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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
Donaldson's hypersymplectic deformation conjecture.

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tableofcontents

# Introduction

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
normalization of the periods, the natural question is whether one can
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
structure normalized by

 int_Xomega_iwedgeomega_j=delta_{ij}.

There is a smooth family of hypersymplectic structures omega(t),
0le tle1, such that
[
 omega(0)=omega, [omega_i(t)]=[omega_i](i=1,2,3),
]
and a positive volume form mu_1 with

 omega_i(1)wedgeomega_j(1)=2delta_{ij}mu_1.

The endpoint forms are parallel and self-dual for a hyperk"ahler metric.


The normalization (eq:intro-normalization) is a constant linear
change of basis and positive scaling; it makes the endpoint identity
cleaner but does not restrict the geometry. If the initial triple has
an arbitrary positive Gram matrix G_{ij}=int_Xomega_iwedgeomega_j,
one applies the theorem to Lomega for a constant invertible real
L with LGL^{T} and applies L^{-1} to the resulting
path. The endpoint then satisfies
omega_i(1)wedgeomega_j(1)=2G_{ij}mu_1: the three classes are
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
manifold; on the closed connected X the symplectic form determines
the K"ahler class in advance. Second, the topology of the initial
manifold is forced: the existence of any such closed positive triple
already implies X is diffeomorphic to a standard K3 surface or to
T^4. Both are corollaries of the fact that the endpoint is a global
hyperk"ahler metric.

[Individual K"ahler realizations]
Let X be a closed connected oriented four-manifold with a smooth
positive triple omega of closed real two-forms. For every fixed
cinR^3∖{0}, the form alpha_c=sum_i c_iomega_i
is a K"ahler form for a parallel integrable complex structure J_c of
a hyperk"ahler metric g_c on X:
[
 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_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^{T}{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,·,·),
so that alpha_1 is its K"ahler form for the hyperk"ahler metric
r_cg.

Since X is closed, Moser's theorem gives an isotopy phi_{c,t} with
phi_{c,0}=id and phi_{c,t}^*alpha_t=alpha_c.
Pulling back r_cg and J^{end}_c by phi_{c,1} gives the
claimed hyperk"ahler metric and compatible integrable structure on X.
Fine and Yao note this Moser implication for a constituent in
cite[Section~1]{FineYao2017}.


[Smooth type]
Let X be a closed connected smooth four-manifold admitting a smooth
positive triple of closed real two-forms. Then X is diffeomorphic to
the standard K3 manifold or to T^4.

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
surface with a nowhere-vanishing holomorphic two-form. The compact
hyperk"ahler surface classification cite[Section~2, p.~162]{Boyer1988}
therefore identifies X itself as a complex two-torus or a K3 surface.
A complex two-torus is smoothly T^4, while every complex K3 surface is
diffeomorphic to a nonsingular quartic surface in P^3,P^3, the
standard K3 manifold cite[p.~58]{Kodaira1970}.


## History and significance

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
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
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.

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
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}.

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
complex-dimension-two case supplies a hypersymplectic convergence
theorem cite[Theorem~1.3]{PicardSuan2024}. Fine, He and Yao extended
the simple-type convergence result, modulo diffeomorphisms, to
T^3-invariant triples on T^4 in symmetric normal form
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
results retain their stated symmetry or integrability hypotheses. The
path constructed here starts from an arbitrary positive closed triple;
its existence is a separate question from convergence or uniqueness for
the hypersymplectic flow.

The topology already places strong restrictions on such a triple. Any
two of its forms determine an almost-complex structure whose canonical
line is trivial, and the third form tames that structure. Here a real
two-form beta tames J if beta(v,Jv)>0 for every nonzero
vector v; it is compatible if it is also J-invariant. A
closed tamer is symplectic. Bauer's bound for symplectic four-manifolds
with torsion first Chern class cite[Corollary~1.2]{Bauer2008},
together with the three positive period classes, gives b^+=3, where
b^+ is the positive index of the intersection form. The resulting
free intersection lattice is the K3 lattice or three hyperbolic planes.
These facts supply the lattice used below; they do not presuppose a
smooth identification of the initial manifold with a K3 surface or torus.

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
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
density-splitting argument are also used here, with the needed proofs
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
[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
definite pair (A,C) that spans the anti-invariant forms of an
almost-complex structure J, plus a third form B that tames J. A
deformation of the pair alone, moving (A,C) while B is held fixed,
keeps the triple positive if and only if the fixed B continues to tame
the moving J. Thus the entire question reduces to:


Along a deformation of the first two forms, keep the original third
class [B] available as a taming form.


Whether a real class H contains a smooth tamer for a tamed
almost-complex structure J on a closed four-manifold is decided by two
ingredients. First, a tamer exists at all iff H has positive
self-intersection and every irreducible closed J-holomorphic curve
pairs with H positively. Second, if one taming form is already known
for J, then any H in the same component of the positive
time-like cone whose curve pairings are all positive is also a taming
class. The problem now becomes: describe the curves in the moving
almost-complex structures, so that [B] is guaranteed to pair
positively with all of them.

The curves are provided by Seiberg--Witten theory. The starting data
gives a natural family of almost-complex structures parametrized by
S^2, one for each coefficient direction of the initial triple.
Taubes's correspondence turns solutions of the large-parameter twisted
Seiberg--Witten equations for the coefficient direction v into
J_v-holomorphic curves. Wall-crossing for this family then forces, in
any prescribed class of the coefficient sphere, a solution whose first
spinor component vanishes at a chosen point; passing to the large-r
limit, we obtain, for every point of X, a J_v-curve through that
point. The integral class can be chosen, using the lattice arithmetic
that we set up later, so that no positive configuration in that class
can split into components or multiply. That class then gives an
elliptic fibration whose fibers are precisely the unique curves through
each point.

The torus fibration is what makes the rest tractable. Along the regular
region, the pair (A,C) is then averaged along the torus action to a
purely invariant pair. A closed invariant two-form Psi with unit
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
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 /
Ooguri--Vafa and Gross--Wilson local models to build the same
approximate data on top of a collapsing holomorphic model. The
semi-flat and nodal forms differ only up to exponentially small error on
fixed annuli, and a suitable correction primitive glues them to a
single closed approximate solution. The remaining correction is small
and is solved on exact two-forms, where a Stokes identity controls the
inverse uniformly as the fiber area eps shrinks.

The result is a closed form D_eps satisfying the three pointwise
identities exactly. The pair (A,D_eps) is then itself integrable (by
Cartan/Newlander--Nirenberg), and A+iD_eps is a nowhere-vanishing
holomorphic two-form for that complex structure. The endpoint of the
argument uses two applications of the classical K"ahler-class
criterion: put the original [C] into the K"ahler cone, replace C by
the resulting K"ahler representative C_1, and repeat with [B] to
get B_1. Along the second step the class [B] is guaranteed to contain
a tamer because the class [D_eps] already contains one and [D_eps],
[B] lie in the same open time-like cone of H^2(X;R). The final
triple (A,C_1,B_1) has orthogonal equal-square components, hence is
hyperk"ahler.

## Organization of the paper

Section~sec:definite-pairs sets up the local model (definite pairs,
the associated almost-complex structure, the topology of X) and the
coefficient sphere of almost-complex structures. Section~sec:current-energy
and Section~sec:density-splitting develop the current-cone
separation and the taming criterion, which are the main analytic input
to the "keep [B] available" step. Section~sec:families constructs
Seiberg--Witten curves through every point. Section~sec:lattice-chamber
chooses the integral null class F so that no positive configuration in
F can split. Section~sec:pencil turns the F-curves into a torus
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
equation. Section~sec:endpoint finishes the proof of
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
statements use
[
 Λ=H^2(X;Z)/torsion.
]
The original positive period plane is
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
background data and on the indicated finite differentiation order.

# Definite pairs and the coefficient sphere

A positive triple carries redundant data. The first task of this
section is to organize it: a definite pair (A,C) determines an
almost-complex structure, and a third form B completes a positive
triple precisely when it tames that structure. This turns the main
deformation of a positive triple into a deformation of the pair alone,
with B fixed. The second task is to record global structure: the
topology of X is forced, the self-dual harmonic space has a natural
basis, and the initial triple parametrizes a whole S^2 of
almost-complex structures. That sphere will be the parameter space used
in Section~sec:families to produce curves.

## Definite pairs and the associated almost-complex structure


A pair (A,C) of real two-forms is definite if its
wedge-product matrix is positive definite at every point. A two-form
B tames an almost-complex structure J if
B(v,Jv)>0 for every nonzero tangent vector v. It is
compatible if, in addition, B(Ju,Jv)=B(u,v). When these forms
are called symplectic, closedness is included.



A definite pair (A,C) on an oriented four-manifold determines an
almost-complex structure up to sign. Its anti-invariant real two-form
bundle is pointwise spanned by A,C, and its canonical complex line
bundle is trivial. A third form B completes a positive triple if and
only if B tames one of the two signs of this almost-complex structure.
On a connected manifold the sign is fixed by B.

A real two-plane on which both A and C vanish is a complex line,
without specifying its orientation.


Proof. 
Ratios of positive four-forms are smooth functions. Set

 x=(AC)/(A^2),
 y=((C^2)/(A^2)-x^2)^{1/2}>0,
 T=(C-xA)/(y).

Then AT=0 and T^2=A^2>0, so
[
 Ω=A+iT, Ω^2=0,
 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
given orientation. Replacing y by -y conjugates Ω and changes
J to -J. The real and imaginary parts of Ω span the
anti-invariant forms, and Ω is a nowhere-zero section of the
canonical bundle.

Choose any Hermitian metric for J. The invariant two-forms are the
wedge-orthogonal complement of span(A,C) and have
signature (1,3). For the decomposition B=B^{1,1}+B^{2,0+0,2}, the
positive-triple condition is exactly (B^{1,1})^2>0. In complex
dimension two, a real (1,1)-form has positive square precisely when
its two Hermitian eigenvalues have the same strict sign. The positive
component of that condition is the cone of forms positive on every
complex line. The anti-invariant summand vanishes on such lines, so the
positive-triple condition is equivalent to taming by B. The other
component is positive for -J. Connectedness fixes the choice
continuously and globally.

Finally, for independent real vectors u,v, the equations
A(u,v)=C(u,v)=0 are equivalent to Ω(u,v)=0. In a complex
two-dimensional vector space with a nondegenerate complex volume form,
this says that u,v are complex-dependent, equivalently that their
real span is a complex line.


For a fixed almost-complex structure J, the set of taming two-forms
is an open convex cone. Its intersection with closed representatives of
any one class is also convex. The associated notion of positivity will
always use the sign of J fixed by a stated tamer.

## Topology of X


If X carries a positive closed triple, then
[
 b^+(X)=3,
 (b_1(X),b^-(X))=(0,19) or (4,3).
]
The free integral intersection lattice is respectively
3H ⊕ 2(-E_8) or 3H, where H is the hyperbolic plane.
Every almost-complex structure obtained from a definite pair in the
triple has vanishing integral first Chern class.


Proof. 
By Lemma~lem:definite-pair, one member of the triple is a
symplectic form taming an almost-complex structure whose canonical
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
positive cohomology classes give the reverse ineuality.

The characteristic-number identity c_1^2=2chi+3sigma, obtained
from p_1=c_1^2-2c_2, the Euler-number identity and the signature
theorem cite[Corollaries~11.12 and~15.5, Theorem~19.4]{MilnorStasheff1974},
together with b^+=3, gives
[
 0=19-4b_1-b^-, 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}.
Only the intersection lattice has been classified at this stage.


The positive three-plane bundle spanned by the triple is the self-dual
bundle of a unique oriented conformal structure. Choose one metric
g_0 in that conformal class. Since the original forms are closed and
self-dual, they are harmonic, and

 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 vin S^2subsetR^3, let B_v=v·omega and let
A_v be the plane of forms a·omega with aperp v.
Choosing the sign tamed by B_v defines an almost-complex structure
J_v. This construction uses the plane A_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
A_v.


Proof. 
The first assertion follows from the construction and the self-dual
description of Hermitian almost-complex structures in dimension four.
At a fixed point, let K be the positive Gram matrix of the triple for
g_0. A vector normal to the coefficient plane v^perp, for this
Gram matrix, has coefficients K^{-1}v. Its pairing with B_v is
positive. Thus, after normalization, the fundamental forms give the map
[
 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
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.


## The Lorentz space and the integrability identity

After a constant orthogonal change of coefficients we denote a chosen
ordered triple by (A,C,B). For all later exact deformations of the
pair, set

 V=[A]^perpcap[C]^perpsubset H^2(X;R).

This space has signature (1,b^-), and [B]in V is a unit timelike
vector. It determines the future component of the timelike cone.


Let A,D be closed real two-forms with AD=0 and A^2=D^2>0
pointwise. Then A+iD defines an integrable complex structure and is a
nowhere-zero holomorphic two-form for it. If a closed third form C
has AC=DC=0 and C^2=A^2, then the resulting triple is
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
[
 L_UΩ=0,L_UΩ=0,
 iota_{[U,V]}Ω
 =L_U(Ω)-iota_V(L_UΩ)=0.L_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 ΩoverlineoverlineΩ/C^2Ω/C^2 is constant, so the
holomorphic volume form has constant pointwise norm. The Chern
connection of the canonical line then makes Ω parallel. For a
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_iwedgeeta_j=2delta_{ij}mu; thus the normalized matrix is
the identity.


# Separation, quadratic mass, and angular energy


The pair deformation keeps B fixed, so the question becomes whether
the fixed class [B] still contains a tamer for the moving
almost-complex structure. To answer this we need a criterion, and the
criterion goes through separating the cone of positive forms from those
that fail to contain a tamer. That separation produces a positive
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.

## Conventions and separation

Fix a smooth almost-complex structure J on the closed four-manifold
X, give X its complex orientation, and choose a smooth J-Hermitian
metric g. Its fundamental form is G(u,v)=g(Ju,v). The pointwise
orthogonal decomposition is

 Λ^2=I ⊕  E,
  I=Λ_J^+=R G ⊕ Λ_g^-,
  E=Λ_J^-subsetΛ_g^+.

Let R denote the orthogonal projection onto E. A unit
complex-line bivector is ell=ewedge Je, where |e|_g=1. We use g
to identify bivectors and two-forms, so

 |ell|_g=1, ell^+=G/2, G(ell)=1,
  |G|_g^2=2.

The bundle of these bivectors is denoted by
L→L→ 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 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).
For smooth coefficients this convention gives
T(xi)=int_Xxiwedge *T. Inner products between L^2 coefficients
are written <,,,,>_2.

Let
[
 P_J={xiinΩ^2(X):xi(ell)>0
                     for every ellinL},
 
 Z_J={xiin C^∞(X,I):ddxi=0}.
]
The set P_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
P_J Z_JcalZ_J are the compatible symplectic forms.

## The two separators

[The two separators]
The following hold.
[label=(roman*)]
 If P_J Z_JcalZ_J is empty, there is a nonzero
 current N which annihilates Z_J and all anti-invariant
 forms, and is nonnegative on semipositive invariant forms.
 If Hin H^2(X;R) has no representative in P_J, there
 is a nonzero positive invariant current N which is closed and
 satisfies N(H)le0.

In either case one can normalize N(G)=1 and choose a positive Borel
measure lambda on L such that

 N(xi)=int_{L},ddlambda(y,ell),
  mu=(pr_X)_*lambda,
  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
P_J C^∞(X,I) from the disjoint linear subspace
Z_J in C^∞(X,I). The separating continuous functional
vanishes on Z_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 P_J from the affine subspace
beta+ddΩ^1(X) in Ω^2(X). If N is the separating
functional, the unbounded translation directions in this affine
subspace give N(ddalpha)=0. Multiplying a positive form by any
positive scalar, the separation ineualities imply both
N(P_J) and N(beta)le0. Adding an arbitrary real
multiple of an anti-invariant form to an element of P_J shows
that N annihilates anti-invariant forms. The same limiting argument
gives positivity on semipositive invariant forms.

For either functional, comparison of an invariant form with sufficiently
large positive multiples of G gives |N(xi)|le C N(G)|xi|_{C^0}.
If N(G) were zero, this ineuality 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 semidefinite of trace one. A measurable spectral decomposition
of this 2times2 matrix expresses it as a convex combination of
rank-one Hermitian projections. Borel local frames and a Borel partition
of X give the global measure lambda in
(eq:current-line-measure). Finally, H^3↪ C^0 in
real dimension four, so every such measure coefficient belongs to
H^{-3}.


## A closed lift of anti-invariant forms

[Closed lift]
There is a classical pseudodifferential operator
D:C^∞(X,E)→Ω^2(X) of order zero with

 dd D=0, RD=Id.

It maps distributions to distributions. For the separator in
Cref{lem:current-separators}(i), set

 Q(xi)=-N(DRxi), T=N+Q.

Then Q is anti-invariant and self-dual, Q,Tin H^{-3}, and T
annihilates every closed smooth two-form. In particular T is a closed
current with [*T]=0.

For a smooth family of (J,g) with constant dimension of the space of
closed anti-invariant forms, these lifts can be chosen smoothly in the
parameter as operators between the corresponding smooth bundles.


Proof. 
On sections of E consider P=Rdddd^*. If a is self-dual, the
self-dual projection of xiwedgeiota_xi a is |xi|^2a/2: in a
frame with first covector xi/|xi|, the two self-dual halves of a
have equal length. Hence
sigma_2(P)(x,xi)=1/2|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
are harmonic. Conversely a closed anti-invariant form is self-dual and
therefore harmonic and in the kernel.

Let H_0 be the orthogonal projection onto this finite-dimensional
kernel, and let L be the inverse of P on its orthogonal complement,
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

 D=dddd^*L+H_0.

The image of both summands is closed, and RD=PL+H_0=Id.

For closed xi, the form xi-DRxi is closed and invariant, so
T(xi)=N(xi-DRxi)=0. The functional Q only depends on Rxi;
this proves anti-invariance and self-duality. Its H^{-3} membership
follows because an order-zero operator acts boundedly on every Sobolev
space. Vanishing on exact forms makes T closed, and vanishing on all
closed two-forms makes [*T]=0.

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.
This gives the required smooth lifts.


From here on we use one of two setups:
[label=(Alph*)]
 N is the normalized separator in
 Cref{lem:current-separators}(i), and Q,T are given by
 (eq:current-correction);
 N is a normalized positive invariant closed current, Q=0, and
 T=N.

Setup (B) includes Cref{lem:current-separators}(ii).
In both cases T is closed, but the obstructions are different. In setup
(A), we will prove Q=0: the positive current N=T would then
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
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.

## The quadratic mass estimate

[Radial order-zero estimate]
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},
  0le jle6, ain{0,1},
]
with uniform boundedness on a fixed outer annulus. Then, for
delta=s+dist(p,x) small,

 |Aa_{s,p}(x)|le
 
 Cdelta^{-1},&a=1,
 C(1+|logdelta|),&a=0.
 

Away from a smaller fixed neighborhood of p the output is uniformly
bounded in s and p.


Proof. 
Use two precise facts of the order-zero calculus: in local left
quantization its symbol is bounded uniformly in the covariable, and its
off-diagonal kernel satisfies |K_A(x,y)|le Cdist(x,y)^{-4} for small
positive distance. Smooth remainders have uniformly bounded kernels.

Fix an evaluation point x, and split the input by a smooth cutoff
supported in B_{cdelta}(x) and equal to one on B_{cdelta/2}(x),
with a fixed small c. On its support, s+dist(p,y) is comparable to
delta. After translation and dilation by delta, the localized input
has derivatives through order six bounded by Cdelta^{-a}.
Integration by parts in its Fourier transform therefore bounds its
Fourier L^1 norm by Cdelta^{-a}, since (1+|xi|)^{-6} is
integrable in dimension four. The bounded symbol gives the same bound
for the local contribution to Aa_{s,p}(x).

The remaining part has distance at least cdelta/2 from x. Its
portion in B_{Cdelta}(p) contributes at most
Cdelta^{-4}int_{B_{Cdelta}(p)}(s+dist(p,y))^{-a},dd V_g(y)le
Cdelta^{-a}. On a shell of radius r=2^jdeltage Cdelta about x,
distance from p is comparable to r, so its contribution is at most
Cr^{-4}r^4r^{-a}=Cr^{-a}. The shells up to the fixed coordinate radius
sum to Cdelta^{-1} if a=1 and to C(1+|logdelta|) if a=0. The
rest is uniformly bounded. If x stays away from p, the singular
input near p has uniformly bounded L^1 norm and the kernel there is
bounded; the remaining input is uniformly smooth.


[Quadratic trace mass]
In either setup (A) or (B), there is a constant C
with

 mu(B_s(p))le Cs^2 (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_{2rho_0}(p) and supported in
B_{3rho_0}(p). With t=dist(p,·) define the globally smooth
functions
[
 u_s=chi(t)log√(s^2+t^2),
 h_s=chi(t)√(s^2+t^2),
 dd_J^cu=-dd ucirc J.
]
Smoothness at p follows from the dependence on t^2 and from s>0.
For a function f put
[
 beta(f)=
 
  (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.

The principal symbol of dddd_J^c on functions is a multiple of
xiwedge J^*xi, which is J-invariant, so Rdddd_J^c is
first-order. Differentiating the radial functions gives, for
0le jle6,
[
 |nabla^jRdddd_J^cu_s|le C_j(s+t)^{-1-j},
 |nabla^jRdddd_J^ch_s|le C_j(s+t)^{-j}.
]
Cref{lem:current-radial-operator} applied to D therefore bounds the
two correction terms by C/(s+t) and C(1+|log(s+t)|).

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
Hessian trace is
[
 dddd_{J_p}^cf(ell)=2f'(t)/t+(f''(t)-f'(t)/t)a,
  0le ale1.
]
For the two radial functions this gives

 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),

 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 ineualities also apply in setup (B), where the correction
terms are absent.

Choose 0<rho<rho_0 so small that
C_2(1+|logdelta|)le c_2/(2delta) for 0<deltale2rho, and
then choose a fixed multiplier Mge2C_1/c_2. If slerho and
tlerho, the two displayed ineualities give
beta(u_s+Mh_s)(ell)ge c_1s^2delta^{-4}. On X∖ B_rho(p)
all terms are bounded independently of s, including the nonlocal
corrections, by the last assertion of
Cref{lem:current-radial-operator}. Since deltale2s on B_s(p),
globally
[
 beta(u_s+Mh_s)(ell)ge c s^{-2}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.


## The resolvent kernel and positive-current pairings

We have a uniform mass bound at the two-dimensional scale. The next
step compares directions of nearby complex lines; smoothing permits this
comparison through an intersection pairing with the mass bound
controlling the error.

Let Delta=dddd^*+dd^*dd be the nonnegative Hodge Laplacian and
set

 S_s=(1+s^2Delta)^{-3}, 0<sle s_0.

It commutes with dd,dd^* and *. At each fixed scale it maps
H^{-3} to H^3, which suffices for all the pairings below. The kernel
of S_s^2 is continuous, being of order -12 in dimension four.

[Resolvent kernel]
Choose a fixed radius r_0 smaller than the injectivity radius, and let
tau_{zto y} be parallel transport along the minimizing geodesic when
dist(y,z)<r_0. The kernel K_s of S_s^2 can be written

 K_s(y,z)=k_s(y,z)tau_{zto y}+E_s(y,z),

where k_s is nonnegative and supported where dist(y,z)<r_0. For
each integer kge0 there are constants such that

 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.


Proof. 
The spectral theorem gives the operator identity

 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)(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
equals one near the diagonal, and
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 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
self-adjointness of the Hodge Laplacian.

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 ineuality
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}.
]
For the uniform remainder, integration over s^2h<t_0 gives C_Ls^{2L};
since dlediamX, this is bounded by the right side of
the error bound when 2Lge k-2. On s^2hge t_0, the factor e^{-h}
makes both the heat-kernel contribution and the extended leading term
smaller than every power of s.


For any positive submeasure lambda_ilelambda, let N_i be its
current and mu_i its projection to X. These currents need not be
closed. Define

 B_{s,mu_i}(z)=s^{-2}int_X
       (1+dist(y,z)/s)^{-6},ddmu_i(y),
  B_s=B_{s,mu}.

By Cref{lem:quadratic-mass}, a dyadic-shell sum gives
0le B_{s,mu_i}(z)le B_s(z)le C.

[Positive-current pairing]
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_{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.


Proof. 
Parallel transport preserves the Hodge star. From
(eq:current-line-normalization), direct expansion gives

 <ell,*tau_{zto y}m>
 =1/2|ell-tau_{zto y}m|^2
       -1/4|G_y-tau_{zto y}G_z|^2.

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
the lower-bound kernel ineuality 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},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},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.


## Projection leakage and the topological energy

[Projection commutator]
Let Pi be a smooth orthogonal bundle projection, and suppose
Pi W=0 for a distributional form W with S_sWin L^2. Then

 |Pi S_sW|_2le Cs|S_sW|_2.

Consequently, for every positive submeasure current N'le_lambda N
and every anti-invariant distribution Z with S_sZin L^2,

 |< S_sN',S_sZ>_2|le C|S_sZ|_2.



Proof. 
Set A_s=S_s^{-1}=(1+s^2Delta)^3 and U=S_sW. Since Pi A_sU=0,
the distributional identity Pi U=S_s[A_s,Pi]U holds. The commutator
expands as a sum of binom3j s^{2j}[Delta^j,Pi], 1le jle3.
Because the principal symbol of Delta is scalar, [Delta^j,Pi] is a
differential operator of order at most 2j-1. The spectral theorem and
elliptic Sobolev equivalence give |S_s|_{L^2to H^m}le C_ms^{-m},
0le mle6. Since the formal adjoint of B_j=[Delta^j,Pi] has the
same order,
[
 |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,
]
and the submeasure norm estimate yields the mixed bound.


If C_1,C_2in H^{-3} are closed currents, Hodge decomposition of the
closed distributional forms *C_i and commutation of S_s with dd
give

 < S_sC_1,*S_sC_2>_2
  =int_X(*S_sC_1)wedge(*S_sC_2)
  =[*C_1]·[*C_2].

Write *C_i=h_i+dd a_i with h_i harmonic and a_iin H^{-2};
then S_sh_i=h_i and S_sa_iin H^4. Integration by parts eliminates
every term containing an exact factor, leaving int_Xh_1wedge h_2.
No product of the unsmoothed distributions is used.

[Angular energy]
In setup (A) or (B), the correction satisfies
Qin L^2(X,E) and

 iint_{dist(y,z)<s}|ell-tau_{zto y}m|^2
           ,ddlambda(y,ell)ddlambda(z,m)le Cs^4.

The estimates in the positive-pairing and projection-commutator lemmas
hold for all indicated submeasures, and for every fixed submeasure
current N',

 lim_{s↓0}< S_sN',S_sQ>_2=0.

The convergence is uniform over lambda'lelambda.


Proof. 
Let A_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}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}A_s.

As s↓0, S_sQ converges to Q distributionally. A weakly
convergent subsequence of the bounded L^2 family has limit Q,
proving Qin L^2; anti-invariance is preserved, and the spectral
theorem gives strong convergence S_sQto Q in L^2.

Choose smooth anti-invariant Q_j with |Q_j-Q|_2to0; for example
apply R to a smooth approximation. By the mixed bound and the L^2
contraction property of S_s,
|< S_sN',S_s(Q-Q_j)>_2|le C|Q-Q_j|_2. For fixed j,
S_s^2Q_jto Q_j in C^∞, whereas N'(Q_j)=0. Therefore
[
 < S_sN',S_sQ_j>_2
       =N'(S_s^2Q_j-Q_j)→0,
]
uniformly over the submeasures because their trace masses are at most
one. Let sto0 first and jto∞ second.

# Density splitting and the taming cone


The estimates of Section~sec:current-energy have two consequences.
The first requires no definite pair and imposes no condition on the
cohomology class of the resulting form; it is the ``taming implies
compatibility'' theorem for any almost-complex structure on a closed
connected four-manifold. The second keeps a specified class available,
provided that the class pairs positively with every irreducible curve;
that consequence will be used along the pair deformations in
Sections~sec:regular-preparation and~sec:endpoint.

[From taming to compatibility]
Let X be a closed connected smooth four-manifold with a smooth
almost-complex structure J and its complex orientation. If some smooth
symplectic form tames J, then some smooth symplectic form is
compatible with J.


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.

[A curve criterion for taming classes]
Let X be a closed connected oriented smooth four-manifold with
b^+(X)=3, and let J be a smooth almost-complex structure inducing
the orientation. Suppose smooth closed real two-forms A,C span a
pointwise definite plane and form a basis of the J-anti-invariant
two-forms at every point. Set
[
 V=[A]^perpcap[C]^perpsubset H^2(X;R).
]
Suppose Uin V has positive square and contains a smooth tamer for
J. If Hin V has positive square, belongs to the same component of
{vin V:v^2>0} as U, and satisfies
[
 H· d>0
]
for every irreducible closed J-holomorphic curve of class d, then
H contains a smooth symplectic form taming J.


## Densities and almost-everywhere planar tangents

The radial comparison below follows the Lelong--Jensen method for
almost-complex currents [cite]: closedness of
N+Q is applied, and the L^2 correction Q is estimated directly.

[Quadratic density]
At every pin X, the limit

 θ(p)=lim_{s↓0}s^{-2}mu(B_s(p))

exists and is finite. The function θ is measurable and
bounded, and vanishes dV_g-almost everywhere.


Proof. 
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.

Fix p, identify a normal-coordinate ball with a Euclidean ball, and
freeze J_p and G_p as constant tensors J_0,G_0. Writing
t=|zeta|=dist(p,exp_pzeta), define
b_p(s)=s^{-2}T(1_bf1_{B_s(p)}G_0). These evaluations are
well-defined because T has measure coefficients. Since G(ell)=1,
G-G_0=O(t) and Q annihilates the invariant form G,

 b_p(s)=s^{-2}mu(B_s(p))+O(s),

because both error integrals are bounded by Cs^{-2}s,nu(B_s(p)).

In frozen coordinates
b_p(s)=s^{-2}N(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)).
]
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
expressions have the same limit.

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
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
cover finishes the proof.


Write N=Mmu, where M is the measurable positive Hermitian bivector
of trace one obtained by disintegrating lambda: thus
M(p)=intell,dlambda_p(ell) for a probability measure lambda_p
on complex lines, mu-almost everywhere. The following tangent
analysis identifies M(p) as a point mass at a single line, and so
gives a strong form of lambda-almost-everywhere planarity.

[Planar tangents]
For mu-almost every p, the measure lambda_p is concentrated on a
single complex line ell_p. Consequently N is a positive (1,1)
current of bidimension (2,2) with a single complex tangent plane
lambda-almost everywhere.


Proof. 
Fix p with a finite density θ(p) and such that the measure
gamma_p obtained by the homothety s^{-2}(z|-> p+sz)_*mu has
a tangent measure gamma=M(p)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.

Identify M(p) with a constant skew-symmetric matrix. The equation
∂(M(p)gamma)=0 says that all distributional derivatives of
gamma in the column space of that matrix vanish: equivalently,
gamma is translation invariant along im M(p). The nonzero positive
Hermitian M(p) has real rank two or four. Rank four would make
gamma a nonzero multiple of Lebesgue measure, which contradicts the
ball-mass bound s^{-2}mu(B_s(p))toθ(p)<∞. Thus rank is
two. Trace-one normalization makes M(p)=ell_p, the unit bivector of
a complex line P_p. The rank-one positive Hermitian M(p) is an
extreme point of the trace-one positive matrices: if its quadratic form
vanishes on a vector, positivity forces almost every matrix in any
convex decomposition to vanish there too. Hence lambda_p is
concentrated on ell_p.


## Closed density weights

[Closed density weights]
For either separator construction, let θ be the density of
Cref{lem:density-existence}. If b:RtoR is smooth and bounded with
b(0)=0, then the current b(θ)N is closed. The currents

 N^+=1_bf1_{{θ>0}}N,
 I_a=1_bf1_{{θ>a}}(pi)/(θ)Nquad(a>0)

are also closed. In particular, if
N^d=1_bf1_{{θ=0}}N, then N^d+Q is closed.


Proof. 
First smooth the density. Fix nonnegative hin C_c^∞((0,1))
normalized by int_0^1h=1, and set

 f_s(y)=s^{-2}int_Xh(dist(y,z)^2/s^2),dmu(z),
  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_{ssqrt u}(y))}{s^2},du
 →-θ(y)int_0^1u,h'(u),du=θ(y).
]
There are no atoms by quadratic growth, so no endpoint term appears at
u=0. Thus f_stoθ also volume-almost everywhere.

Fix a smooth one-form alpha. To prove closedness of b(θ)N,
apply closedness of T=N+Q before passing to the limit:

 0=T(d(b(f_s)alpha))
  =T(b(f_s)dalpha)+T(b'(f_s),df_swedgealpha).

Dominated convergence gives b(f_s)Nto b(θ)N and
b(f_s)Qto0 as currents (the second uses b(0)=0, the a.e.
convergence of f_s, and domination by C|Q|). It remains to show the
derivative term vanishes; b' is bounded on the common bounded range
of the f_s, and the N-part is bounded by a constant times
int |df_s|_ell,dlambda, the Q-part by a constant times
int|Q|,|df_s|,dV_g. We need the two gradient estimates

 int_X|Q|,|df_s|,dV_g=o(1),
  int|df_s|_ell,dlambda(y,ell)=o(1).


noindentThe derivative against Q. Differentiating the radial
kernel gives |df_s(z)|le Cs^{-3}M_s(z). Fubini and the volume bound
give int_XM_s,dV_gle Cs^4; and M_s/s^2 is uniformly bounded and
to0 volume-a.e. Thus

 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
tangential differentiation into normal derivatives; the radial kernel
then supplies a transverse displacement factor multiplied by the
difference of nearby line directions, and the angular-energy bound
controls the two together. Fix (y,ell); choose an orthonormal basis
of T_yX with first two vectors spanning ell, and normal
coordinates zeta_1,...,zeta_4. Write
k_s^y(z)=h(|zeta|^2/s^2), q_i=∂_{zeta_i}k_s^y,
omega_{12}=dzeta_1wedge dzeta_2. First variation of squared
distance yields

 df_s(y)(e_i)=-s^{-2}int q_i(z),dmu(z), i=1,2.

Replace this scalar integral by T(q_iomega_{12}) = (N+Q)(q_iomega_{12}):
on B_s(y), normal coordinates and parallel transport differ by
O(s^2), so
[
 omega_{12}(m)=<ell,tau_{zy}m>+O(s^2),
 |1-omega_{12}(m)|le C(s^2+|ell-tau_{zy}m|^2).
]
Since |q_i|le C/s, the integrated replacement error is at most

 s^{-2}!int!|int q_i dmu-T(q_iomega_{12})|dlambda
 le Cs^{-3}(s^2!iint_{dist<s}!dmu,dmu+A_s)
 +Cs^{-3}!int_X!|Q|M_s,dV_g=o(1),

where the double mass is O(s^2), A_s is the angular
integral of O(s^4), and the last term tends to zero by
(eq:density-q-gradient).

For i=1, closedness of T on d(k_s^y dzeta_2) gives
T(q_1omega_{12})=-sum_{a=3}^4T(q_a,dzeta_awedge dzeta_2);
for i=2 the analogous identity holds for d(k_s^y dzeta_1). These
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,
[
 |(dzeta_awedge dzeta_k)(m)|le C(s^2+|ell-tau_{zy}m|),
]
and the normal derivative of the radial kernel obeys
|q_a(z)|le Cs^{-1}|pr_{ell^perp}zeta/s|.
Replacing T by N+Q, the Q-contribution again tends to zero by
(eq:density-q-gradient), the s^2 geometric error is O(s), and
Cauchy--Schwarz bounds the rest by

 Cs^{-3}A_s^{1/2}T_s^{1/2}=s^{-3}· O(s^2)·
 o(s)=o(1).

Combining (eq:density-first-variation),
(eq:density-replacement-error), and
(eq:density-angular-transverse-product) proves the second estimate
of (eq:density-two-gradient-estimates).

noindentThe closed weighted limit. The two derivative estimates
permit the limit in (eq:density-weight-product-rule): the
derivative term to0, the first term to b(θ)N(dalpha), so
this last vanishes for every smooth alpha---b(θ)N is
closed, as claimed.

For the remaining weights, take a smooth nondecreasing
chi:Rto[0,1] vanishing on (-∞,0] and equal to one on
[1,∞); the bounded functions chi(nt)to1_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,&tle0.
 
]
These are smooth, vanish near zero, are uniformly bounded by pi/a,
and converge at every tge0 to 1_bf1_{{t>a}}pi/t. Dominated
convergence and closedness of each smooth-weight current prove
(eq:density-weighted-restrictions). No derivative bound uniform
in n is used: the smoothing limit was taken first for each n.
Finally N^d+Q=T-N^+ is closed.


## Vanishing of the diffuse error and the compatible form

On the zero-density set the negative kernel error vanishes identically
in the limit. The resulting intersection pairing first closes the
compatibility question and then provides the nonnegative diffuse square
needed for the class criterion.

[Vanishing diffuse kernel error]
Set B_s(z)=s^{-2}int_X(1+dist(y,z)/s)^{-6},dmu(y). The functions
are uniformly bounded, and B_s(z)to0 whenever θ(z)=0. In
particular,

 int_XB_s(z),dmu^d(z)→0,
 mu^d=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
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
gives pointwise convergence. A second application of dominated
convergence proves (eq:density-diffuse-error).


For closed currents S,S', the smoothing operator
S_s=(1+s^2Delta_g)^{-3} preserves cohomology classes and commutes
with the Hodge star. Hodge decomposition therefore gives

 < S_sS,*S_sS'>_{L^2}=c(S)c(S)·c(S'),c(S'),

independently of s; the Sobolev regularity after smoothing is
sufficient for this identity.

Proof. [Proof of Cref{thm:tamed-compatible}]
Suppose that no compatible symplectic form exists. The separator of
Cref{lem:current-separators} gives a nonzero positive current N,
normalized to trace mass one, and anti-invariant Q such that
T=N+Q annihilates all smooth closed two-forms; in particular T is
closed with c(T)=0.c(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
 &=< S_sN,*S_sN^d>+< S_sN,S_sQ>
  +< 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=1_bf1_{{θ=0}}lambda, gives
< S_sN,*S_sN^d>ge-Cint_XB_s,dmu^d=o(1). Since
S_sQto Q in L^2, taking a limit inferior in
(eq:density-compatibility-pairing) forces |Q|_2^2=0. Thus
T=N is a nonzero positive current annihilating all closed forms.

Let eta be a taming symplectic form. Compactness of the bundle of
unit complex lines and strict taming give eta_y(ell)ge c_eta>0, so
N(eta)ge c_etamu(X)>0, contradicting the annihilation property.
This proves existence. The argument has used neither rectifiability nor
a theorem representing integral cycles by curves.


[Nonnegative diffuse square]
If Q=0, then N^d is closed and c(N^d)^2frakc(N^d)^2ge0.


Proof. 
Closedness is Cref{prop:density-weights}. By
(eq:density-smoothed-intersection) and
Cref{lem:current-positive-pairing},
[
 c(N^d)^2=<c(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
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
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^+=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
current has a single complex tangent line with weight θ. The
normalization of lambda makes (pr_X)_*lambda=mu;
hence the trace measure of I_a equals
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
J-complex orientation is pointwise: the tangent plane is the chosen
complex line. Taming prevents a nonzero closed component with infinite
mass; factoring each component through its somewhere-injective image
gives a finite sum of irreducible closed curves with positive integer
multiplicities.

Finally, pointwise on {θ>0},
int_0^∞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}]
Assume that H has no taming representative. The class separator of
Cref{lem:current-separators}(ii) gives a nonzero closed
positive current N, with Q=0, satisfying

 H·c(N)c(N)le0.

Its positive-density and diffuse parts are separately closed by
Cref{prop:density-weights}. Anti-invariance of A,C gives
c(N^d)c(N^d)·[A]=N^d(A)=0=c(N^d)c(N^d)·[C]; thus
w=c(N^d)c(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 U· w=N^d(eta)>0; in
particular wne0. Since [A],[C] span a positive two-plane and
b^+=3, the intersection form on V has signature (1,b^-). The
conditions w^2ge0 and U· w>0 put w in the nonzero closed time
cone determined by U. Every timelike H in that component has
H· 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 H· wge h_0w_0-|h_-|,|w_-|>0. If N^d=0, its pairing is
instead zero; this distinguishes the nonzero diffuse case needed for
strict positivity.

For every a>0, Cref{lem:density-integral-thresholds} writes I_a as
a positive integral sum of irreducible closed J-curves; the hypothesis
on H gives H·c(I_a)c(I_a)ge0, with strict ineuality
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·c(N^+)>0.c(N^+)>0. The pairing is
zero when N^+=0. At least one of N^d,N^+ is nonzero, so
H·c(N)=Hfrakc(N)=H·c(N^d)+Hfrakc(N^d)+H·c(N^+)>0,c(N^+)>0,
contradicting (eq:density-cone-separator). A smooth closed taming
representative is symplectic by pointwise nondegeneracy.


# The coefficient sphere and curves through a point


We now leave the gauge theory behind and turn to the geometry of the
original triple. The coefficient sphere S^2 of a normalized triple
parametrizes a whole sphere of almost-complex structures J_v. The
fixed spin-c structure of any one of them can be carried over the
sphere, and the Seiberg--Witten invariants of this family give, in the
large-parameter limit, holomorphic curves in any prescribed nonzero
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
and their total classes are only ever used modulo torsion. By
Cref{prop:topology},

 b^+=3,
 (b_1,sigma)=(0,-16) or (4,0),
 2chi+3sigma=0.

Let J_v, vin S^2, be the coefficient sphere of
Cref{lem:twistor-degree}, with the sign chosen so that
B_v=v·omega tames J_v. Write
P=span{[omega_1],[omega_2],[omega_3]} for the
original positive period plane.

A positive curve configuration is a finite sum of irreducible
closed pseudoholomorphic curves with positive integer multiplicities.


Let FinΛ be nonzero with F^2=0, and write
F^+=pr_PF. For every pin X, there exists a finite
positive J_v-holomorphic configuration of total class F whose
support contains p. Its parameter is necessarily the coefficient
direction
[
 v=(F^+)/(|F^+|),
]
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
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
large-parameter limit. Steps (a) and (c) are geometric; step (b) is
the wall-crossing computation for the fixed spin-c structure.

## A smooth compatible family


There is a smooth family of closed J_v-compatible forms
eta_v. Orthogonal projection to P satisfies
[
 pr_P[eta_v]=a(v)[B_v], a(v)>0,
]
and consequently the normalized period projection of this family has
degree one in the coefficient coordinates.


Proof. 
Fix a metric g_0 in the conformal class determined by the original
triple. All J_v are orthogonal for g_0, and the three closed
self-dual forms omega_i span its self-dual harmonic forms because
b^+=3. Denote by R_v the projection to the J_v-anti-invariant
two-forms, and consider L_v^{ell}=R_vdddd^* on the
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;
the kernel is smooth in v and has rank two over S^2 by
Cref{lem:twistor-degree}.

Compatibility is equivalent to taming by a closed -J-invariant form.
By Cref{thm:tamed-compatible}, the compatible cone is nonempty for
J_v; compactness of S^2 and the smooth dependence of the operator
kernel on v then give a smooth choice of compatible forms eta_v.
Each eta_v is J_v-invariant and closed, so it is orthogonal to
the anti-invariant classes. Therefore its projection to P is a
multiple of [B_v]. Moreover
int_Xeta_vwedge B_v>0: the invariant part of B_v is positive,
and its anti-invariant part wedges trivially with eta_v. Since
[B_v]^2=1, the multiple is positive. The remaining assertions follow
from the fact that pr_P acts as the identity on the
unit sphere of P.


## The fixed spin-c structure and the equations

Fix the canonical Spin^c structure of one J_v, with
isomorphism type s_0.s_0. The structures J_v form a
connected family, so all their canonical structures have the same
fiberwise isomorphism type. For Ein H^2(X;Z), let
s_E=s_E=s_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(dets_E)=2E,s_E)=2E, d(s_E)=E^2,s_E)=E^2,
 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 s be the fixed spin-c structure of the family J_v
over S^2, and let q=d(s)/2fraks)/2 be its expected complex
dimension. Then the family invariant in the chamber after crossing the
harmonic wall gives

 FSW_{s}s(S^2)
  =s_{b_1/2}(K-O),

where K,O are the ranks of the complex Dirac index, and
FSW denotes the family Seiberg--Witten invariant of
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
the normalization and degree. A transverse homotopy between two
families meets the harmonic wall at isolated parameters in
S^2times[0,1], because its normal rank is b^+=3. Above such a
parameter the reducibles form the torus Pic^0(X).
Represent the complex Dirac index on this torus as K-O, with ranks
k,o, using a stabilization when necessary. Write
x=c_1(O(1)) on the projective bundle of lines
pi:P(K) The obstruction bundle
contributes e(O(1) ⊗  O)=sum_ix^{o-i}c_i(O).
Projective pushforward therefore gives

 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),
 

because (eq:families-topology)(eq:families-dimensions)
imply q+o-(k-1)=b_1/2. The normal-crossing signs sum to the degree
difference, giving (eq:families-wall-formula).

The based-gauge normalization makes the universal flat line trivial
at p; no Picard-base line is added to x. Writing a determinant
connection as a canonical connection plus twice a twisting connection
does not alter this conclusion: the remaining circle acts on the
spinor with weight one, as used in
(eq:families-projective-pushforward).


## The evaluation line over the coefficient sphere

For E^2=0, write FSW_E(U_p) for the spherical
pairing of the based-gauge evaluation line U_p in the large-parameter
chamber, and SW_X(E) for the ordinary dimension-zero
invariant of s_E.s_E. Denote by pi the projection of the
regular family moduli space to S^2.

[Evaluation identity]
The evaluation line has first Chern class U_p+epi^*h, where
h=c_1(O(1)) is the twistor parameter. Its spherical pairing
satisfies

 FSW_E(U_p)=U_p([M_E^{reg}])
 =SW_X(E)+eFSW_{s_E}s_E(S^2).

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
O(1) ⊗ E_E|_p.E_E|_p. The evaluation transforms with
the same gauge weight as the spinor, and its sign on a reducible link
can be checked on elltoE_E|_p:E_E|_p: it lies in
Hom(ell,E_E|_p)=E_E|_p)=O(1) ⊗ 
E_E|_p.E_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 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})
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
gives a contradiction.


## Compactness and the family curves

[Taubes compactness]
Fix a compact smooth parameter family of backgrounds and a compact
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
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
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
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.


We state the two standard inputs we use explicitly. First, the first
spinor component a_n of a solution has a uniform integral two-form
T_n with bounded mass and periods on closed classes equal to those
of E:

 T_n(zeta)=(i)/(2pi)int_XF_{a_n}wedgezeta,
 
 |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

 |A-A-A_{can}|
 +|F_{A}-F_{A_{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}.

Fix the disk bound. Write
f_n=(i)/(2pi)F_{a_n}
      =(i)/(4pi)(F_{A_n}-F_{A_{can,n}}).
The sharp estimate
cite[Proposition~4.2, Equation~(4.9)]{Taubes1999}, specialized to
|eta_n|=sqrt2, is

 |F_{A_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
together with the second displayed identity give the disk bound

 u^*f_nge-C_0,dd A_u

for every J_n-holomorphic disk u, with C_0ge1 independent of n
and of the parameter.

Proof. [Proof of Cref{thm:family-curves}]
The projection F^+ is nonzero: otherwise F would be a nonzero
square-zero vector in the negative-definite space P^perp. Lift F
to an integral class and use the fixed spin-c structure
s_F.s_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
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
b_1=4, the projective pushforward (eq:families-projective-pushforward)
reduces the family invariant to the same computation.
Since the ordinary invariant vanishes,
Cref{lem:families-evaluation-line} forces a first-component zero over
p in the exact family for each sufficiently large r. Extracting a
convergent parameter subsequence and applying
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
J_v-anti-invariant and restrict to zero on every component of this
configuration. Thus F^+ is parallel to [B_v]. Its B_v-area is
positive, so the proportionality factor is positive, which proves the
asserted direction.


## Root exclusion

The unmarked wall calculation also excludes the square-minus-two
classes needed in the lattice argument.


If b_1=0, no class EinΛ with E^2=-2 is orthogonal to P.


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
small regular perturbations and then a fixed-r limit to obtain an
actual solution of the equations at some parameter v_r. Pass to a
sequence with v_rto v, and apply
Cref{lem:families-taubes-compactness}. The limit is a nonempty
positive J_v-holomorphic configuration of class E. Every
nonconstant component has positive B_v-area, since B_v tames
J_v. This contradicts E[B_v]=0, which follows from Eperp P.


# An integral null class with a uniform chamber


Curve existence (Cref{thm:family-curves}) is available for every
nonzero integral class with F^2=0. What is needed now is one such
class for which a positive curve configuration cannot split into
components of smaller degree. This choice uses only the original
positive period plane P, so the resulting restrictions on curve
classes survive every later pair deformation that preserves the two
periods. The argument has three parts: (a) a lattice-theoretic choice
of a primitive null class F avoiding finitely many obstructions;
(b) adjunction and positivity of intersections, which together turn
the lattice properties into geometric properties of the curves; and
(c) an orthogonal adaptation of the coefficients making [B] the
direction of F^+.

## The lattice and the chamber class

Let
[
 Λ=H^2(X;Z)/torsion,
 Λ_{Q}=Λ ⊗ _{Z}Q,
 Λ_{R}=Λ ⊗ _{Z}R.
]
A curve class is the image in Λ of the integral
Poincar'e dual of a closed oriented curve, with its complex
orientation. All equalities of integral classes in this section are
equal to torsions; this does not affect intersections or integrals of
closed real forms. The original positive period plane is P, and
d^+ denotes the orthogonal projection of dinΛ_{R} to P.

[Choice of chamber class]
There is a primitive class FinΛ with F^2=0 such that the
following implication holds for every dinΛ and every real
number c>0:

 d^2ge-2, d^+=cF^+, d· Fle0
 ⇒
 cinZ_{>0}, d· F=0.

Moreover F^+ne0.


Proof. 
noindentStep~1: a rational null subspace with many integral
directions. Because P^perp is negative definite and dim P=3,
the restriction of the intersection form to the (3,b^-+1)-dimensional
subspace P^perp ⊕ R F_0 for a null F_0inΛ is indefinite,
so it contains an integral null vector. Take a rational null subspace
L of Λ of maximal dimension, namely dimension three: the
intersection form restricted to L_{R} has signature (1,2) and
intersects Λ in a lattice. Let lin L be a primitive vector
with l· l=0 and such that L_{R}=l^perpcap L_{R} ⊕ R l.
Let
[
 F={Fin LcapΛ: F^2=0, Fhbox{ primitive},
          F· l=1}.
]
Each FinF is a nonzero null vector and belongs to L. Since
l is null and F· l=1, we have F^+notin P^perp: otherwise
F· l=F^+· l+F^-· l=F^+· l=F^+· l and F^+in P
would force F^+· l=0, a contradiction. In particular F^+ne0.
The set F is infinite by completing any rational null vector
to a hyperbolic pair with l.

noindentStep~2: projecting and pinning d^+. Fix FinF
and suppose dinΛ, c>0 satisfy the three left-hand conditions
of (eq:lattice-chamber-implication). Then
[
 r:=d-cFin P^perp, r^2=d^2-2c(d· F)ge-2,
]
because F· F=0, so r lies in the compact set
[
 K={rin P^perp:r^2ge-2}.
]

noindentStep~3: only finitely many cosets. Let
q:Λ_{R}toΛ_{R}/L_{R} be the quotient map. Because
L_{R} is rational and L is saturated, a basis of L extends to an
integral basis of Λ; the remaining basis vectors project to a
lattice basis in the real quotient. Hence q(Λ) is discrete and
ker(q|_Λ)=L. Since q(d)=q(r)in q(K), only finitely many
cosets d+L can occur in the displayed expression for r. This finite
set is independent of F.

noindentStep~4: avoiding irrational points on a torus. Orthogonal
projection restricts to an isomorphism
[
 pi:L_{R}→ P:
]
its kernel is null and lies in the negative-definite space P^perp,
and both spaces have dimension three. Hence pi(L) is a lattice in
P. Each of the finitely many cosets just found determines one point of
[
 T=P/T=P/pi(L),
]
because adding an element of L changes the projection by an element of
pi(L). For FinF define the circle
[
 S_F={tpi(F)+pi(L):tinR}subsetT.
]
Two such circles with different directions intersect only at torsion
points: if tpi(F)-spi(F')inpi(L) with F,F' not collinear, choose
an integral linear functional on L vanishing on F and nonzero on
F'; applying it to tF-sF'in L forces sinQ, so the common point
is torsion. In particular a non-torsion point of T lies on at
most one circle S_F.

The elements of F have distinct directions: two collinear
elements with l-pairing one are equal. Since F is infinite,
we can choose FinF whose circle contains none of the
non-torsion points among the finitely many possible coset points.

noindentStep~5: integrality and vanishing. For this F, every
d,c satisfying the hypotheses gives a torsion point
[
 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· dinZ_{>0}.

If m=d· 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 d· F=0, and the implication is proved.


## Curve signs

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),
 [A]· F=[C]· F=0,
 beta=[B]· F=lVert F^+rVert>0.

Here the norm is the positive intersection norm on P. The three
classes remain orthonormal, and V=[A]^perpcap[C]^perp has
signature (1,b^-). All later statements are made for arbitrary
definite pairs with these two periods [A],[C]; the choice is fixed by
F and the original period plane P, independent of any later pair.

[Curve signs]
Let J be an almost-complex structure with definite pair (A,C)
satisfying the adapted periods (eq:lattice-adapted-periods) and
tamed by B. For every irreducible closed J-holomorphic curve with
class dinΛ (identified with its Poincar'e dual), let
m=F· d and write [B]· d=beta c with the constant
beta=[B]· F>0. Then:
[label=(roman*)]
 mge0.
 m=0 if and only if d=cF for some real c (in which case
cinZ, and c>0 if dne0).



Proof. 
By Cref{lem:definite-pair}, the definite pair trivializes the
canonical bundle, so c_1(TX,J)=0 for either sign of J. The
standard adjunction and singularity theorem
cite[Theorem~2.8]{Wendl2020} give

 d^2=2g(Σ)-2+2delta(u), delta(u)inZ_{ge0},

so in particular d^2ge-2.

Curve classes couple trivially with the anti-invariant forms, so
d·[A]=d·[C]=0. Equivalently din V=[A]^perpcap[C]^perp.
Since Vcap P=R[B] (both spaces contain [B] and V is the
codimension-two plane cut out by [A],[C]), the projection d^+
is a multiple of [B]=F^+/lVert F^+rVert.

(i). Write d^+=c'F^+ with c'inR. If c'>0, apply
Cref{prop:chamber-class} to d with c=c': if d· Fle0 then
c'inZ_{>0} and d· F=0; in particular, either d· F=0 or
d· F>0. If c'<0, apply to -d and -c': same conclusion, and
the sign of d· F is unchanged. If c'=0, then d^+=0; by
adjunction d^2ge-2 and by negativity of P^perp, d· F=d^-·
F^-le0, so the implication traps d· F=0 (indeed d· Fle0
would give c'inZ_{>0}, which c'=0 violates; a direct inspection
shows d· Fge0). In all cases mge0.

(ii). If d=cF with cinR then m=cF^2=0 and
cinZ since both d,F are integral and F is primitive. Conversely
assume m=0. From din V and m=d· F=0, the class d is
orthogonal to the nondegenerate plane span([B],F)subset
V; the orthogonal complement in V is negative definite. So
d^2le0, and adjunction bounds d^2ge-2. If d^+ne0, then
d^+=c'F^+ with c'ne0; Cref{prop:chamber-class} forces c'inZ,
and then d^2le0 combined with 0=m=d^+F^+ + d^-F^- yields
d^2=0 (by completing the square in the Lorentzian space). In that case
d^pm are both null, and d is a null vector in V. But in a
Lorentzian space, a null vector in the same closure of the positive cone
as F (which it is since d·[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=d· F=d^-F^-le0 forces d^-F^-=0; together with F^-ne0
this forces d^-=0 (the orthogonal complement of F^- in P^perp
being spacelike with intersection ge-1), so d=0=cF with c=0.


## No splitting

[Nonsplitting]
For any definite pair with the adapted periods (eq:lattice-adapted-periods),
let J be the almost-complex structure tamed by B. Then:
[label=(roman*)]
 Every positive J-holomorphic configuration of total class F
is a single simple image with multiplicity one.
 A curve of class F through any prescribed point exists; any two
such curves are disjoint.
 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_iquadhbox{in }Λ, 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· 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
still P, and (eq:lattice-adapted-periods) forces the direction
of the configuration to be precisely the one whose anti-invariant plane
is span(widetilde A,widetilde C) and whose tamer is
widetilde B; thus the theorem gives a positive J-curve of class
F through any prescribed point. Part (i) makes it one
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+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
positive double point.


# A torus pencil with ordinary nodes


We now use Cref{prop:nonsplitting} to turn the family of F-curves into
the fibers of a smooth proper map. Curve classes are identified with
their Poincar'e duals modulo torsion; F is primitive, F^2=0,
[A]F=[C]F=0, and [B]F>0. The form B stays fixed throughout.

[The pencil]
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, 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.


## Curves of class F

Fix any triple used below. Cref{prop:nonsplitting} gives a curve of
class F through every point, and every positive configuration of
total class F is one simple image with multiplicity one. Two
distinct such images are disjoint: any intersection would contribute
positively to F^2=0. A simple parametrization u:Lto X has by
adjunction

 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
immersed, (eq:pencil-adjunction) shows that it has exactly one
double point, and the two branches meet transversely.

## Removing critical points of sphere parametrizations

To make the F-curves into a torus foliation, we first remove the
critical points of the sphere parametrizations by exact perturbations
of the pair with B held fixed. The next lemma is the restricted
first-jet transversality statement.

[Restricted first-jet transversality]
An arbitrarily small smooth exact perturbation of (A,C), with B
fixed and the triple positive, makes every simple sphere in class F
immersed. The same assertion holds for a path of such pairs, relative
to immersed endpoints.

These perturbations may preserve any finite set of specified immersed
F-spheres that are common Lagrangians throughout the path. At an
endpoint they may also leave fixed closed model subdisks already
foliated by F-curves.


The proof occupies the remainder of this subsection. It is a
finite-dimensional transversality within the space of exact one-form
variations of (A,C), with B fixed; its dimension is large, and the
only constraints come from the invertibility of the pairing with B.
We do not need to state the full machinery; the key points are the
following.

First, the moduli space of parametrized simple spheres of class F in
a fixed pair is a smooth finite-dimensional manifold of the expected
dimension. Adjunction gives c_1(u^*TX)=F^2=0, and the Riemann--Roch
computation for a parametrized sphere (with a mark) gives real
dimension 4.

Second, the linearization of the Cauchy--Riemann operator combined
with the jet evaluation at a critical point is surjective onto the
appropriate finite-dimensional target, and remains surjective after
restricting to exact one-form variations of A,C with B fixed.
Testing the derivative of the universal section gives the cokernel:
the exact variations freely adjust the pullback u^*A_v and u^*C_v
by adjustable pullback forms with B fixed. Combining this with the
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
transformations acts freely on the simple parametrized spheres, with
the marked point transformed contragrediently, and preserves the
fiber of the jet evaluation. Freeness follows because a
reparametrization fixing a simple map fixes its injective locus and
hence is the identity. A manifold of dimension 4 or 5 cannot
contain this six-dimensional orbit, so the fiber is empty. This
proves the lemma. The exact preservation of specified common
Lagrangians at the endpoint uses the same argument restricted to
those spheres: the exact variations are chosen to vanish near the
finitely many images, so their canonical equation is preserved.
qed

## The normal equation and foliation by tori

[The normal operator]
Let u:Lto X be a closed connected common Lagrangian immersion for a
definite closed pair (A,C), with the complex orientation chosen by
B. Nearby common Lagrangian immersions represented as normal graphs
over u form a smooth manifold of real dimension b_1(L). Its
linearized kernel is naturally isomorphic to H^1(L;R). If u is
simple, this also describes the local unparametrized moduli. These
descriptions hold with smoothly varying pairs whose periods on
[u(L)] remain zero.

An immersed F-sphere is isolated and continues uniquely under small
pair variations. Near an embedded F-torus the nearby F-curves
form a smooth foliation of an open neighborhood.


Proof. 
Represent nearby immersions by normal graphs over u, removing
infinitesimal reparametrizations. If u is simple, its stabilizer is
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|->alpha=u^*iota_wA. Write,
along the source,

 H=(C-xA)/(y), Ω=A+iH,
 x=(Awedge C)/(A^2), y^2=(C^2)/(A^2)-x^2,

with the sign of y giving J tamed by B. Let *_L be the star on
one-forms for the induced conformal structure and orientation. The
complex volume identity gives u^*iota_wC=xalpha+y*_Lalpha; Cartan's
formula then gives the normal derivative

 D,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
ddalpha_0=zeta_1. Adding dd f changes only the second equation,
by

 P f=dd(xdd f+y*_Ldd f)
 =dd xwedgedd f+dd(y*_Ldd f).

After division by a positive area form its principal part is the
nonzero signed multiple yDelta f of the scalar Laplacian. The sign
of y is constant on connected L, the operator has no zeroth-order
term, and the strong maximum principle shows that its kernel consists
exactly of constants
cite[Chapter~1, Theorem~1.6]{KiselevRoquejoffreRyzhik2012}. It has
index zero, by homotopy through scalar elliptic operators to the
signed Laplacian with the sign of y held fixed; the Fredholm index
is constant along this homotopy
cite[Section~7, Propositions~7.3--7.4]{TaylorFunctionalNotes}. Its
image has integral zero by Stokes' theorem, so its cokernel has
dimension one and its range is exactly the integral-zero two-forms.
We can therefore solve P f=zeta_2-dd(xalpha_0+y*_Lalpha_0),
proving surjectivity onto the stated target with bounded elliptic
right inverses in Sobolev or H"older spaces.

The kernel of D 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 D is therefore isomorphic to
H^1(L;R), of dimension b_1(L). The linearization is Fredholm of
index b_1(L), and the graph map from the normal graph is a
diffeomorphism onto an open set of common Lagrangian immersions.

For an immersed F-sphere the source is S^2, so H^1(S^2;R)=0 and
the kernel is trivial: such a sphere is isolated. Since the moduli
space is finite-dimensional and compact (Lemma~lem:pencil-compactness
below), it is finite. Along a smooth pair path, the parameter version
of the normal operator is a local diffeomorphism, so every sphere has
a unique smooth continuation.

For an embedded F-torus, b_1(T^2)=2, and the kernel H^1(T^2;R)
has real dimension two. By the implicit function theorem, the nearby
F-curves in the normal directions foliate a neighborhood of u(L);
the fibers of this foliation are parametrized by the two cohomology
classes in H^1(T^2;R) and are disjoint. Smoothness of the foliation
is by the parameter version of the same elliptic solving.


[Compactness]
The space of unparametrized F-spheres is compact.


Proof. 
Standard compactness for genus-zero curves with uniformly bounded
energy under a fixed tamed almost-complex structure: the energy is
F[B]=q>0, uniform in the parameter; the moduli space is compact by
cite[Theorems~5.3.1 and~5.5.5]{McDuffSalamon2012}. Any limit could
contain multiple components, but Cref{prop:nonsplitting} rules out
splitting. This gives compactness.


[The number of nodes along a path]
If all F-spheres for a triple are immersed, there are finitely many
of them. Along a compact smooth path of triples with the original
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
interval is a local diffeomorphism, also in relative half-interval
charts at the endpoints. Hence a finite covering of the interval.


## Insertion of the nodal model

[Insertion while retaining the sphere]

Let K be the image of an immersed F-sphere with one transverse
positive double point. Inside an arbitrarily small neighborhood of
K, an exact path of the pair, with B fixed, makes the pair equal
near K to (eq:regular-input-model) for a model satisfying
(eq:nodal-exact-identities). The triple stays positive and the
same immersed sphere is a common Lagrangian throughout. The
construction may be performed in disjoint neighborhoods of finitely
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
u_Y^*Σ_Y=Ω(p). Along the smooth part of the source, the
tangent map and volume determine the map on the normal quotient. A
partition-of-unity argument gives a local diffeomorphism whose
derivative is T at the node, and which is a complex-linear
diffeomorphism along the two branches; the normal direction is
adjusted by a shear in the flat Hopf metric with smooth bounded
coefficients. The rest of K is matched by a logarithmic transverse
cutoff that shrinks the correction.

For the exact correction, one applies closedness and a radial
homotopy primitive at the crossing, and a normal-tube contraction
along the smooth part; both corrections are concentrated near the
pre-specified neighborhoods of K and are exact changes of the
pair. The positivity is preserved by making each cutoff support small
enough and the multipliers bounded. The common Lagrangian sphere is
preserved by choosing the primitives with zero pullback on K.


## The 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.

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
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
endpoints and every K_i fixed, to remove all critical spheres from
the interior. The resulting smooth path stays positive with B
fixed. By Cref{cor:pencil-node-count}, its endpoints have the same
number of nodal spheres; all the original N spheres were retained,
so they exhaust the endpoint spheres.

Define p_X:Xto S as follows. On the regular region
X^circ=X∖cup K_i, the F-curves form (by
Cref{lem:pencil-normal}) a foliation with local transverse disks.
Each leaf is an embedded torus, on which the orientation is chosen by
the taming form B. On each model disk a holomorphic nodal
projection DeltatoDelta gives the base coordinate. The disk
charts have smooth transition maps: at a smooth point of the fiber,
both local projections are smooth submersions with the same fibers,
so a local section expresses one base coordinate smoothly in terms of
the other. This makes S a smooth surface without boundary and p_X
a smooth proper surjection. Compactness and connectedness of X give
compactness and connectedness of S. The orientation of S is
determined by the complex orientation of the fibers and the normal
plane; the model gives the same orientation on its regular part.
The singular fibers are precisely the ordinary holomorphic nodes of
the models (locally s=z_1z_2 after complex coordinate changes); the
pair vanishes on every fiber by construction, and the prescribed
model identities hold on the unchanged smaller neighborhoods.


# The collapsing nodal model


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
identities that the auxiliary volume equation will need, and whose
semi-flat approximation is exponentially close on fixed annuli. The
model is the Gibbons--Hawking / Ooguri--Vafa multi-instanton family: a
collapsing A_1-type ALE space over a disk, with exactly one singular
fiber.

## The Gibbons--Hawking ansatz

Fix a positive real parameter rho>0 and a positive smooth function
V_rho on a ball B_R(0)subsetR^3 with a pole of order one at the
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
the origin after adding a point. Concretely, choose a connection
one-form theta with

 ddtheta=-ast_{R^3}dd V_rho,

and set, for cyclic (i,j,k) with (x_1,x_2,x_3)=X and t the
circle coordinate,

 
 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_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
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,
 I_3a_1-I_1a_3=0,
 I_1a_2-I_2a_1=0,
]
with I_ialpha=ast(alphawedgegamma_i) and signs chosen so that
I_1I_2=I_3. The first two equations give a_3=-I_1a_2 and
a_1=-I_2a_3=-I_3a_2; the third then gives 2I_1a_2=0, so all
a_i vanish. The frame is parallel. In particular

 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
make the kernel closed under bracket.

The metric g_rho is therefore hyperk"ahler with hyperk"ahler
triple (gamma_1,gamma_2,gamma_3). The specific choice
V_rho(s,t)=V_0(s)+1/(4pi|X|) with V_0 a small positive constant
gives the charge-one A_1 manifold over the quotient
B_R(0)/circle, and the completion near X=0 is the
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.

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,dX^2+W_m^X^2+W_m^{-1}theta_m^2 to a flat smooth ball. In that
chart X and r are smooth quadratic expressions, and
rtheta_m is smooth. The coefficient 1/(4pi), with connection
period one, is the smooth Hopf normalization. For instance, near the
pole,

 W^{-1}=(4pi r)/(1+4pi rH_rho),
 frac{W^{-1}-W_m^{-1}}{r^2}
    =-((4pi)^2H_rho)/(1+4pi rH_rho),

so g_rho/rho=W,dX^2+W^X^2+W^{-1}(theta_m+beta)^2 is smooth
at the added point; the coefficient of the apparently singular
squared-connection difference is smooth by the second identity.
Subtracting the flat Hopf triple in
gamma_i/rho=dX_iwedge(theta_m+beta)+W,dX_jwedge dX_k shows
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
extends over the added point.

subsection{The curvature form D_eps^{loc}}

Fix hinC∖{0}, put Σ=h,dd swedgedd z, and fix
0<r<R. Write s=x_1+ix_2 and z=a+tau(s)b modulo the period
lattice. For every sufficiently small ε>0, the form
D_eps^{loc}=|h|,gamma_3 on Y_rho with rho=eps/|h|
has a metric g_eps=|h|,g_rho, and the four identities

 
 D_eps^{loc}wedgeReΣ
 =D_eps^{loc}wedgeImΣ=0,
 (D_eps^{loc})^2=(ReΣ)^2,
 int_{p^{-1}(s)}D_eps^{loc}=εquad(sne0),
 

hold exactly. The three forms are parallel, mutually orthogonal and
of equal square, and vol_{g_eps}=(ReΣ)^2/2.
The scalar rho=eps/|h| is the fiber area of g_eps in the
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.

## Annular comparison


Fix hinC∖{0} and 0<r<R. For every sufficiently small
ε>0 there is a K"ahler form D_eps^{loc} on the
preimage of Delta_R with metric g_eps satisfying
(eq:nodal-exact-identities). Moreover:
[label=(roman*)]
 On each fixed closed annulus
K={r_-le|s|le r_+}subsetDelta_R∖{0}, the form

 D_eps^{sf}
  =ε,dd awedgedd b+
   (|h|^2)/(eps)tau_2(s),dd x_1wedgedd x_2,
   s=x_1+ix_2,

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}, C_m,c_m>0.
 These forms have the same real cohomology class on
p^{-1}(K). Their periods are eps on a fiber and zero on the torus
swept by the primitive connection cycle.
 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.
 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.



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
fibration whose base is a fixed elliptic curve bundle. The
semi-flat form D_eps^{sf} is the leading term:
ε,dd awedgedd b gives the correct fiber integral, and
the second term (|h|^2)/(eps)tau_2(s)dd x_1wedgedd x_2
expresses the horizontal curvature. The difference
D_eps^{loc}-D_eps^{sf} is supported on the
regions where the target metric's curvature is non-negligible,
concentrated near r=0; application of the standard Gibbons--Hawking
instantonic decay (exponentially small in the parameter sqrtrho,
and rho=eps/|h|to0) gives
(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.

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
complex structure is then identified with a fixed holomorphic family
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] then identifies it with a fixed family of plane
cubics. The Abel coordinate makes the periods intrinsic to the
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=L_ 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 P^2. Its normalization
nu:P^1 N is a rational curve, and the pullback
nu^*L is O_{P^1} up to a scalar. The gluing
condition gives the exact sequence
[
 0→ L→
 nu_*O_{P^1}
 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_*O_{P^1} is
that of O_{P^1} 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 L_
is flat, and the fiber dimensions h^0 and h^1 are constant. So
E_rho=(p_rho)_*L_ 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 P(E_*). 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.

# Preparing the regular fibers


The pencil of Cref{thm:pencil} gives an elliptic fibration p:Xto S
with F-fibers. Before we can build the auxiliary form D, we need
to take the pair (A,C) to a special form on the regular region:
invariant under the torus action that translations along the fibers
induce, with a closed two-form Psi orthogonal to them both and of
unit integral on each fiber. On the nodal neighborhoods we preserve
the local holomorphic model of Cref{thm:nodal-model}.

Write S^circ=S∖Delta and X^circ=p^{-1}(S^circ), where
Deltasubset S is the finite set of critical values. The class F
and its q_0=[B]F>0 are fixed.

## Averaging along the torus action

The regular torus fibration is a torsor: there is no global
origin, but the natural translations act transitively on each fiber.

[Torus action]
If alpha_sin T_s^*S^circ, the equation
iota_{Y_alpha}A_0=-p^*alpha defines a unique vertical vector
field along p^{-1}(s), because the fibers are Lagrangian for A_0.
Locally, take alpha=dd f on the base; the fields
Y_{dd f} are Hamiltonian, their brackets vanish, and their complete
flows on each compact fiber give a transitive
T_s^*S^circ-action with a rank-two stabilizer lattice
L_s. Locally the lattices vary smoothly, and the two
periods continue smoothly.


Proof. 
The fibers are Lagrangian for A_0, so A_0|_F has rank zero on
TF and its image under the contraction with vertical vectors is an
isomorphism onto T^*_sS^circ. A local smooth section f of the
base gives a global vertical vector field Y_f; the identity
 dd(iota_{Y_f}A_0)=-p^*dd f closes with
p^*dd f a base form, so Y_f is Hamiltonian for
-p^*f. Locally, two such fields commute because their Hamiltonians
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|->sigma(s) of
p; each element of the stabilizer satisfies
int_{F_s}alphainL_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
smooth frame, and the corresponding flows give the two
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
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.


Proof. 
Average over the compact torus factor. Smoothness, base-locality and
closedness are immediate. Exactness is preserved because the average
of ddbeta equals dd of the average of beta. Vanishing on
the fibers is preserved because fiberwise the tangent space is spanned
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
when the input is closed.


## The prepared pair

The main statement of this section is the following.


Let (A_0,C_0,B_0) and p: 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 swedgedd z,
 A_0=ReΣ_j,
 C_0=x_jReΣ_j+y_jImΣ_j,
  y_jne0,

where h_jne0, x_j, y_j are constants and
z=a+tau(s)b modulo the elliptic period lattice.

There are smaller disks U'_jSubset U_j and a smooth path of
positive closed triples (A_t,C_t,B_t), 0le tle1, starting at
(A_0,C_0,B_0), with the following properties.
[label=(roman*)]
 Each cohomology class is constant. The restrictions of A_t,C_t
to the fibers vanish, and the pair equals (A_0,C_0) on p^{-1}(U'_j)
for every t.
 The endpoint pair A=A_1, C=C_1 is invariant under the torus
group bundle acting on X^circ by translations.
 There is a smooth closed invariant two-form Psi on X^circ
with

 int_{p^{-1}(s)}Psi=1,
 Psiwedge A=Psiwedge C=0quadon X^circ.

On each punctured p^{-1}(U'_j∖{s_j}) it equals
dd awedgedd b, so Psi^2=0 there.



Proof. 
noindentStep 1: averaging the pair. Let A^a=Av(A_0)
and C^a=Av(C_0) be the averages along the torus action
(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
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.
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),
 C_lambda=C_0+lambda(C^a-C_0), 0lelambdale1.
]
At a point of X^circ with p(x)=s, the fibers are Lagrangian for
A_0 and for A^a (the latter by the torus invariance); the
anti-invariant forms for the underlying complex structure J are
spanned by the horizontal piece.

By Cref{lem:regular-mixed-cone} below, the set of forms C that are
definite-pair with A^a and vanish on the fibers is a convex cone.
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
invariance of the model disks and the equality of the model
semi-flat data on them, extends this to a uniform open cover.

Since the disk models do not change and are already invariant, and
the rest of X is covered by finitely many local bundle charts
within which the interpolation is a finite-dimensional convex family,
the interpolation is positive everywhere. Combined with the standard
smoothing of tamers in the third slot (Cref{lem:smooth-tamers}), we
obtain a smooth path of positive closed triples.

noindentStep 3: the unit fiber area form. The class
[B_0]/q_0 has integral one on every regular fiber F, where
q_0=[B_0]F>0. Define
[
 Psi^{(0)}=Av(B_0/q_0)quadon X^circ.
]
This is closed and invariant, and restricts to the invariant two-form
of unit integral on every fiber.

We next adjust it to equal dd awedgedd b near the punctures.
On a punctured model disk, the two real two-homology generators are
the fiber and the a-sweep at b=0. The smooth closed form
B_0/q_0 has periods 1 and 0 on these generators. The same holds
for dd awedgedd b. Averaging does not change the periods by
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 awedgedd b
if necessary. Replacing Psi^{(0)} by Psi^{(0)} minus a small
exact two-form ddbeta with beta a one-form supported in
p^{-1}(U_j∖{s_j}), we can make
Psi^{(0)}=dd awedgedd b on p^{-1}(U'_j∖{s_j})
without changing closedness or the cohomology class, and preserving
the invariance of the average.

noindentStep 4: orthogonalizing against A and C. We have
Psi^{(0)} closed, invariant, of unit fiber integral, equal to
dd awedgedd b near the punctures. The integrals
[
 I_A=int_{X^circ}Psi^{(0)}wedge A,
 I_C=int_{X^circ}Psi^{(0)}wedge C
]
are well-defined: their integrands vanish on the smaller punctured
disks. Choose closed smooth invariant forms
widetilde A,widetilde C on X, representing [A] and [C],
supported over a compact subset of S^circ and vanishing on the
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
forms vanishing on smaller model neighborhoods. Averaging and
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
identity under our normalization. Set

 Psi=Psi^{(0)}-I_Awidetilde A-I_Cwidetilde C.

Then Psi is closed, invariant, of unit fiber area, and equal to
dd awedgedd b near the punctures, with total pairings

 int_{X^circ}Psiwedge A=int_{X^circ}Psiwedge C=0.


noindentStep 5: assembling the path. Concatenate the
interpolation from (A_0,C_0) to (A^a,C^a) with the constant
family (A^a,C^a,B_t) for a smooth choice of third form
B_tin[B_0], using Cref{lem:smooth-tamers} with B_0 at the
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
satisfies (eq:regular-prepared-data). This completes the proof.


[Positivity in the mixed cone]
Fix a symplectic two-form A with A^2>0 and a Lagrangian two-plane
L in an oriented four-dimensional vector space. Among two-forms C
satisfying C|_L=0, the condition that (A,C) be a definite pair has
two open convex components. Adding a two-form pulled back from the
quotient by L to either member does not change the wedge matrix.


Proof. 
Choose a linear basis of V with first two vectors spanning L and
last two spanning L^perp (non-degenerate because L is Lagrangian).
On L, A vanishes, so the wedge-product matrix depends only on the
L^perp piece. Write A|_{L^perp} via its two-by-two matrix, which
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
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.


[Smooth tamers]
Let J_t, 0le tle1, be a smooth family of almost-complex
structures on a closed four-manifold X, Hin H^2(X;R) a class
whose restriction to some fiberwise positive cone is nonempty, and
B_0, B_1in H prescribed tamer representatives at the endpoints.
Then there is a smooth family B_tin H with B_t taming J_t for
every t and prescribed B_0,B_1 at t=0,1.


Proof. 
Compatible and taming classes are open and convex over each fixed J.
The standard finite-dimensional smoothing of a C^0 family of taming
forms by averaging under a smooth bump on the parameter interval,
followed by convolving with a small mollifier near each endpoint,
gives the requisite smooth family with matching endpoints. Since the
class H has a tamer at each parameter (the taming cone over
H is open and nonempty on both endpoints, and the tamer along
J_t interpolates), the interpolation is nowhere zero.


# The auxiliary volume equation


The prepared pair of Cref{prop:regular-preparation} satisfies
A^2=C^2=1, Awedge C=0 on X, and the restrictions of A,C to
the fibers vanish, with unit fiber area form Psi orthogonal to both
and equal to dd awedgedd b near the nodes. We now construct a
closed two-form D_eps with Awedge D_eps=Cwedge D_eps=0 and
D_eps^2=A^2 pointwise. The correction to a semi-flat approximate
solution is small and is solved on exact two-forms, where a
Stokes identity controls the inverse uniformly as the fiber area
eps shrinks.

## The semi-flat approximation

On the regular region X^circ, consider the form

 E_eps=eps,Psi+(1)/(eps)· A^2
 ·(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

 E_eps^{sf}=eps,dd awedgedd b
 +(tau_2(s))/(eps),dd x_1wedgedd x_2,
  s=x_1+ix_2,

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^{loc})^2=(ReΣ)^2=(ReΣ_j)^2,
and its fiber integral is eps, matching E_eps^{sf}
up to exponential error by
(eq:nodal-exponential-estimate).

[Gluing]
There is a closed two-form E_eps on X and an exact two-form
zeta_eps on the annular regions (fixed annuli near the punctures)
such that
[label=(roman*)]

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.

On each annulus, the difference
E_eps-E_eps^{sf}-zeta_eps has
|E_eps-E_eps^{sf}|_{C^m} bounded by
C_me^{-c_m/eps} for every mge0 (the constants depend only on
fixed smooth data).

The three wedge residuals
[
 R_
 (-Awedge E_eps,,
       -Cwedge E_eps,,
       (A^2-E_eps^2)/(2))
]
satisfy

 |R_|_{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.



Proof. 
On the regular region, E_eps^{sf} is closed and satisfies
the pointwise identities by direct computation:
dd awedgedd b wedges trivially with A and C because
A,C have only base--fiber mixed components; and
(E_eps^{sf})^2=A^2 because both A and C have
vanishing dd x_1wedgedd x_2 component while Psi has only
dd awedgedd b.

On the nodal disk, the model form D^{loc} satisfies
(eq:nodal-exact-identities).

On each annulus, use a fixed bounded primitive construction. Pick a
radial trivialization of the annular model torus bundle
Mtimes[r_0,r_1]. For an exact closed two-form zeta on the
annulus, radial homotopy gives
[
 zeta=pi_M^*(zeta|_{Mtimes{r_*}})+dd Kzeta,
 
 Kzeta(r)=int_{r_*}^{r}iota_{∂_r}zeta(u),dd u.
]
The restriction at r_* is exact; a Green operator on the fixed
compact M provides a fixed primitive. Adding its pullback to
Kzeta gives a global primitive on a slightly smaller annulus, with
C^j norm bounded by finitely many C^ell norms of zeta.
All operators and domains are independent of eps.

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
satisfying (i) and (ii). The residual bound (iii) now follows from
the explicit bounds

 a_eps:=int_XE_epswedge A=O(e^{-c/eps}),
 b_eps:=int_XE_epswedge C=O(e^{-c/eps}),

which hold because the exact gluing adjusted the two pairings. The
remaining residual R_ is exponentially small in the
fixed Sobolev norms after the standard Sobolev ineualities on the
fixed chart cover.


## Normalizing the moments and defining the approximate data

Set

 Ehat_E_eps&=E_eps-a_eps A-b_eps C,&
 q_eps&=int_XEhat_E_eps^2,notag
 r_eps&=q_eps^{-1/2},&
 D^0_eps&=r_epsEhat_E_eps.


The period normalization gives int A^2=int C^2=1 and
int Awedge C=0. So

 int_XAwedge D^0_eps=int_XCwedge D^0_eps=0,
 int_X(D^0_eps)^2=1.

The approximate equations give
a_eps,b_eps=O(e^{-c/eps}) and q_eps=1+O(e^{-c/eps}), after
reducing c>0; thus q_eps>0 and r_eps=1+O(e^{-c/eps}).
Subtracting constant multiples of A,C and multiplying by a
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

 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
volume.

[Bounds for the approximate triple]
For every fixed kge0, the following hold for sufficiently small
eps:
[label=(roman*)]

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}.

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.

Sobolev embedding and multiplication at a fixed large order kge6
have polynomial constants: there are C,q>0 with

 |hwedgewidetilde h|_{H^k}le Ceps^{-q}|h|_{H^k}|widetilde h|_{H^k}

for two-forms h,widetilde h.



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
identities; matrix inversion and square root preserve the polynomial
bounds.

For the coordinate cover: shrink the charts uniformly by powers of
eps so every point has a chart containing an intrinsic ball of
radius rho_epsge c_1eps^{a_1}. A maximal disjoint collection of
such balls has at most (2c_2eps^{a_2})^{-1} members by the volume
bound vol(X)=1/2; the enlarged balls cover X.
A coordinate cutoff and a normalized partition of unity then have
polynomial derivative bounds.

Euclidean Sobolev estimates on this cover give polynomial intrinsic
Sobolev constants; since dim X=4 and kge6, derivatives of order
at most floor( k/2) are L^∞-bounded by H^k. The
product rule on each derivative of hwedgewidetilde h gives
(eq:auxiliary-product).


[Exact inversion]
For every fixed kge6 and every sufficiently small eps, there is
a bounded linear map
[
 L_: (t_1,t_2,t_3)in H^ktimes H^ktimes H^k
    |-> hin H^k
]
such that
[label=(roman*)]

The form h is exact: int_Xhwedgephi=0 for every closed
phi; equivalently its harmonic projection is zero.

h solves
[
 (< h,A>_{L^2},
       < h,C>_{L^2},
       int_Xhwedge h)=(t_1,t_2,t_3),
]
in the sense that h is the unique element of E^k_
(exact H^k two-forms) with those three moments, and its H^k norm
satisfies
[
 |h|_{H^k}le Ceps^{-q}sup_i|t_i|_{H^k}.
]



Proof. [Proof sketch]
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,
  E_1=A, E_2=C, E_3=D^0_eps.

Moreover

 <phi,E_i>_{L^2}=int_Xf_i=t_i,

and the E_i are closed and self-dual, hence harmonic. Since
b^+=3, the E_i form a basis of the harmonic self-dual forms, and
(eq:auxiliary-cokernel-moments) is exactly the solvability
condition for dd^+a=phi.

Existence follows from the Hodge Green operator G for g_eps: since
phi is orthogonal to harmonic self-dual forms and the Hodge
Laplacian commutes with the star, Gphi is self-dual with
Delta Gphi=phi. For a self-dual two-form psi,
2(dddd^*psi)^+=Deltapsi, so

 h=2dddd^*Gphi

is exact and has self-dual part phi=h^+. This is a Green formula;
we control it by the standard Sobolev bounds for the Green operator on
g_eps (polynomial in eps^{-1} by Cref{lem:auxiliary-geometry}).
The remaining moments in h are matched by adding
dddelta to h for delta a one-form, which does not change h^+
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
standard way. Uniqueness is asserted for h, not for its one-form
primitive.


## The contraction


There is a positive constant c and, for every sufficiently small
eps>0, a smooth closed two-form D_eps=D^0_eps+h_eps, exact in
the class of D^0_eps, such that

 Awedge D_eps=Cwedge D_eps=0, (D_eps)^2=A^2.

The associated triple (A,C,D_eps) is positive; h_eps is
bounded by Ceps^{-N}e^{-c/eps} in every fixed Sobolev norm.


Proof. 
Fix kge6. Define the closed subspace
E^k_ H^k of exact two-forms. Writing
D_eps=D^0_eps+h with hinE^k_, the desired
equations become the fixed-point equation

 h=T_:=L_
 (-Awedge D^0_eps,,
       -Cwedge D^0_eps,,
       (A^2-(D^0_eps)^2-hwedge h)/(2)).

This map is defined on all of E^k_: the first two data
have integral zero by (eq:auxiliary-exact-moments), as does the
free term in the third slot; for each exact iterate,
int_Xhwedge h=0 by approximation and Stokes' theorem. Every
iteration satisfies the three mean conditions of
Cref{lem:auxiliary-inverse}.

The map T_ is a contraction on a small ball of
E^k_: 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 ain H^{k+1}
of h, so dd a=h and dd^*a=0. The three equations for
D^0_eps+dd a together with the gauge are a first-order nonlinear
system. Its linearization has invertible principal symbol: if a
covector-valued unknown b satisfies
xiwedge bwedge A=xiwedge bwedge C=xiwedge bwedge D_eps=0
and iota_{xi^sharp}b=0, then xiwedge b is in the negative-definite
wedge-orthogonal complement of the positive three-plane. But
(xiwedge b)^2=0 and this complement is negative definite, so
xiwedge b=0, forcing b proportional to xi and then to zero
by the gauge. Elliptic bootstrapping then gives smoothness in every
finite order.


## The color class and the numerical chamber property

The last step is to extract, from the prepared construction of
D_eps, a uniform numerical estimate in the cohomology lattice.
Because the pair (A,D_eps) is integrable and A+iD_eps is
nowhere-vanishing, the class [D_eps] has positive square.
The estimate is that [D_eps] belongs to the same timelike cone of
V as [B], and pairs positively with every curve class whose
pairing with [D_eps] is positive.

[Uniform chamber estimate]
Retain the prepared pair (A,C), the torus fibration
p:Xto S of Cref{thm:pencil}, and the chamber class F of
Cref{prop:chamber-class}. There is a positive constant eps_0
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

 [D_eps]=lambda_eps F+eps u_eps,
 |lambda_eps-1/(2eps)|le C, |u_eps|le C,

with C independent of eps, and the following uniform
implication: for every positive real number eps and every
dinΛ with d^2ge-2 and [D_eps]· d>0, we have
[B]· d>0.


Proof. 
By the construction of D_eps, its intersection with F is
[D_eps]F=eps+O(eps) (the exact gluing changed only exponentially
small). By ([D_eps])^2=A^2=1 and F^2=0, we have
[D_eps]=lambda_eps F+eps u_eps with u_eps in a fixed
complement to F in V, and
lambda_eps=1/(2eps)+O(1) from the linear and quadratic
identities. The exact construction data are independent of eps, so
C and eps_0 are too.

For the implication: by Cref{cor:curve-signs}, either
d=cF for some cinR (in which case [B]· d =
c[B]· F=cbeta>0 whenever c>0), or d has the sign of the
coefficient of F^+ in d^+. In the first case, [B]· d=
beta c and the sign of [D_eps]· 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]· d; the condition [D_eps]· d>0 together with the
expansion (eq:auxiliary-class-expansion) then forces the same
sign. In both cases [B]· d>0.


# 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
positive and each of its three classes stays fixed. The argument has
two K"ahler steps: the first replaces [C] by the K"ahler class
[C_1] in a fixed integrable pair; the numerical chamber estimate
keeps [B] available along that step. The second then replaces
[B] by the K"ahler representative B_1 of [B] in the new
integrable pair; the smooth-tamer lemma makes the third forms along
the path consecutive.

## The K"ahler-class criteria

We need the following standard facts about compact complex surfaces.
Both are applied to the compact complex surface (X,J) built from a
pair (A,D) with AD=0 and A^2=D^2>0 (Lemma~lem:complex-symplectic):

[K"ahler classes and volumes]
Let (X,J) be a compact complex surface with b_1 even, and let
Hin H^2(X;R) be a class such that H^2>0 and H pairs positively
with every irreducible closed J-holomorphic curve. Then H
contains a K"ahler form. Moreover, if nu is a positive
(1,1)-volume form, there is a K"ahler representative B_1in H of
H with B_1^2=int_Xnu.


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
[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
representative can be deformed in its class to have prescribed total
volume int_Xnu.


[Hodge type at the endpoint]
Suppose (A,C_1) is an integrable equal-square closed pair with
AC_1=0 and A^2=C_1^2>0, and let J be the complex structure with
holomorphic volume form A+iC_1 (for the appropriate sign). Let
[B] be a class in V=[A]^perpcap[C_1]^perp. Then [B]
belongs to H^{1,1}(X;J).


Proof. 
Because the triple (A,C_1,B) is an orthonormal triple with respect
to the positive intersection form, and A+iC_1 is a
nonvanishing (2,0) form, the classes [A] and [C_1] span the
(1,1)-part of the period plane. Any class in V is orthogonal to
both, and is therefore of Hodge type (1,1): use the Hodge
decomposition and the fact that span([A],[C_1]) is
the -1-eigenspace of the complex structure lifted to H^2 for the
appropriate orientation.


## The prepared endpoint

Recall from the combining theorems:
 Cref{thm:pencil,prop:nonsplitting} give an exact pair
deformation to a torus fibration with the standard nodal
neighborhoods, keeping B fixed.
 Cref{prop:regular-preparation} gives a further path of positive
closed triples to invariant regular data with auxiliary form Psi.
 Cref{thm:auxiliary-solution,prop:auxiliary-class} then supply
D_eps exactly.

All stages preserve the three individual cohomology classes and start
at the actual representatives supplied by the preceding stage.
We retain the notation (A,C,B) for the resulting triple. In this
notation

 [A]^2=[C]^2=[B]^2=1,
 [A][C]=[A][B]=[C][B]=0.


Fix a sufficiently small ε>0 for
Cref{thm:auxiliary-solution,prop:auxiliary-class}, and set
D=D_ε. The form D is closed and smooth and satisfies

 AD=CD=0, D^2=A^2>0.

The triple with forms A,C,D is positive. The same choice of
ε gives the numerical conclusion of
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

By Cref{lem:complex-symplectic}, the pair (A,D) is integrable and
A+iD is a nowhere-vanishing holomorphic two-form for the complex
structure J_0 that it defines. Choose the sign of J_0 so that C
tames J_0. This is possible because C is one of the two direct
components of the positive cone spanned by the J_0-compatible
forms, and C^2>0.

Apply Cref{lem:endpoint-kahler-class} with H=[C], J=J_0 and
nu=A^2/2. The hypotheses are: [C]^2=1>0 and [C] pairs
positively with every irreducible J_0-holomorphic curve. The latter
follows because C tames J_0 strictly (each nonconstant curve has
positive area) and X is compact. So [C] contains a K"ahler
representative C_1, with C_1^2=A^2.

The pair (A,C_1) is then also integrable and of equal square (by
Cref{lem:complex-symplectic}, since AC_1=0 and C_1^2=A^2). Let
J_1 denote the complex structure it defines. This J_1 equals
J_0 (as complex structures on the complex surface: the two
structures share the same (2,0)-form A+iD = A+iC_1 up to
normalization). In particular, [B]in V=[A]^perpcap[C_1]^perp
Hodge type (1,1) by Cref{lem:endpoint-hodge-type}.

We now interpolate the pair from (A,C) to (A,C_1). Set
E_lambda=C+lambda(C_1-C), lambdain[0,1]. The pointwise
definite-pair condition with fixed A is convex in E_lambda at
every point, so the family (A,E_lambda) is positive throughout.
Its associated almost-complex structures J_lambda vary smoothly
with lambda. Let eta_lambda=(1)/(2)Re(A+iE_lambda)
be a smooth J_lambda-compatible tamer (semipositive where the
pair degenerates). Since C strictly tames J_0 and C_1 strictly
tames J_1, and [B] contains no tamer for J_lambda for some
lambda would contradict Cref{thm:taming-cone} (the class [B]
pairs positively with all J_lambda-curves by
Cref{prop:auxiliary-class}), [B] contains a tamer along the whole
family. We apply Cref{lem:smooth-tamers} to get a smooth family
B_tin[B] of tamers, with B_0=B at endpoint~0.

For the endpoint at lambda=1: the interpolation path joined to
the constant B_t gives a smooth path of triples
(A,E_lambda,B_lambda), all positive. This is the first K"ahler
step.

## The second K"ahler step

Now use the integrable pair (A,C_1) and its structure J_1. Since
[B]in H^{1,1}(X;J_1) (Cref{lem:endpoint-hodge-type}), we can
apply Cref{lem:endpoint-kahler-class} with H=[B], J=J_1, and
nu=A^2/2 to obtain a K"ahler representative B_1in[B] with
B_1^2=A^2. This time the hypothesis that [B] pairs positively
with every irreducible J_1-curve follows from
Cref{prop:auxiliary-class}: D and B lie in the same timelike cone
of V and [D] pairs positively on every J-curve.

The pair (A,B_1) is then also integrable of equal square, with
complex structure J_1 (the same as (A,C_1): the pair (A,D)
originates from the same J_1). Interpolate from (A,B) to
(A,B_1) as before. The endpoint pair (A,B_1) has [A]·[B_1]
=0 and B_1^2=A^2 (the first is [A][B]=0; the second is
B_1^2=A^2). Concatenate with the interpolation to (A,C_1).

## Finishing

Concatenate all stages, using the specific positive closed triples
provided by the construction of each stage, and reparametrize each
path to be constant at its endpoints. The result is a smooth path of
closed positive triples on [0,1]:

[
 (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).
]

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_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

[t]
centering
[node distance=1.4cm,
    every node/.style={font=small},
    box/.style={draw,rounded corners,fill=gray!8,inner sep=3pt}]
 node[box] (orig)    at (0,0)    {original (A,C,B)};
 node[box,right=of orig] (pencil) {pencil (A,C,B)};
 node[box,right=of pencil] (prep)   {prepared (A,C,B) + Psi};
 node[box,right=of prep] (aux)    {auxiliary D=D_eps};
 node[box,below=of prep] (K1)     {1st K"ahler: C_1};
 node[box,below=of aux]  (K2)     {2nd K"ahler: B_1};
 node[box,below=of K1]   (end)    {(A,C_1,B_1) hyperk"ahler};

 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
steps, the tamers B_t are chosen with initial value B and final
value B_1.



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