0^∞\mathbf 1_{\{θ>a\}}\pi/θ\,da=\pi$; for a
-smooth two-form $\alpha$, Fubini is justified by
-\[
- \int_0^∞|I_a(\alpha)|\,da\le\pi\|\alpha\|_{C^0}\mu^+(X).
-\]
+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).
+]
This proves (eq:density-layer-cake) with its stated normalization.
-Proof. [Proof of \Cref{thm:taming-cone}]
+Proof. [Proof of Cref{thm:taming-cone}]
Assume that $H$ has no taming representative. The class separator of
-\Cref{lem:current-separators}(ii) gives a nonzero closed
+Cref{lem:current-separators}(ii) gives a nonzero closed
positive current $N$, with $Q=0$, satisfying
- H\cdot\mathfrak c(N)\le0.
+ 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
-$\mathfrak c(N^d)\cdot[A]=N^d(A)=0=\mathfrak c(N^d)\cdot[C]$; thus
-$w=\mathfrak c(N^d)$ lies in $V$ and $w^2\ge0$ by
-\Cref{lem:density-diffuse-square}.
-
-If $N^d\ne0$, choose a taming form $\eta$ in $U$. Strict taming on the
-compact bundle of unit complex lines gives $U\cdot w=N^d(\eta)>0$; in
-particular $w\ne0$. Since $[A],[C]$ span a positive two-plane and
+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 $w^2\ge0$ and $U\cdot w>0$ put $w$ in the nonzero closed time
+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
-$H\cdot w>0$: writing $H=h_0u+h_-$, $w=w_0u+w_-$ with a unit timelike
-$u$ and orthogonal negative parts, $h_0>|h_-|$, $w_0\ge|w_-|$, $w_0>0$,
-so $H\cdot w\ge h_0w_0-|h_-|\,|w_-|>0$. If $N^d=0$, its pairing is
+$Hcdot w>0$: 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
+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 $H\cdot\mathfrak c(I_a)\ge0$, with strict inequality
-whenever $I_a\ne0$. If $N^+\ne0$, the layer identity has positive total
-mass, so the set of $a$ for which $I_a\ne0$ has positive Lebesgue
+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 $H\cdot\mathfrak c(N^+)>0$. The pairing is
+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
-$H\cdot\mathfrak c(N)=H\cdot\mathfrak c(N^d)+H\cdot\mathfrak c(N^+)>0$,
+$Hcdotmathfrak c(N)=Hfrakc(N)=Hcdotmathfrak c(N^d)+Hfrakc(N^d)+Hcdotmathfrak c(N^+)>0$,frakc(N^+)>0$,
contradicting (eq:density-cone-separator). A smooth closed taming
representative is symplectic by pointwise nondegeneracy.
@@ -1601,40 +1601,40 @@ large-parameter limit, holomorphic curves in any prescribed nonzero
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
+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
and their total classes are only ever used modulo torsion. By
-\Cref{prop:topology},
+Cref{prop:topology},
- b^+=3,\qquad
- (b_1,\sigma)=(0,-16)\ or\ (4,0),\qquad
- 2\chi+3\sigma=0.
+ b^+=3,qquad
+ (b_1,sigma)=(0,-16) or (4,0),qquad
+ 2chi+3sigma=0.
-Let $J_v$, $v\in S^2$, be the coefficient sphere of
-\Cref{lem:twistor-degree}, with the sign chosen so that
-$B_v=v\cdot\omega$ tames $J_v$. Write
-$P=span\{[\omega_1],[\omega_2],[\omega_3]\}$ for the
+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
original positive period plane.
A positive curve configuration is a finite sum of irreducible
closed pseudoholomorphic curves with positive integer multiplicities.
-Let $F\inΛ$ be nonzero with $F^2=0$, and write
-$F^+=pr_PF$. For every $p\in X$, there exists a finite
+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
@@ -1645,10 +1645,10 @@ the wall-crossing computation for the fixed spin-c structure.
There is a smooth family of closed $J_v$-compatible forms
-$\eta_v$. Orthogonal projection to $P$ satisfies
-\[
- pr_P[\eta_v]=a(v)[B_v],\qquad a(v)>0,
-\]
+$eta_v$. Orthogonal projection to $P$ satisfies
+[
+ pr_P[eta_v]=a(v)[B_v],qquad a(v)>0,
+]
and consequently the normalized period projection of this family has
degree one in the coefficient coordinates.
@@ -1656,24 +1656,24 @@ 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
+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 $L_v^{ell}=R_v\dd\dd^*$ on the
+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
-\Cref{lem:closed-lift}, with kernel the closed anti-invariant forms;
+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
-\Cref{lem:twistor-degree}.
+Cref{lem:twistor-degree}.
Compatibility is equivalent to taming by a closed $-J$-invariant form.
-By \Cref{thm:tamed-compatible}, the compatible cone is nonempty for
+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
+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
-$\int_X\eta_v\wedge B_v>0$: the invariant part of $B_v$ is positive,
-and its anti-invariant part wedges trivially with $\eta_v$. Since
+$int_Xeta_vwedge B_v>0$: 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 $P$.
@@ -1682,28 +1682,28 @@ 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 $\mathfrak s_0$. The structures $J_v$ form a
+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 $E\in H^2(X;\Z)$, let
-$\mathfrak s_E=\mathfrak s_0 ⊗ L_E$, where
+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(\det\mathfrak s_E)=2E,\qquad d(\mathfrak s_E)=E^2,\qquad
- ind_{\C}\not D_E=(4E^2-\sigma)/(8).
+ c_1(detmathfrak s_E)=2E,fraks_E)=2E,qquad d(mathfrak s_E)=E^2,fraks_E)=E^2,qquad
+ ind_{C}not D_E=(4E^2-sigma)/(8).
Fix a homology orientation once and for all.
## Wall-crossing at the harmonic wall
[Wall-crossing]
-Let $\mathfrak s$ be the fixed spin-c structure of the family $J_v$
-over $S^2$, and let $q=d(\mathfrak s)/2$ be its expected complex
+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
dimension. Then the family invariant in the chamber after crossing the
harmonic wall gives
- FSW_{\mathfrak s}(S^2)
+ FSW_{mathfrak s}fraks(S^2)
=s_{b_1/2}(K-O),
where $K,O$ are the ranks of the complex Dirac index, and
@@ -1713,20 +1713,20 @@ the universal spinor evaluation.
Proof.
The critical wall-crossing theorem and its section-class version are
-\cite[Theorem~4.10 and Proposition~4.11]{LiLiu2001}; we only check
+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
-$S^2\times[0,1]$, because its normal rank is $b^+=3$. Above such a
+$S^2times[0,1]$, 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
-$x=c_1(\mathcal O(1))$ on the projective bundle of lines
-$\pi:\mathbb P(K)\toPic^0(X)$. The obstruction bundle
-contributes $e(\mathcal O(1) ⊗ O)=\sum_ix^{o-i}c_i(O)$.
+$x=c_1(mathcal O(1))$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^q\sum_ix^{o-i}c_i(O))
- &=\sum_is_{q+o-(k-1)-i}(K)c_i(O)\notag\\
+ pi_*(x^qsum_ix^{o-i}c_i(O))
+ &=sum_is_{q+o-(k-1)-i}(K)c_i(O)notag
&=s_{b_1/2}(K-O),
@@ -1747,37 +1747,37 @@ spinor with weight one, as used in
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 $\mathfrak s_E$. Denote by $\pi$ the projection of the
+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 $U_p+e\pi^*h$, where
-$h=c_1(\mathcal O(1))$ 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}(S^2).
+ =SW_X(E)+eFSW_{mathfrak s_E}fraks_E(S^2).
-For every prescribed point $p$, if $FSW_E(U_p)\ne0$
+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
-$\mathcal O(1) ⊗ \mathcal E_E|_p$. 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 $\ell\to\mathcal E_E|_p$: it lies in
-$Hom(\ell,\mathcal E_E|_p)=\mathcal O(1) ⊗
-\mathcal E_E|_p$. 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 $\mathcal E_E|_{\{p\}\times S^2}$; its first Chern
-class is $U_p+e\pi^*h$. Pairing $\pi^*h$ with the family moduli cycle
+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
@@ -1788,7 +1788,7 @@ 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 $r\to∞$.
+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
configuration of the prescribed class; the support contains all limit
@@ -1797,11 +1797,11 @@ points of first-component zeros.
Proof.
This is the standard compactness argument of Taubes
-\cite[Proposition~5.2 and Section~6]{Taubes1996}, with the essential
+cite[Proposition~5.2 and Section~6]{Taubes1996}, with the essential
uniform estimate (eq:families-taubes-off-zeros) below. The
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
+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
approximation tolerance are fixed.
@@ -1812,65 +1812,65 @@ 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)/(2\pi)\int_XF_{a_n}\wedge\zeta,
- \qquad
- |T_n|\le C(r_n)
+ 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
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
canonical connection is
- |\mathcal A-\mathcal A_{can}|
- +|F_{\mathcal A}-F_{\mathcal A_{can}}|
- \le Cr^{-1}+Cr\exp(-\sqrt r\,
- \dist(x,\alpha^{-1}(0))/C)
+ |mathcal A-calA-mathcal A_calA_{can}|
+ +|F_{mathcal A}calA-F_{mathcal A_calA_{can}}|
+ le Cr^{-1}+Crexp(-sqrt r,
+ dist(x,alpha^{-1}(0))/C)
when the distance is at least $r^{-1/2}$
-\cite[Proposition~6.1]{Taubes1999}.
+cite[Proposition~6.1]{Taubes1999}.
Fix the disk bound. Write
-$f_n=(i)/(2\pi)F_{a_n}
- =(i)/(4\pi)(F_{\mathcal A_n}-F_{\mathcal A_{can,n}})$.
+$f_n=(i)/(2pi)F_{a_n}
+ =(i)/(4pi)(F_{mathcal A_n}calA_n-F_{mathcal A_calA_{can,n}})$.
The sharp estimate
-\cite[Proposition~4.2, Equation~(4.9)]{Taubes1999}, specialized to
-$|\eta_n|=\sqrt2$, is
+cite[Proposition~4.2, Equation~(4.9)]{Taubes1999}, specialized to
+$|eta_n|=sqrt2$, is
- |F_{\mathcal A_n}^-|\le(r_n)/(2\sqrt2)(1-|\alpha_n|^2)+C.
+ |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_n\ge-C_0\,\dd A_u
+ u^*f_nge-C_0,dd A_u
-for every $J_n$-holomorphic disk $u$, with $C_0\ge1$ 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}]
+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 $F$
+square-zero vector in the negative-definite space $P^perp$. Lift $F$
to an integral class and use the fixed spin-c structure
-$\mathfrak s_F$.
+$mathfrak s_F$.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
+$SW_X(F)ne0$, 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.
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
+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
+Cref{lem:families-evaluation-line} forces a first-component zero over
$p$ 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
+Cref{lem:families-taubes-compactness} gives a configuration through
$p$ in some $J_v$, with total class $F$.
The two closed forms whose coefficients are perpendicular to $v$ are
@@ -1886,27 +1886,27 @@ The unmarked wall calculation also excludes the square-minus-two
classes needed in the lattice argument.
-If $b_1=0$, no class $E\inΛ$ 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
-\Cref{lem:families-wall-crossing}, its large-parameter unmarked
-family invariant is $\pm1$. For each sufficiently large $r$, take
+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 $v_r\to v$, and apply
-\Cref{lem:families-taubes-compactness}. The limit is a nonempty
+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 $E\perp P$.
+$J_v$. 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
+Curve existence (Cref{thm:family-curves}) is available for every
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
@@ -1922,131 +1922,131 @@ direction of $F^+$.
## The lattice and the chamber class
Let
-\[
- Λ=H^2(X;\Z)/torsion,\qquad
- Λ_{\Q}=Λ ⊗ _{\Z}\Q,\qquad
- Λ_{\R}=Λ ⊗ _{\Z}\R.
-\]
+[
+ Λ=H^2(X;Z)/torsion,qquad
+ Λ_{Q}=Λ ⊗ _{Z}Q,qquad
+ Λ_{R}=Λ ⊗ _{Z}R.
+]
A curve class is the image in $Λ$ of the integral
-Poincar\'e dual of a closed oriented curve, with its complex
+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 $d\inΛ_{\R}$ to $P$.
+$d^+$ denotes the orthogonal projection of $dinΛ_{R}$ to $P$.
[Choice of chamber class]
-There is a primitive class $F\inΛ$ with $F^2=0$ such that the
-following implication holds for every $d\inΛ$ and every real
+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^2\ge-2,\qquad d^+=cF^+,\qquad d\cdot F\le0
- \quad⇒\quad
- c\in\Z_{>0},\qquad d\cdot F=0.
+ d^2ge-2,qquad d^+=cF^+,qquad dcdot Fle0
+ quad⇒quad
+ cinZ_{>0},qquad 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 $\dim P=3$,
+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 $F_0\inΛ$ is indefinite,
+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 $l\in L$ be a primitive vector
-with $l\cdot l=0$ and such that $L_{\R}=l^\perp\cap L_{\R} ⊕ \R l$.
+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
-\[
- \mathcal F=\{F\in L\capΛ:\ F^2=0,\ F\hbox{ primitive},\
- F\cdot l=1\}.
-\]
-Each $F\in\mathcal F$ is a nonzero null vector and belongs to $L$. Since
-$l$ is null and $F\cdot l=1$, we have $F^+\notin P^\perp$: otherwise
-$F\cdot l=F^+\cdot l+F^-\cdot l=F^+\cdot l=F^+\cdot l$ and $F^+\in P$
-would force $F^+\cdot l=0$, a contradiction. In particular $F^+\ne0$.
-The set $\mathcal F$ is infinite by completing any rational null vector
+[
+ 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 $F\in\mathcal F$
-and suppose $d\inΛ$, $c>0$ satisfy the three left-hand conditions
+noindentStep~2: projecting and pinning $d^+$. Fix $Finmathcal F$calF$
+and suppose $dinΛ$, $c>0$ satisfy the three left-hand conditions
of (eq:lattice-chamber-implication). Then
-\[
- r:=d-cF\in P^\perp,\qquad r^2=d^2-2c(d\cdot F)\ge-2,
-\]
-because $F\cdot F=0$, so $r$ lies in the compact set
-\[
- K=\{r\in P^\perp:r^2\ge-2\}.
-\]
-
-\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
+[
+ r:=d-cFin P^perp,qquad r^2=d^2-2c(dcdot F)ge-2,
+]
+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
+$ker(q|_Λ)=L$. 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
+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
+[
+ 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
-\[
- \mathbb T=P/\pi(L),
-\]
+[
+ mathbb T=P/bbT=P/pi(L),
+]
because adding an element of $L$ changes the projection by an element of
-$\pi(L)$. For $F\in\mathcal F$ define the circle
-\[
- \mathbb S_F=\{t\pi(F)+\pi(L):t\in\R\}\subset\mathbb T.
-\]
+$pi(L)$. For $Finmathcal F$calF$ 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 $t\pi(F)-s\pi(F')\in\pi(L)$ with $F,F'$ not collinear, choose
+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 $s\in\Q$, so the common point
-is torsion. In particular a non-torsion point of $\mathbb T$ lies on at
-most one circle $\mathbb S_F$.
+$F'$; 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 $\mathcal F$ have distinct directions: two collinear
-elements with $l$-pairing one are equal. Since $\mathcal F$ is infinite,
-we can choose $F\in\mathcal F$ whose circle contains none of the
+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
non-torsion points among the finitely many possible coset points.
-\noindentStep~5: integrality and vanishing. For this $F$, every
+noindentStep~5: integrality and vanishing. For this $F$, every
$d,c$ satisfying the hypotheses gives a torsion point
-\[
- d^++\pi(L)=c\pi(F)+\pi(L).
-\]
-So there is an integer $k>0$ with $kcF\in L$. Pairing with $l$ shows
-$kc\in\Z$, so $c\in\Q$. It follows that $r=d-cF\in P^\perp\capΛ_\Q$,
-and the lattice property of $P^\perp$ now gives
+[
+ 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
- c=l\cdot d\in\Z_{>0}.
+ c=lcdot dinZ_{>0}.
-If $m=d\cdot F<0$, then $m$ is a negative integer and
-\[
- r^2=d^2-2cm\ge-2+2c\ge0.
-\]
-Negative definiteness of $P^\perp$ forces $r=0$, which would imply
-$m=0$. Contradiction; hence $d\cdot F=0$, and the implication is proved.
+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.
## 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
- [B]=(F^+)/(\lVert F^+\rVert),\qquad
- [A]\cdot F=[C]\cdot F=0,\qquad
- \beta=[B]\cdot F=\lVert F^+\rVert>0.
+ [B]=(F^+)/(lVert F^+rVert),qquad
+ [A]cdot F=[C]cdot F=0,qquad
+ beta=[B]cdot F=lVert F^+rVert>0.
Here the norm is the positive intersection norm on $P$. The three
-classes remain orthonormal, and $V=[A]^\perp\cap[C]^\perp$ has
+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.
@@ -2055,59 +2055,59 @@ $F$ and the original period plane $P$, independent of any later pair.
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 $d\inΛ$ (identified with its Poincar\'e dual), let
-$m=F\cdot d$ and write $[B]\cdot d=\beta c$ with the constant
-$\beta=[B]\cdot F>0$. Then:
-[label=(\roman*)]
-\item $m\ge0$.
-\item $m=0$ if and only if $d=cF$ for some real $c$ (in which case
-$c\in\Z$, and $c>0$ if $d\ne0$).
+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 $dne0$).
Proof.
-By \Cref{lem:definite-pair}, the definite pair trivializes the
+By Cref{lem:definite-pair}, the definite pair trivializes 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
+cite[Theorem~2.8]{Wendl2020} give
- d^2=2g(Σ)-2+2\delta(u),\qquad \delta(u)\in\Z_{\ge0},
+ d^2=2g(Σ)-2+2delta(u),qquad delta(u)inZ_{ge0},
-so in particular $d^2\ge-2$.
+so in particular $d^2ge-2$.
Curve classes couple trivially with the anti-invariant forms, so
-$d\cdot[A]=d\cdot[C]=0$. Equivalently $d\in V=[A]^\perp\cap[C]^\perp$.
-Since $V\cap P=\R[B]$ (both spaces contain $[B]$ and $V$ is the
+$dcdot[A]=dcdot[C]=0$. 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 $c'\in\R$. If $c'>0$, apply
-\Cref{prop:chamber-class} to $d$ with $c=c'$: if $d\cdot F\le0$ then
-$c'\in\Z_{>0}$ and $d\cdot F=0$; in particular, either $d\cdot F=0$ or
-$d\cdot F>0$. If $c'<0$, apply to $-d$ and $-c'$: same conclusion, and
-the sign of $d\cdot F$ is unchanged. If $c'=0$, then $d^+=0$; by
-adjunction $d^2\ge-2$ and by negativity of $P^\perp$, $d\cdot F=d^-\cdot
-F^-\le0$, so the implication traps $d\cdot F=0$ (indeed $d\cdot F\le0$
-would give $c'\in\Z_{>0}$, which $c'=0$ violates; a direct inspection
-shows $d\cdot F\ge0$). In all cases $m\ge0$.
-
-\noindent(ii). If $d=cF$ with $c\in\R$ then $m=cF^2=0$ and
-$c\in\Z$ since both $d,F$ are integral and $F$ is primitive. Conversely
-assume $m=0$. From $d\in V$ and $m=d\cdot F=0$, the class $d$ is
-orthogonal to the nondegenerate plane $span([B],F)\subset
+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
-$d^2\le0$, and adjunction bounds $d^2\ge-2$. If $d^+\ne0$, then
-$d^+=c'F^+$ with $c'\ne0$; \Cref{prop:chamber-class} forces $c'\in\Z$,
-and then $d^2\le0$ combined with $0=m=d^+F^+ + d^-F^-$ yields
+$d^2le0$, 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
+$d^pm$ 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 $d\cdot[B]=\beta c'>0$) is a positive
-multiple of $F$; hence $d=cF$ for some $c\in\Z_{>0}$. If $d^+=0$, then
-$d\in P^\perp$ and $d^2\le0$; adjunction gives $-2\le d^2\le0$, and
-$m=d\cdot F=d^-F^-\le0$ 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
@@ -2115,47 +2115,47 @@ being spacelike with intersection $\ge-1$), so $d=0=cF$ with $c=0$.
[Nonsplitting]
For any definite pair with the adapted periods (eq:lattice-adapted-periods),
let $J$ be the almost-complex structure tamed by $B$. Then:
-[label=(\roman*)]
-\item Every positive $J$-holomorphic configuration of total class $F$
+[label=(roman*)]
+item 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
+item 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
+item Each such curve is either an embedded torus or an immersed rational
curve with exactly one transverse positive double point.
Proof.
For (i), write a positive configuration as
-\[
- F=\sum_{i=1}^rn_id_i\quad\hbox{in }Λ,\qquad n_i\in\Z_{>0},
-\]
+[
+ F=sum_{i=1}^rn_id_iquadhbox{in }Λ,qquad n_iinZ_{>0},
+]
where $d_i$ are the classes of its distinct irreducible images. Taming
-excludes zero real component classes. By \Cref{cor:curve-signs},
-$m_i=d_i\cdot F\ge0$, 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 $c_i\in\Z_{>0}$, giving
-\[
- \sum_ic_in_i=1.
-\]
+excludes zero real component classes. By Cref{cor:curve-signs},
+$m_i=d_icdot Fge0$, 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
+[
+ sum_ic_in_i=1.
+]
There is exactly one summand, with $n_1=c_1=1$.
-For (ii), apply \Cref{thm:family-curves}. Its period plane is
+For (ii), apply Cref{thm:family-curves}. Its period plane is
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
+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
simple images would contribute positively.
-For (iii), adjunction becomes $0=F^2=2g(Σ)-2+2\delta(u)$,
-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
@@ -2165,24 +2165,24 @@ 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, $F^2=0$,
+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 $a_t,c_t$, $0\le t\le1$, 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,qquad 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
smooth proper map
-\[
+[
p_X:X→ S
-\]
+]
to a closed connected oriented surface. Each regular fiber is an
embedded torus. The singular fibers, if any, are finitely many immersed
rational curves, each with one transverse positive double point.
@@ -2190,20 +2190,20 @@ rational curves, each with one transverse positive double point.
## Curves of class $F$
-Fix any triple used below. \Cref{prop:nonsplitting} gives a curve of
+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
distinct such images are disjoint: any intersection would contribute
-positively to $F^2=0$. A simple parametrization $u:L\to X$ has by
+positively to $F^2=0$. A simple parametrization $u:Lto X$ has by
adjunction
- g(L)+\delta(u)=1.
+ g(L)+delta(u)=1.
Consequently it is either an embedded torus or a rational curve of
defect one. The latter may initially have a critical point. The
singularity theorem for a simple curve makes its critical and
noninjective points a finite set on the normalization
-\cite[Propositions~B.26 and~B.41 and Theorem~2.3]{Wendl2020}. If it is
+cite[Propositions~B.26 and~B.41 and Theorem~2.3]{Wendl2020}. If it is
immersed, (eq:pencil-adjunction) shows that it has exactly one
double point, and the two branches meet transversely.
@@ -2250,7 +2250,7 @@ parametrized moduli calculation yields a Fredholm projection of
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
@@ -2261,16 +2261,16 @@ proves the lemma. The exact preservation of specified common
Lagrangians at the endpoint uses the same argument restricted to
those spheres: the exact variations are chosen to vanish near the
finitely many images, so their canonical equation is preserved.
-\qed
+qed
## The normal equation and foliation by tori
[The normal operator]
-Let $u:L\to X$ be a closed connected common Lagrangian immersion for a
+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
+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.
@@ -2285,50 +2285,50 @@ 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 $w\mapsto\alpha=u^*\iota_wA$. Write,
+normal bundle with $T^*L$ by $wmapstoalpha=u^*iota_wA$. Write,
along the source,
- H=(C-xA)/(y),\quad Ω=A+iH,\quad
- x=(A\wedge C)/(A^2),\quad y^2=(C^2)/(A^2)-x^2,
+ H=(C-xA)/(y),quad Ω=A+iH,quad
+ x=(Awedge C)/(A^2),quad y^2=(C^2)/(A^2)-x^2,
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 $u^*\iota_wC=x\alpha+y*_L\alpha$; Cartan's
+complex volume identity gives $u^*iota_wC=xalpha+y*_Lalpha$; Cartan's
formula then gives the normal derivative
- \mathcal D\alpha=(\dd\alpha,\dd(x\alpha+y*_L\alpha)).
+ mathcal DcalDalpha=(ddalpha,dd(xalpha+y*_Lalpha)).
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
-$\dd\alpha_0=\zeta_1$. 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 P f=\dd(x\dd f+y*_L\dd f)
- =\dd x\wedge\dd f+\dd(y*_L\dd f).
+ 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 $y\Delta f$ of the scalar Laplacian. The sign
+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
+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
is constant along this homotopy
-\cite[Section~7, Propositions~7.3--7.4]{TaylorFunctionalNotes}. Its
+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 $\mathcal P f=\zeta_2-\dd(x\alpha_0+y*_L\alpha_0)$,
+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.
+right inverses in Sobolev or H"older spaces.
-The kernel of $\mathcal D$ is thus $\{\alpha:\dd\alpha=0\}$ together
+The kernel of $mathcal D$calD$ 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 $\mathcal D$ is therefore isomorphic to
-$H^1(L;\R)$, of dimension $b_1(L)$. The linearization is Fredholm of
+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
diffeomorphism onto an open set of common Lagrangian immersions.
@@ -2355,8 +2355,8 @@ 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
-\cite[Theorems~5.3.1 and~5.5.5]{McDuffSalamon2012}. Any limit could
-contain multiple components, but \Cref{prop:nonsplitting} rules out
+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.
@@ -2369,7 +2369,7 @@ periods and with all $F$-spheres immersed, their number is constant.
Proof.
The unparametrized sphere space is compact and discrete, so finite.
Over a path, the total space is compact and (by the parameter
-version of \Cref{lem:pencil-normal}) the projection to the path
+version of Cref{lem:pencil-normal}) the projection to the path
interval is a local diffeomorphism, also in relative half-interval
charts at the endpoints. Hence a finite covering of the interval.
@@ -2391,8 +2391,8 @@ many such spheres.
Proof. [Proof sketch]
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
+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
@@ -2413,33 +2413,33 @@ preserved by choosing the primitives with zero pullback on $K$.
## The pencil
-Proof. [Proof of \Cref{thm:pencil}]
+Proof. [Proof of Cref{thm:pencil}]
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-jets}) and the number of these spheres is finite (by
+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.
-Apply the path version of \Cref{lem:pencil-jets}, keeping its
+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 $B$
-fixed. By \Cref{cor:pencil-node-count}, its endpoints have the same
+fixed. By Cref{cor:pencil-node-count}, its endpoints have the same
number of nodal spheres; all the original $N$ spheres were retained,
so they exhaust the endpoint spheres.
-Define $p_X:X\to S$ 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.
+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 $\Delta\to\Delta$ gives the base coordinate. The disk
+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
@@ -2457,9 +2457,9 @@ model identities hold on the unchanged smaller neighborhoods.
# The collapsing nodal model
-The pencil of \Cref{thm:pencil} has ordinary nodal fibers. In this
+The pencil of Cref{thm:pencil} has ordinary nodal fibers. In this
section we build, on a neighborhood of one such fiber, a complete local
-K\"ahler model whose forms satisfy the pointwise equal-square
+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
@@ -2468,102 +2468,102 @@ fiber.
## The Gibbons--Hawking ansatz
-Fix a positive real parameter $\rho>0$ and a positive smooth function
-$V_\rho$ on a ball $B_R(0)\subset\R^3$ 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
+produces a positive definite hyperk"ahler metric on the total space of
+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
- \dd\theta=-\ast_{\R^3}\dd V_\rho,
+ 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
circle coordinate,
- g_\rho&=V_\rho(dx_1^2+dx_2^2+dx_3^2)+V_\rho^{-1}\theta^2,\\
- \gamma_i&=dx_i\wedge\theta+V_\rho\,dx_j\wedge dx_k.
+ g_rho&=V_rho(dx_1^2+dx_2^2+dx_3^2)+V_rho^{-1}theta^2,
+ gamma_i&=dx_iwedgetheta+V_rho,dx_jwedge dx_k.
The signs are chosen from
-$\dd(dx_i\wedge\theta)=-∂_iV_\rho\,dx_1\wedge dx_2\wedge dx_3$
-and $\dd(V_\rho\,dx_j\wedge dx_k)=∂_iV_\rho\,dx_1\wedge dx_2\wedge dx_3$.
+$dd(dx_iwedgetheta)=-∂_iV_rho,dx_1wedge dx_2wedge dx_3$
+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
frame signs, the equations for closedness are
-\[
- I_2a_3-I_3a_2=0,\qquad
- I_3a_1-I_1a_3=0,\qquad
+[
+ I_2a_3-I_3a_2=0,qquad
+ I_3a_1-I_1a_3=0,qquad
I_1a_2-I_2a_1=0,
-\]
-with $I_i\alpha=\ast(\alpha\wedge\gamma_i)$ and signs chosen so that
+]
+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+i\gamma_2=ds\wedge(\theta-iV_\rho\,\dd t),
+ 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
-K\"ahler. This kernel is involutive: closedness and Cartan's formula
+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
-$V_\rho(s,t)=V_0(s)+1/(4\pi|X|)$ with $V_0$ a small positive constant
+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
smooth because the Hopf chart of the charge-one instanton completes
-$W_m\,d\mathbf X^2+W_m^{-1}\theta_m^2$ to a flat smooth ball. In that
-chart $\mathbf X$ and $r$ are smooth quadratic expressions, and
-$r\theta_m$ is smooth. The coefficient $1/(4\pi)$, 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 $mathbf X$bfX$ 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}=(4\pi r)/(1+4\pi rH_\rho),\qquad
- \frac{W^{-1}-W_m^{-1}}{r^2}
- =-((4\pi)^2H_\rho)/(1+4\pi rH_\rho),
+ W^{-1}=(4pi r)/(1+4pi rH_rho),qquad
+ frac{W^{-1}-W_m^{-1}}{r^2}
+ =-((4pi)^2H_rho)/(1+4pi rH_rho),
-so $g_\rho/\rho=W\,d\mathbf X^2+W^{-1}(\theta_m+\beta)^2$ 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
-$\gamma_i/\rho=dX_i\wedge(\theta_m+\beta)+W\,dX_j\wedge dX_k$ 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 $h\in\C∖\{0\}$, put $Σ=h\,\dd s\wedge\dd z$, and fix
-$0<r<R$. Write $s=x_1+ix_2$ and $z=a+\tau(s)b$ modulo the period
+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
+$D_eps^{loc}=|h|,gamma_3$ 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})^2=(ReΣ)^2,\\
- \int_{p^{-1}(s)}D_\eps^{loc}=ε\quad(s\ne0),
+ D_eps^{loc}wedgeReΣ
+ =D_eps^{loc}wedgeImΣ=0,qquad
+ (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.
@@ -2571,34 +2571,34 @@ scaling $|h|$ that we are using.
## Annular comparison
-Fix $h\in\C∖\{0\}$ 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
-$K=\{r_-\le|s|\le r_+\}\subset\Delta_R∖\{0\}$, the form
+[label=(roman*)]
+item On each fixed closed annulus
+$K={r_-le|s|le r_+}subsetDelta_R∖{0}$, the form
- D_\eps^{sf}
- =ε\,\dd a\wedge\dd b+
- (|h|^2)/(\eps)\tau_2(s)\,\dd x_1\wedge\dd x_2,
- \qquad s=x_1+ix_2,
+ D_eps^{sf}
+ =ε,dd awedgedd b+
+ (|h|^2)/(eps)tau_2(s),dd x_1wedgedd x_2,
+ qquad s=x_1+ix_2,
-is well-defined and closed. For every integer $m\ge0$, 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.
+ |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
+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
swept by the primitive connection cycle.
-\item The following polynomial estimates hold on bounded branches:
-$g_\eps/\eps$ has polynomial bounds in $(1+|\log\eps|)$ and $\eps^{-1}$
+item 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
+item 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.
@@ -2607,40 +2607,40 @@ 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:
-$ε\,\dd a\wedge\dd b$ gives the correct fiber integral, and
-the second term $(|h|^2)/(\eps)\tau_2(s)\dd x_1\wedge\dd x_2$
+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 $\sqrt\rho$,
-and $\rho=\eps/|h|\to0$) gives
+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 $p:Y_\rho\to\Delta_R$
-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 $\tau_2(s)=Im\tau_\rho(s)$ 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
fiber, and Grauert's base-change theorem
-\cite[Section~7, Satz~5]{Grauert1960} and its correction
+cite[Section~7, Satz~5]{Grauert1960} and its correction
[cite] then identifies it with a fixed family of plane
cubics. The Abel coordinate makes the periods intrinsic to the
model.
@@ -2648,49 +2648,49 @@ 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 $L=\mathcal L_\rho$ be the relative polarization on
-$p:Y_\rho\to\Delta_R$. The nodal fiber $N=p^{-1}(0)$ is the image of
-a nodal plane cubic in $\mathbb P^2$. Its normalization
-$\nu:\mathbb P^1\to N$ is a rational curve, and the pullback
-$\nu^*L$ is $\mathcal O_{\mathbb P^1}(3)$ 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
-\[
+[
0→ L→
- \nu_*\mathcal O_{\mathbb P^1}(3)
- \xrightarrow{ev_0-\lambdaev_∞}
- \C_{node}→0,
-\]
-with $\lambda$ the appropriate modular parameter of the nodal cubic.
+ nu_*mathcal O_calO_{mathbb P^1}bbP^1(3)
+ xrightarrow{ev_0-lambdaev_∞}
+ C_{node}→0,
+]
+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 $\nu_*\mathcal O_{\mathbb P^1}(3)$ is
-that of $\mathcal O_{\mathbb P^1}(3)$. In particular $h^0(N,L)=3$ and
+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 $\mathcal L_\rho$
+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
-$E_\rho=(p_\rho)_*\mathcal L_\rho$ is a holomorphic rank-three bundle,
+$E_rho=(p_rho)_*mathcal L_calL_rho$ 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 $\mathbb P(E_\rho^*)$. At the nodal
+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 $p:X\to S$
+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}.
+the local holomorphic model of Cref{thm:nodal-model}.
-Write $S^\circ=S∖\Delta$ and $X^\circ=p^{-1}(S^\circ)$, where
-$\Delta\subset S$ is the finite set of critical values. The class $F$
+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
@@ -2699,34 +2699,34 @@ The regular torus fibration is a torsor: there is no global
origin, but the natural translations act transitively on each fiber.
[Torus action]
-If $\alpha_s\in T_s^*S^\circ$, the equation
-$\iota_{Y_\alpha}A_0=-p^*\alpha$ defines a unique vertical vector
+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
+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
-$\mathcal L_s$. 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
+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
+ $dd(iota_{Y_f}A_0)=-p^*dd f$ 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 $s\mapsto\sigma(s)$ of
+To identify the lattice, use a local section $smapstosigma(s)$ of
$p$; each element of the stabilizer satisfies
-$\int_{F_s}\alpha\in\mathcal L_s$ for a small piecewise-smooth disk
+$int_{F_s}alphainmathcal L_s$calL_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.
@@ -2734,7 +2734,7 @@ structure-preserving translations.
[Invariant smooth pairs]
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 +2743,12 @@ 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 $\dd\beta$ 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.
@@ -2757,71 +2757,71 @@ when the input is closed.
The main statement of this section is the following.
-Let $(A_0,C_0,B_0)$ and $p\colon X\to S$ 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 $U_j$
-about the critical values, use the model of \Cref{thm:nodal-model}
+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 $U_j$
+about the critical values, use the model of Cref{thm:nodal-model}
with
- Σ_j=h_j\,\dd s\wedge\dd z,\quad
- A_0=ReΣ_j,\quad
+ Σ_j=h_j,dd swedgedd z,quad
+ A_0=ReΣ_j,quad
C_0=x_jReΣ_j+y_jImΣ_j,
- \quad y_j\ne0,
+ quad y_jne0,
-where $h_j\ne0$, $x_j$, $y_j$ 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 $U'_j\Subset U_j$ and a smooth path of
-positive closed triples $(A_t,C_t,B_t)$, $0\le t\le1$, starting at
+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 $A_t,C_t$
+[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 $X^\circ$
+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 $X^circ$
with
- \int_{p^{-1}(s)}\Psi=1,\qquad
- \Psi\wedge A=\Psi\wedge C=0\quadon X^\circ.
+ int_{p^{-1}(s)}Psi=1,qquad
+ Psiwedge A=Psiwedge C=0quadon X^circ.
-On each punctured $p^{-1}(U'_j∖\{s_j\})$ it equals
-$\dd a\wedge\dd b$, 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 $A^a=Av(A_0)$
+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
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 $C^a=C_0$.
-\noindentStep 2: the positive cone along the interpolation.
+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
-\[
- A_\lambda=A_0+\lambda(A^a-A_0),\qquad
- C_\lambda=C_0+\lambda(C^a-C_0),\qquad 0\le\lambda\le1.
-\]
-At a point of $X^\circ$ with $p(x)=s$, the fibers are Lagrangian for
+[
+ A_lambda=A_0+lambda(A^a-A_0),qquad
+ C_lambda=C_0+lambda(C^a-C_0),qquad 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
spanned by the horizontal piece.
-By \Cref{lem:regular-mixed-cone} below, the set of forms $C$ that are
+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.
@@ -2829,75 +2829,75 @@ Since the disk models do not change and are already invariant, and
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
+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
+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
-\[
- \Psi^{(0)}=Av(B_0/q_0)\quadon X^\circ.
-\]
+[
+ 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 $\dd a\wedge\dd b$ 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 $\dd a\wedge\dd b$. Averaging does not change the periods by
-\Cref{lem:regular-averaging}. Both closed forms with the same periods
+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
punctured mapping torus is generated by the fiber and the sweep, the
-same holds after subtracting a constant multiple of $\dd a\wedge\dd b$
-if necessary. Replacing $\Psi^{(0)}$ by $\Psi^{(0)}$ minus a small
-exact two-form $\dd\beta$ with $\beta$ a one-form supported in
-$p^{-1}(U_j∖\{s_j\})$, we can make
-$\Psi^{(0)}=\dd a\wedge\dd b$ 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
-$\dd a\wedge\dd b$ near the punctures. The integrals
-\[
- I_A=\int_{X^\circ}\Psi^{(0)}\wedge A,\quad
- I_C=\int_{X^\circ}\Psi^{(0)}\wedge C
-\]
+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_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
+$dd$ 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
+$widetilde A,widetilde C$ is the cohomology Gram matrix, i.e. the
identity under our normalization. Set
- \Psi=\Psi^{(0)}-I_A\widetilde A-I_C\widetilde C.
+ Psi=Psi^{(0)}-I_Awidetilde A-I_Cwidetilde C.
-Then $\Psi$ is closed, invariant, of unit fiber area, and equal to
-$\dd a\wedge\dd b$ 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}\Psi\wedge A=\int_{X^\circ}\Psi\wedge C=0.
+ int_{X^circ}Psiwedge A=int_{X^circ}Psiwedge C=0.
-\noindentStep 5: assembling the path. Concatenate the
+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
-$B_t\in[B_0]$, using \Cref{lem:smooth-tamers} with $B_0$ at the
+$B_tin[B_0]$, 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.
@@ -2911,24 +2911,24 @@ 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).
+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
+$L^perp$ 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
+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.
+$L^perp$ (i.e., is pulled back from $L^perp$) does not change the
+$L^perp$ component.
[Smooth tamers]
-Let $J_t$, $0\le t\le1$, be a smooth family of almost-complex
-structures on a closed four-manifold $X$, $H\in H^2(X;\R)$ 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
-$B_0$, $B_1\in H$ prescribed tamer representatives at the endpoints.
-Then there is a smooth family $B_t\in H$ with $B_t$ taming $J_t$ for
+$B_0$, $B_1in H$ 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 $t=0,1$.
@@ -2946,71 +2946,71 @@ $J_t$ interpolates), the interpolation is nowhere zero.
# The auxiliary volume equation
-The prepared pair of \Cref{prop:regular-preparation} satisfies
-$A^2=C^2=1$, $A\wedge C=0$ on $X$, and the restrictions of $A,C$ to
-the fibers vanish, with unit fiber area form $\Psi$ orthogonal to both
-and equal to $\dd a\wedge\dd b$ near the nodes. We now construct a
-closed two-form $D_\eps$ with $A\wedge D_\eps=C\wedge D_\eps=0$ and
-$D_\eps^2=A^2$ pointwise. The correction to a semi-flat approximate
+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
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).
+ 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 a\wedge\dd b
- +(\tau_2(s))/(\eps)\,\dd x_1\wedge\dd x_2,
- \qquad s=x_1+ix_2,
+ E_eps^{sf}=eps,dd awedgedd b
+ +(tau_2(s))/(eps),dd x_1wedgedd x_2,
+ qquad s=x_1+ix_2,
-with the volume form $\nu=A^2/2$ and $\tau_2=Im\tau$.
-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
+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
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
+$D_eps^{loc}$ 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 $E_\eps^{sf}$
+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
+[label=(roman*)]
+item
+$E_eps$ 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 $U'_j$.
+item
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 $m\ge0$ (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
+item
The three wedge residuals
-\[
- \mathcal R_\eps=
- (-A\wedge E_\eps,\,
- -C\wedge E_\eps,\,
- (A^2-E_\eps^2)/(2))
-\]
+[
+ mathcal R_calR_eps=
+ (-Awedge E_eps,,
+ -Cwedge E_eps,,
+ (A^2-E_eps^2)/(2))
+]
satisfy
- \|\mathcal R_\eps\|_{H^k}
- \le C\eps^{-s}e^{-c/\eps}
+ |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
fixed construction data.
@@ -3018,43 +3018,43 @@ 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:
-$\dd a\wedge\dd b$ wedges trivially with $A$ and $C$ because
+$dd awedgedd b$ 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 $\dd x_1\wedge\dd x_2$ component while $\Psi$ has only
-$\dd a\wedge\dd b$.
+$(E_eps^{sf})^2=A^2$ 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
(eq:nodal-exact-identities).
On each annulus, use a fixed bounded primitive construction. Pick a
radial trivialization of the annular model torus bundle
-$M\times[r_0,r_1]$. 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
-\[
- \zeta=\pi_M^*(\zeta|_{M\times\{r_*\}})+\dd K\zeta,
- \qquad
- K\zeta(r)=\int_{r_*}^{r}\iota_{∂_r}\zeta(u)\,\dd u.
-\]
+[
+ 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
-$K\zeta$ 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 $\eps$.
+$Kzeta$ 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_\eps\wedge A=O(e^{-c/\eps}),\quad
- b_\eps:=\int_XE_\eps\wedge C=O(e^{-c/\eps}),
+ a_eps:=int_XE_epswedge A=O(e^{-c/eps}),quad
+ b_eps:=int_XE_epswedge C=O(e^{-c/eps}),
which hold because the exact gluing adjusted the two pairings. The
-remaining residual $\mathcal R_\eps$ 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.
@@ -3063,57 +3063,57 @@ fixed chart cover.
Set
- \widehat E_\eps&=E_\eps-a_\eps A-b_\eps C,&
- q_\eps&=\int_X\widehat E_\eps^2,\notag\\
- r_\eps&=q_\eps^{-1/2},&
- D^0_\eps&=r_\eps\widehat E_\eps.
+ widehatwidehat E_E_eps&=E_eps-a_eps A-b_eps C,&
+ q_eps&=int_Xwidehatwidehat E_E_eps^2,notag
+ r_eps&=q_eps^{-1/2},&
+ D^0_eps&=r_epswidehatwidehat E_E_eps.
-The period normalization gives $\int A^2=\int C^2=1$ and
-$\int A\wedge C=0$. So
+The period normalization gives $int A^2=int C^2=1$ and
+$int Awedge C=0$. So
- \int_XA\wedge D^0_\eps=\int_XC\wedge D^0_\eps=0,\qquad
- \int_X(D^0_\eps)^2=1.
+ int_XAwedge D^0_eps=int_XCwedge D^0_eps=0,qquad
+ 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 $r_\eps=1+O(e^{-c/\eps})$.
+$a_eps,b_eps=O(e^{-c/eps})$ 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).
+ 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 $k\ge0$, the following hold for sufficiently small
-$\eps$:
-[label=(\roman*)]
-\item
-The metric $g_\eps$ has a finite coordinate cover whose cardinality,
+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,
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
+$eps^{-1}$.
+item
+The frame $(A,C,D^0_eps)$ and the inverse of its pointwise Gram
matrix have polynomial intrinsic derivative bounds through order $k$.
The Gram matrix itself is uniformly positive and bounded.
-\item
-Sobolev embedding and multiplication at a fixed large order $k\ge6$
+item
+Sobolev embedding and multiplication at a fixed large order $kge6$
have polynomial constants: there are $C,q>0$ with
- \|h\wedge\widetilde h\|_{H^k}\le C\eps^{-q}\|h\|_{H^k}\|\widetilde h\|_{H^k}
+ |hwedgewidetilde h|_{H^k}le Ceps^{-q}|h|_{H^k}|widetilde h|_{H^k}
-for two-forms $h,\widetilde h$.
+for two-forms $h,widetilde h$.
@@ -3121,83 +3121,83 @@ 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 $\rho_\eps\ge c_1\eps^{a_1}$. A maximal disjoint collection of
-such balls has at most $(2c_2\eps^{a_2})^{-1}$ 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 $k\ge6$, derivatives of order
-at most $\lfloor k/2\rfloor$ are $L^∞$-bounded by $H^k$. The
-product rule on each derivative of $h\wedge\widetilde h$ 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 $k\ge6$ and every sufficiently small $\eps$, there is
+For every fixed $kge6$ and every sufficiently small $eps$, there is
a bounded linear map
-\[
- \mathcal L_\eps:\ (t_1,t_2,t_3)\in H^k\times H^k\times H^k
- \mapsto h\in H^k
-\]
+[
+ mathcal L_calL_eps: (t_1,t_2,t_3)in H^ktimes H^ktimes H^k
+ mapsto hin H^k
+]
such that
-[label=(\roman*)]
-\item
-The form $h$ is exact: $\int_Xh\wedge\varphi=0$ for every closed
-$\varphi$; equivalently its harmonic projection is zero.
-\item
+[label=(roman*)]
+item
+The form $h$ is exact: $int_Xhwedgevarphi=0$ for every closed
+$varphi$; equivalently its harmonic projection is zero.
+item
$h$ solves
-\[
+[
(< h,A>_{L^2},
< h,C>_{L^2},
- \int_Xh\wedge h)=(t_1,t_2,t_3),
-\]
-in the sense that $h$ is the unique element of $\mathcal E^k_\eps$
+ 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
satisfies
-\[
- \|h\|_{H^k}\le C\eps^{-q}\sup_i\|t_i\|_{H^k}.
-\]
+[
+ |h|_{H^k}le Ceps^{-q}sup_i|t_i|_{H^k}.
+]
Proof. [Proof sketch]
-The self-dual part $\phi=h^+$ is uniquely specified by requiring
-$f_i=t_i\nu$; 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.
+ phi=sum_{j=1}^3(M^{-1}t)_jE_j,
+ qquad E_1=A, E_2=C, E_3=D^0_eps.
Moreover
- <\phi,E_i>_{L^2}=\int_Xf_i=t_i,
+ <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
(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, $G\phi$ is self-dual with
-$\Delta G\phi=\phi$. For a self-dual two-form $\psi$,
-$2(\dd\dd^*\psi)^+=\Delta\psi$, 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=2\dd\dd^*G\phi
+ 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}).
+$g_eps$ (polynomial in $eps^{-1}$ by Cref{lem:auxiliary-geometry}).
The remaining moments in $h$ are matched by adding
-$\dd\delta$ to $h$ for $\delta$ a one-form, which does not change $h^+$
+$dddelta$ 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 stated Sobolev bound follows from the elliptic estimate in the
@@ -3209,52 +3209,52 @@ primitive.
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
+$eps>0$, a smooth closed two-form $D_eps=D^0_eps+h_eps$, exact in
+the class of $D^0_eps$, such that
- A\wedge D_\eps=C\wedge D_\eps=0,\qquad (D_\eps)^2=A^2.
+ Awedge D_eps=Cwedge D_eps=0,qquad (D_eps)^2=A^2.
-The associated triple $(A,C,D_\eps)$ is positive; $h_\eps$ is
-bounded by $C\eps^{-N}e^{-c/\eps}$ 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 $k\ge6$. Define the closed subspace
-$\mathcal E^k_\eps\subset H^k$ of exact two-forms. Writing
-$D_\eps=D^0_\eps+h$ with $h\in\mathcal E^k_\eps$, 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_\eps(h):=\mathcal L_\eps
- (-A\wedge D^0_\eps,\,
- -C\wedge D^0_\eps,\,
- (A^2-(D^0_\eps)^2-h\wedge h)/(2)).
+ h=mathcal T_calT_eps(h):=mathcal L_calL_eps
+ (-Awedge D^0_eps,,
+ -Cwedge D^0_eps,,
+ (A^2-(D^0_eps)^2-hwedge h)/(2)).
-This map is defined on all of $\mathcal E^k_\eps$: 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,
-$\int_Xh\wedge h=0$ 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 $\mathcal T_\eps$ is a contraction on a small ball of
-$\mathcal E^k_\eps$: 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
-$h\wedge h$ is a contraction with a factor $\le1/2$ on a ball of
-radius $C\eps^{-N}e^{-c/\eps}$. The Banach fixed-point theorem gives
-a unique smooth $h_\eps$, and
+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
+(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
(eq:auxiliary-exact-equations) holds exactly.
-To see spatial smoothness, choose a Coulomb primitive $a\in H^{k+1}$
-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
-$\xi\wedge b\wedge A=\xi\wedge b\wedge C=\xi\wedge b\wedge D_\eps=0$
-and $\iota_{\xi^\sharp}b=0$, then $\xi\wedge b$ is in the negative-definite
+$xiwedge bwedge A=xiwedge bwedge C=xiwedge bwedge D_eps=0$
+and $iota_{xi^sharp}b=0$, then $xiwedge b$ is in the negative-definite
wedge-orthogonal complement of the positive three-plane. But
-$(\xi\wedge b)^2=0$ and this complement is negative definite, so
-$\xi\wedge b=0$, 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.
@@ -3262,97 +3262,97 @@ 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
+$D_eps$, 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.
+pairing with $[D_eps]$ is positive.
[Uniform chamber estimate]
Retain the prepared pair $(A,C)$, the torus fibration
-$p:X\to S$ of \Cref{thm:pencil}, and the chamber class $F$ of
-\Cref{prop:chamber-class}. There is a positive constant $\eps_0$
+$p:Xto S$ 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 $[D_\eps]$
-of \Cref{thm:auxiliary-solution} satisfies
+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/(2\eps)|\le C,\qquad |u_\eps|\le C,
+ [D_eps]=lambda_eps F+eps u_eps,qquad
+ |lambda_eps-1/(2eps)|le C,qquad |u_eps|le C,
-with $C$ independent of $\eps$, and the following uniform
-implication: for every positive real number $\eps$ and every
-$d\inΛ$ with $d^2\ge-2$ 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
+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
-$\lambda_\eps=1/(2\eps)+O(1)$ 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 $c\in\R$ (in which case $[B]\cdot d =
-c[B]\cdot F=c\beta>0$ 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 $d^2\ge-2$, and \Cref{cor:curve-signs} gives the sign of
-$[B]\cdot d$; the condition $[D_\eps]\cdot d>0$ together with the
+$lambda_eps=1/(2eps)+O(1)$ 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
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
+# The hyperk"ahler endpoint
-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
+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
+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
+$[B]$ 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
+## 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):
-[K\"ahler classes and volumes]
+[K"ahler classes and volumes]
Let $(X,J)$ be a compact complex surface with $b_1$ even, and let
-$H\in H^2(X;\R)$ be a class such that $H^2>0$ and $H$ pairs positively
+$Hin H^2(X;R)$ 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 $B_1\in H$ of
-$H$ with $B_1^2=\int_X\nu$.
+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.
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
+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
-normalization is Yau's theorem [cite]: the K\"ahler
+normalization is Yau's theorem [cite]: the K"ahler
representative can be deformed in its class to have prescribed total
-volume $\int_X\nu$.
+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 $V=[A]^\perp\cap[C_1]^\perp$. Then $[B]$
+$[B]$ be a class in $V=[A]^perpcap[C_1]^perp$. Then $[B]$
belongs to $H^{1,1}(X;J)$.
@@ -3371,94 +3371,94 @@ appropriate orientation.
Recall from the combining theorems:
-\item \Cref{thm:pencil,prop:nonsplitting} give an exact pair
+item 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.
+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.
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
notation
- [A]^2=[C]^2=[B]^2=1,\qquad
+ [A]^2=[C]^2=[B]^2=1,qquad
[A][C]=[A][B]=[C][B]=0.
Fix a sufficiently small $ε>0$ for
-\Cref{thm:auxiliary-solution,prop:auxiliary-class}, and set
+Cref{thm:auxiliary-solution,prop:auxiliary-class}, and set
$D=D_ε$. The form $D$ is closed and smooth and satisfies
- AD=CD=0,\qquad D^2=A^2>0.
+ AD=CD=0,qquad D^2=A^2>0.
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
+Cref{prop:auxiliary-class} for every later definite pair with period
+classes $[A],[C]$. These choices precede the K"ahler steps that
follow.
-## The first K\"ahler step
+## The first K"ahler step
-By \Cref{lem:complex-symplectic}, the pair $(A,D)$ is integrable and
+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
+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
+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
+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, $[B]\in V=[A]^\perp\cap[C_1]^\perp$
-Hodge type $(1,1)$ by \Cref{lem:endpoint-hodge-type}.
+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
-$E_\lambda=C+\lambda(C_1-C)$, $\lambda\in[0,1]$. 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
+$E_lambda=C+lambda(C_1-C)$, $lambdain[0,1]$. 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
-$B_t\in[B]$ of tamers, with $B_0=B$ at endpoint~0.
-
-For the endpoint at $\lambda=1$: the interpolation path joined to
+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.
+
+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
+$(A,E_lambda,B_lambda)$, all positive. This is the first K"ahler
step.
-## The second 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 $B_1\in[B]$ with
+$[B]in H^{1,1}(X;J_1)$ (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
+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 $[A]\cdot[B_1]
+$(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 $(A,C_1)$.
@@ -3469,56 +3469,56 @@ 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 $[0,1]$:
-\[
+[
(A_0,C_0,B_0)
- \xrightarrow{pencil} (A,C,B)
- \xrightarrow{prepared} (A,C,B)
- \xrightarrow{aux.\ D} (A,C,D)
- \xrightarrow{1st K\"ahler} (A,C_1,B)
- \xrightarrow{2nd K\"ahler} (A,C_1,B_1).
-\]
+ xrightarrow{pencil} (A,C,B)
+ xrightarrow{prepared} (A,C,B)
+ xrightarrow{aux. D} (A,C,D)
+ xrightarrow{1st K"ahler} (A,C_1,B)
+ xrightarrow{2nd K"ahler} (A,C_1,B_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
-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
-\[
- \eta_i\wedge\eta_j=2\delta_{ij}\mu_1;
-\]
-the forms are parallel and self-dual for a hyperk\"ahler metric with
+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_iwedgeeta_j=2delta_{ij}mu_1;
+]
+the forms are parallel and self-dual for a hyperk"ahler metric with
this volume. Undoing the initial orthogonal change of coefficients
preserves the displayed identity and restores the original ordered
-classes. This completes the proof of \Cref{thm:main}.
-\qed
+classes. This completes the proof of Cref{thm:main}.
+qed
[t]
-\centering
+centering
[node distance=1.4cm,
- every node/.style={font=\small},
+ 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};
-
- \draw[->] (orig) -- (pencil);
- \draw[->] (pencil) -- (prep);
- \draw[->] (prep) -- (aux);
- \draw[->,dashed] (aux) -- (K1);
- \draw[->] (K1) -- (K2);
- \draw[->,dashed] (aux) -- (K2);
- \draw[->] (K1) -- (end);
- \draw[->] (K2) -- (end);
+ 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);
+ draw[->] (prep) -- (aux);
+ draw[->,dashed] (aux) -- (K1);
+ draw[->] (K1) -- (K2);
+ draw[->,dashed] (aux) -- (K2);
+ draw[->] (K1) -- (end);
+ draw[->] (K2) -- (end);
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
+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 $B_1$.
@@ -3526,268 +3526,268 @@ value $B_1$.
{99}
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