# Constructions of symplectic forms on 4-manifolds

Peter Lambert-Cole  Department of Mathematics, University of Georgia, Athens,
GA 30602<EMAIL_ADDRESS>
(Date: March 2024)

###### Abstract.

Given a symplectic 4-manifold $(X,\omega)$ with rational symplectic form,
Auroux constructed branched coverings to $(\mathbb{CP}^{2},\omega_{FS})$. By
modifying a previous construction of Lambert-Cole–Meier–Starkston, we prove
that the branch locus in $\mathbb{CP}^{2}$ can be assumed holomorphic in a
neighborhood of the spine of the standard trisection of $\mathbb{CP}^{2}$.
Consequently, the symplectic 4-manifold $(X,\omega)$ admits a cohomologous
symplectic form that is Kähler in a neighborhood of the 2-skeleton of $X$. We
define the Picard group of holomorphic line bundles over the holomorphic
2-skeleton. We then investigate Hodge theory and apply harmonic spinors to
construct holomorphic sections over the Kähler subset.

###### 2010 Mathematics Subject Classification:

##  1\. Introduction

###  1.1. Motivation

The motivation for this work is the geography problem for symplectic
4-manifolds and the symplectic BMY inequality. The Bogomolev-Miyaoka-Yau
inequality for complex surfaces states that if $X$ is a surface of general
type, then

(1.1) $c_{1}^{2}\leq 3c_{2}$

Miyaoka’s proof proceeds via complex algebraic geometry [Miy77], while Yau’s
proof derives from his solution to the Calabi conjecture [Yau77]. It is
conjectured that the same inequality holds for closed symplectic 4-manifolds.

######  Conjecture 1.1.

Let $(X,\omega)$ be a closed, minimal symplectic 4-manifold satisfying
$c_{1}(X)^{2}\geq 0$. Then $c_{1}(X)^{2}\leq 3c_{2}(X)$.

One aim of this paper is to investigate whether it is possible to mimic these
methods from complex algebraic geometry in the symplectic case. The approach
we take it to construct explicit symplectic forms that appear well-adapted to
the smooth topological problem. We give two constructions, of a so-called
Kähler trisection and a symplectic 1-cocycle. The underlying approach is the
same in both cases, but produces slightly different geometric structures.

###  1.2. Construction

The constructions in this paper are slight variations of the construction in
[LCMS21], with modifications according to the geometric structures of
interest.

Given an integral symplectic form $\omega$ on $X$, Auroux constructed branched
coverings $\pi_{k}:X\rightarrow\mathbb{CP}^{2}$ for $k\gg 0$ [Aur00]. The
symplectic form $k[\omega]$ can be recovered by pulling back the Fubini-Study
form $\omega_{FS}$ by $\pi_{k}$ and perturbing to be nondegenerate. The smooth
topology of the branched covering depends a priori on the integral cohomology
class of $\omega$ and is independent of choices for $k$ sufficiently large.
Auroux and Katzarkov strengthened this result, so that the branch locus in
$\mathbb{CP}^{2}$ is a so-called quasiholomorphic curve [AK00]. Such a branch
locus can be algebraically encoded by a factorization of the full twist in the
braid group. In [LC23], it was shown that the branch locus $\mathcal{R}$ in
$\mathbb{CP}^{2}$ can be smoothly isotoped into transverse bridge position
with respect to the standard trisection of $\mathbb{CP}^{2}$. The trisection
of $\mathbb{CP}^{2}$ then pulls back to a trisection of $X$, along with the
relevant geometric structure.

###  1.3. Holomorphic trisections and Kähler-like forms

An almost-Kähler manifold is a triple $(X,\omega,J)$ where $\omega$ is a
symplectic form and $J$ is an almost-complex structure satisfying the
compatibility condition $\omega(JX,JY)=\omega(X,Y)$ everywhere. The pair
$(\omega,J)$ determine a Riemannian metric $g(X,Y)=\omega(X,JY)$ on $X$. The
pair $(\omega,J)$ determines a Kähler structure if $J$ is integrable.

Our first main result is a construction of almost-Kähler structures that are
integrable over the interesting part of the smooth topology of $X$.

######  Corollary 1.2.

Let $(X,\omega)$ be a closed, symplectic 4-manifold with rational symplectic
form. There exists an almost-Kahler structure $(X,\omega^{\prime},J)$ with
$[\omega^{\prime}]=[\omega]\in H^{2}_{DR}(X)$ and a smooth handle
decomposition of $X$ such that $J$ is integrable over the 2-skeleton.

This follows immediately from the next theorem, as a trisection determines an
‘inside-out’ smooth handle decomposition contained in the spine of the
trisection.

######  Theorem 1.3.

Let $(X,\omega)$ be a closed, symplectic 4-manifold with rational symplectic
form. There exists an almost-Kahler structure $(X,\omega^{\prime},J)$ with
$[\omega^{\prime}]=[\omega]\in H^{2}_{DR}(X)$ and a trisection $\mathcal{T}$
of $X$ such that $J$ is integrable in a neighborhood of the spine
$\mathcal{S}$ of $\mathcal{T}$.

Moreover, there exists a decomposition $X=\nu(\mathcal{S})\cup Z$ such that

  1. (1)

$(\nu(\mathcal{S}),\omega^{\prime},J)$ is Kähler with strictly pseudoconcave
boundary,

  2. (2)

$Z$ is a (disconnected) subcritical Weinstein domain.

There is a decomposition of $\mathbb{CP}^{2}$ as in Theorem 1.3 depicted in
terms of its moment polytope in Figure 1. The standard trisection of
$\mathbb{CP}^{2}=Z_{1}\cup Z_{2}\cup Z_{3}$ arises by taking the unit bidisk
in each of the three standard affine charts. Letting
$Z^{\prime}_{\lambda}\subset Z_{\lambda}$ be a small shrinking with strictly
pseudoconvex boundary, we let $Z=Z^{\prime}_{1}\cup Z^{\prime}_{2}\cup
Z^{\prime}_{3}$ and $\nu(\mathcal{S})$ be its complement, which has strictly
pseudoconcave boundary. Pulling this decomposition back by a symplectic
branched covering yields the decomposition of Theorem 1.3.

Figure 1. The decomposition of Theorem 1.3 applied to the moment polytope of
$\mathbb{CP}^{2}$

Since every subcritical Weinstein domain embeds symplectically in
$(\mathbb{C}^{2},\omega_{std})$, we have the following corollary.

######  Corollary 1.4.

Let $(X,\omega)$ be a closed, symplectic 4-manifold with rational symplectic
form. There exists an almost-Kahler structure $(X,\omega^{\prime},J)$ with
$[\omega^{\prime}]=[\omega]\in H^{2}_{DR}(X)$ such that $J$ is integrable,
except on three trivial symplectic cylinders of the form $\left([a,b]\times
Y_{\lambda},d(e^{t}\alpha)\right)$, where $Y_{\lambda}$ is $S^{3}$ or
$\\#_{k}S^{1}\times S^{2}$ for some $k>0$.

######  Remark 1.5.

The subtlety that $[\omega^{\prime}]$ and $[\omega]$ are merely cohomologous
is due to the fact that we appeal to the result of [LC23], where the branch
locus is only smoothly isotoped into transverse bridge position, as opposed to
the Hamiltonian isotopy used in [LCMS21].

######  Remark 1.6.

Viewed with respect to the inward-pointing coorientation, Theorem 1.3 states
that each boundary component admits a strongly pseudoconvex CR-structure. If
this CR-structure can be holomorphically embedded in $\mathbb{C}^{2}$, then
the Kähler structure could be extended across the subcritical Weinstein
domain. However, generic CR-structures on 3-manifolds do not holomorphically
embed in $\mathbb{C}^{2}$ and furthermore the existence of non-Kähler
symplectic 4-manifolds implies that these boundary components cannot always be
filled in holomorphically.

Since $J$ is integrable over the spine, we can attempt to mimic objects from
algebraic geometry. In particular, we introduce the Picard group
$\text{Pic}(X,J,\mathcal{T})$ of holomorphic line bundles (Definition 5.1). In
fact, every isomorphism class of complex line bundle admits a holomorphic
representative in some neighborhood of the 2-skeleton.

######  Theorem 1.7.

Let $(X,\omega)$ be a closed symplectic 4-manifold with rational symplectic
form. There exists an almost-Kähler structure $(X,\omega^{\prime},J)$, a
trisection $\mathcal{T}$ of $X$, and decomposition $X=\nu(\mathcal{S})\cup Z$
as in Theorem 1.3 such that for any $\alpha\in H^{2}(X;\mathbb{Z})$, there is
some open subneighborhood $\nu(\mathcal{S})^{\prime}\subset\nu(\mathcal{S})$
of the 2-skeleton and a holomorphic line bundle $L$ over
$\nu(\mathcal{S})^{\prime}$ such that $c_{1}(L)=\alpha$.

Neither Corollary 1.2 nor Corollary 1.4 require a smooth trisection
decomposition, only Auroux’s branched covering result. One only needs to apply
Lemma 4.7 to the 1-skeleton of the branch curve to realize the holomorphic
structures. However, constructing a holomorphic structure on every smooth line
bundle as in Theorem 1.7 does require the smooth trisection.

We then attempt Hodge theory over the integrable region $\nu(\mathcal{S})$,
viewed as a Kähler manifold with pseudoconcave boundary. We can recover the
Hodge-Riemann bilinear relations in subcritical dimensions (Theorem 5.2). To
extract more information, we use harmonic spinors. Every holomorphic line
bundle in the Picard group can be extended to a complex line bundle on $X$.
After choosing a Hermitian connection $A$, we obtain a
$\text{Spin}^{\mathbb{C}}$ Dirac operator
$\mathcal{D}_{A}=\mathcal{D}^{+}_{A}+\mathcal{D}^{-}_{A}$ on the bundle
$\Lambda^{0,*}(L)$; the index of this operator is given by the classical
Riemann-Roch formula. Over the integrable region $\nu(\mathcal{S})$, the space
of even harmonic spinors $\text{ker}(\mathcal{D}^{+}_{A})$ decomposes into
$L$-valued $(0,0)$\- and $(0,2)$-forms. In particular, if $\chi(L)>0$, we
obtain holomorphic sections of $L$ or its Serre dual $K-L$ over
$\nu(\mathcal{S})$. The vanishing locus of such a section is a symplectic
surface in $\nu(\mathcal{S})$ whose boundary is transverse to the contact
structure on $\partial\nu(\mathcal{S})$.

###  1.4. Symplectic cone

The symplectic cone of a smooth 4-manifold $X$ is the subset of
$H^{2}_{DR}(X)$ that can be realized by symplectic forms. Similarly, the
Kähler cone of a complex surface $X$ is the subset of $H^{1,1}(X)$ that can be
realized by Kähler forms. Determining the symplectic cone of a general
4-manifold remains an open problem in general, although it is understood for
some classes of 4-manifolds. Seiberg-Witten invariants give some restrictions
on the symplectic cone. And determining the symplectic cone of
$X\\#k\overline{\mathbb{CP}}^{2}$ is connected to the problem of symplectic
ball packing [Bir01]. In the Kähler case, the Nakai-Moishezon criterion for
ampleness of holomorphic line bundles gives information about the Kähler cone.
There exists a symplectic Nakai-Moishezon-type criterion in the case when
$b_{2}^{+}=1$ [Bir97, Bir01, LL01], but there is no corresponding general
theorem.

Using harmonic spinors and Theorem 1.7, we can construct partial holomorphic
sections of over the spine $\nu(\mathcal{S})$. In particular, if $D^{2}>0$
then either $nD$ or $K-nD$ admit many holomorphic sections over the spine
$\nu(\mathcal{S})$. The projective holomorphic embedding

$\iota:\nu(\mathcal{S})\hookrightarrow\mathbb{CP}^{N}$

determined by this linear system gives a Kähler form $\iota^{*}(\omega_{FS})$
over the spine $\nu(\mathcal{S})$ that dominates the contact structures along
the boundary $\partial\nu(\mathcal{S})$. In other words, it gives
$\nu(\mathcal{S})$ the structure of a weak symplectic cap with $\omega$ in a
prescribed cohomology class.

If this form satisfies the stronger condition of being contact-type along the
boundary — in other words, being a strong symplectic cap — then the boundary
components can be filled in with symplectic 1-handlebodies to obtain a
symplectic form $\omega$ on the closed 4-manifold $X$ with $\omega$ in a
prescribed integral cohomology class of positive square.

However, it is not immediately obvious how to ensure this stronger condition
or how to incorporate the restrictions from Seiberg-Witten theory.

###  1.5. Symplectic 1-cocycles

The notion of a Weinstein trisection of $(X,\omega)$ was introduced in
[LCM22]. It is a trisection decomposition $X=X_{1}\cup X_{2}\cup X_{3}$ such
that there exists a subcritical Weinstein structure on each sector
$X_{\lambda}$ compatible with the global symplectic form. Lambert-Cole, Meier
and Starkston later showed that every closed, symplectic 4-manifold admits a
Weinstein trisection [LCMS21]. A converse condition was introduced in [LC22a]
to ensure that a given trisection admits a global symplectic structure.

######  Definition 1.8.

Let $X$ be a closed, oriented 4-manifold and let $\mathcal{T}$ be a trisection
of $X$ with central surface $\Sigma$ and 3-dimensional sectors
$H_{1},H_{2},H_{3}$. A symplectic 1-cocycle for $(X,\mathcal{T})$ is a triple
$(\widetilde{\beta}_{1},\widetilde{\beta}_{2},\widetilde{\beta}_{3})$ of
closed, nonvanishing 1-forms on the three handlebodies such that the
restrictions $\\{\beta_{\lambda}=\widetilde{\beta}_{\lambda|_{\Sigma}}\\}$
satisfy the following conditions

  1. (1)

the cocycle condition:

$\beta_{1}+\beta_{2}+\beta_{3}=0$

  2. (2)

the pairwise positivity condition

$\beta_{\lambda}\wedge\beta_{\lambda+1}\geq 0$

  3. (3)

they all vanish at the same $2g-2$ points $b_{1},\dots,b_{2g-2}$ in $\Sigma$
and there exists a smooth function $f:\Sigma\rightarrow\mathbb{R}$ and
neighborhoods $\\{b_{i}\in U_{i}\\}$ such that $f|_{U_{i}}=\pm 1$ and
$\sum_{i}f(b_{i})=0$ and

$\widetilde{\beta}_{\lambda}=\beta_{\lambda}+d(sf)$

on a collar neighborhood of $\partial H_{\lambda}=\Sigma$.

Recall that a boundary component of a symplectic manifold is strongly concave
if there exists an inward-pointing Liouville vector field $\rho$. This is
equivalent to the criterion that the form $\alpha=\omega(\rho,-)$, which is a
primitive for $\omega$, is a negative contact form on that boundary component
(with respect to the outward-pointing coorientation).

######  Theorem 1.9.

Let $X$ be a closed, oriented 4-manifold and let $\mathcal{T}$ be a trisection
of $X$ with central surface $\Sigma$. Let
$(\widetilde{\beta}_{1},\widetilde{\beta}_{2},\widetilde{\beta}_{3})$ be a
symplectic 1-cocycle for $(X,\mathcal{T})$. Then

  1. (1)

there is a symplectic form $\omega$ on $\nu(H_{1}\cup H_{2}\cup H_{3})$ and
three inward-pointing Liouville vector fields for $\omega$ along each of the
three boundary components $\widehat{Y}_{1},\widehat{Y}_{2},\widehat{Y}_{3}$.

  2. (2)

if each of the induced contact structures
$(\widehat{Y}_{\lambda},\widehat{\xi}_{\lambda})$ are tight, then there exists
a global symplectic form $\omega$ on $X$.

In this paper, we prove two important existence statements about symplectic
1-cocycles and trisections. In particular, we show that the criterion in
[LC22a] is completely general.

######  Theorem 1.10.

Let $(X,\omega)$ be a closed, symplectic 4-manifold. Then

  1. (1)

there exists a trisection $\mathcal{T}$ and symplectic 1-cocycle for
$(X,\mathcal{T})$ such that all three induced contact structures on
$\partial\nu(H_{1}\cup H_{2}\cup H_{3})$ are tight.

  2. (2)

every stabilization of $\mathcal{T}$ admits a symplectic 1-cocycle such that
the induced contact structures on $\partial\nu(H_{1}\cup H_{2}\cup H_{3})$ are
tight.

###  1.6. Comparison

A key fact is that any two smooth trisections
$\mathcal{T}_{1},\mathcal{T}_{2}$ of $X$ admit a common stabilization
$\mathcal{T}^{\prime}$. The branch locus in the symplectic constructions
described here can be perturbed to achieve a smooth stabilization of the
trisection in the cover. Consequently, we can directly compare two distinct
symplectic structures on some $X$ via a common trisection.

######  Theorem 1.11.

Let $\omega_{1},\omega_{2}$ be two rational symplectic forms on $X$. There
exists a single smooth trisection $\mathcal{T}$ of $X$ such that the
conclusions of Theorem 1.3 and Theorem 1.10 hold for both $\omega_{1}$ and
$\omega_{2}$, with respect to $\mathcal{T}$.

###  1.7. Acknowledgements

I am deeply indebted to Phil Engel for explaining Miyaoka’s proof to me in an
intelligble way and for the initial suggestion to consider harmonic spinors in
this context.

##  2\. Trisections

Trisections are 4-dimensional generalization of Heegaard splittings of
3-manifolds. In this section, we will describe the basic terminology and
background results of the theory. At this point, there is quite a voluminous
literature defining the basic building blocks of the theory:

  1. (1)

trisections of 4-manifolds [GK16]

  2. (2)

relative trisections of 4-manifolds with boundary [CGPC18]

  3. (3)

bridge trisections of surfaces [MZ17, MZ18]

  4. (4)

constructing branched coverings via surfaces in bridge position [LCM22,
LCMS21]

  5. (5)

bridge trisections for properly embedded surfaces in 4-manifolds with boundary
[Mei20]

The two key ideas we exploit in this paper are:

  1. (1)

Trisections pull back under branched coverings when the branch locus is in
bridge position. This has been used to construct trisections of projective
surfaces [LCM22] and symplectic 4-manifolds [LCMS21].

  2. (2)

In a simple branched covering, an (interior) perturbation of the branch locus
downstairs results in an (interior) stabilization of the trisection upstairs.

###  2.1. Trisections

Let $X$ be a closed, smooth 4-manifold.

######  Definition 2.1.

A $(g;k_{1},k_{2},k_{3})$-trisection $\mathcal{T}$ of $X$ is a decomposition
$X=Z_{1}\cup Z_{2}\cup Z_{3}$ such that for each $\lambda\in\mathbb{Z}_{3}$,

  1. (1)

Each $Z_{\lambda}$ is diffeomorphic to $\natural_{k_{\lambda}}(S^{1}\times
B^{3})$ for $k_{\lambda}\geq 0$,

  2. (2)

Each double intersection $H_{\lambda}=Z_{\lambda-1}\cap Z_{\lambda}$ is a
3-dimensional, genus $g$ handlebody, and

  3. (3)

the triple intersection $\Sigma=Z_{1}\cap Z_{2}\cap Z_{3}$ is a closed, genus
$g$ surface.

We refer to $\Sigma$ as the central surface of the trisection, the genus $g$
of $\Sigma$ as the genus of the trisection, and the union $H_{1}\cup H_{2}\cup
H_{3}$ as the spine of the trisection.

A trisection of $X$ can be encoded by a trisection diagram that describes how
each $H_{\lambda}$ is attached to $\Sigma$ [GK16, Definition 3]. In
particular, since $H_{\lambda}$ is a 1-handlebody, the gluing is determined by
how a complete system of compressing disks is attached to $\Sigma$. These
compressing disks are attached along a cut system of curves in $\Sigma$ and
these curves can be depicted on $\Sigma$ itself.

###  2.2. Trisections of spheres

The $n$-sphere has a standard trisection, obtained by pulling back a
trisection on the unit disk $D$ in $\mathbb{R}^{2}$.

######  Definition 2.2.

The standard trisection $D=D^{s}_{1}\cup D^{s}_{2}\cup
D^{s}_{3}\subset\mathbb{R}^{2}$ of the unit disk is defined by choosing the
following subsets (in radial coordinates):

$\displaystyle D^{s}_{1}$
$\displaystyle=\left\\{0\leq\theta\leq\frac{2\pi}{3}\right\\},$ $\displaystyle
D^{s}_{2}$
$\displaystyle=\left\\{\frac{2\pi}{3}\leq\theta\leq\frac{4\pi}{3}\right\\},$
$\displaystyle D^{s}_{3}$ $\displaystyle=\left\\{\frac{4\pi}{3}\leq\theta\leq
2\pi\right\\}.$

It is symmetric under rotation by $2\pi/3$.

######  Definition 2.3 ([LCM21]).

The standard trisection of $S^{k}$ for $k\geq 2$ is the decomposition
$\mathcal{T}_{std}=\\{\pi^{-1}(D^{s}_{i})\cap S^{k}\\}$ where
$\pi:\mathbb{R}^{k+1}\rightarrow\mathbb{R}^{2}$ is a coordinate projection and
$D^{s}_{1}\cup D^{s}_{2}\cup D^{s}_{3}$ is the standard trisection of the unit
disk in $\mathbb{R}^{2}$.

######  Example 2.4.

The standard trisection of $S^{2}$. The triple intersection of
$\mathcal{T}_{std}$ consists of the North and South poles; the double
intersections are three meridians connecting the poles; and the sectors are
the three bigons bounded by the pairs of meridians.

######  Example 2.5.

The standard trisection of $S^{3}$. The triple intersection consists of a
single $S^{1}$, unknotted in $S^{3}$; each double intersection is a $D^{2}$
Seifert disk for the unknot; and each sector of the trisection is a 3-ball. In
addition, we can give two other useful interpretations:

  1. (1)

The lens space $L(1,3)$. Here, $S^{3}$ is decomposed into three lenses: i.e
take $D^{2}\times[0,1]$ and collapse the vertical boundary to a circle. These
lenses are glued sequentially along their boundaries.

  2. (2)

The genus 0 open book decomposition of $S^{3}$. Recall that $S^{3}$ has an
open book decomposition $(B,\phi)$ where the binding $B$ is the unknot and the
fibration $\phi:S^{3}\smallsetminus\nu(B)\cong S^{1}\times D^{2}\rightarrow
S^{1}$ is trivial. This induces a trisection as follows. Choose any three
distinct pages
$\left\\{\phi^{-1}(\theta_{1}),\phi^{-1}(\theta_{2}),\phi^{-1}(\theta_{3})\right\\}$
of the open book decomposition. Their union becomes the spine of the
trisection and the binding $B$ is the codimension 2 stratum of the trisection.
The complement of the spine consists of three open 3-balls.

The following lemma is immediate from the definition.

######  Lemma 2.6.

Let $\mathcal{T}_{std}=(S_{1},S_{2},S_{3})$ be the standard trisection of
$S^{k}$. Then each $S_{i}$ is diffeomorphic to $B^{k}$; each double
intersection $S_{i}\cap S_{j}$ is diffeomorphic to $B^{k-1}$ and the central
surface is diffeomorphic to $S^{k-2}$.

In particular, the standard trisection of $S^{4}$ is a $(0;0)$-trisection.

######  Example 2.7.

$S^{4}$ also admits an unbalanced $(1;1,0,0)$-trisection. It can be described
by a trisection diagram on $T^{2}$ with attaching curves
$\alpha_{1},\alpha_{2}$ parallel and the third attaching curve $\alpha_{3}$
geometrically dual to both. This implies that $Y_{1}=H_{1}\cup-H_{2}$ is
homeomorphic to $S^{1}\times S^{2}$ and bounds $S^{1}\times B^{3}$, while
$Y_{2}=H_{2}\cup-H_{3}$ and $Y_{3}=H_{3}\cup-H_{1}$ are homeomorphic to
$S^{3}$ and bound $B^{4}$. By permuting the indices, we also obtain unbalanced
$(1;0,1,0)$ and $(1;0,0,1)$-trisections of $S^{4}$.

###  2.3. Connected sum and stabilization

######  Definition 2.8.

Let $X$ and $Y$ be closed, smooth, oriented 4-manifolds with trisections
$\mathcal{T}_{X},\mathcal{T}_{Y}$ given by the decompositions:

$X=X_{1}\cup X_{2}\cup X_{3}\qquad\text{and}\qquad Y=Y_{1}\cup Y_{2}\cup
Y_{3}$

The connected sum of $\mathcal{T}_{X}$ and $\mathcal{T}_{Y}$ is the trisection
$\mathcal{T}_{Z}$ of $Z=X\\#Y$ obtained by performing the connected sum along
points $x\in\Sigma_{X}\subset X$ and $y\in\Sigma_{Y}\subset Y$ in the central
surfaces of the trisections. Each sector is precisely
$Z_{\lambda}=X_{\lambda}\natural Y_{\lambda}$. If $X$ has a
$(g;k_{1},k_{2},k_{3})$-trisection and $Y$ has a
$(h;l_{1},l_{2},l_{3})$-trisection, then this construction gives a
$(g+h;k_{1}+l_{1},k_{2}+l_{2},k_{3}+l_{3})$-trisection of $Z$.

Recall from Example 2.7 that $S^{4}$ admits an (unbalanced) trisection of
genus 1, where the sector $X_{\lambda}\cong S^{1}\times B^{3}$ and the
remaining sectors are 4-balls. Let $\mathcal{T}_{\lambda}$ denote this
trisection.

######  Definition 2.9.

A sector-$\lambda$ stabilization of a trisection $\mathcal{T}$ of $X$ is the
trisection $\mathcal{T}^{\prime}$ obtained by taking the connected sum of
$\mathcal{T}$ with the trisection $\mathcal{T}_{\lambda}$ of $S^{4}$

Stabilization increases the trisection genus by 1. Note that in [GK16,
Definition 8], stabilizations are assumed to be balanced and increase the
genus by 3; however this operation is equivalent to three unbalanced
stabilizations, one in each sector.

######  Remark 2.10.

There is an equivalence between trisection decompositions and handle
decompositions [GK16, Section 4]. Under this equivalence, an unbalanced
stabilization of a trisection translates to adding a pair of canceling
handles.

######  Theorem 2.11 ([GK16]).

Let $X$ be a closed, smooth, oriented 4-manifold. Then

  1. (1)

$X$ admits a trisection,

  2. (2)

any two trisections of $X$ become ambient isotopic after sufficiently many
stabilizations, and

  3. (3)

$X$ is determined up to diffeomorphism by the spine of any trisection
$\mathcal{T}$.

######  Remark 2.12.

All of the results in [GK16] assume balanced trisections, meaning
$k_{1}=k_{2}=k_{3}$, although in practice it is often useful to consider
unbalanced trisections. It is always possible to balance a trisection by
stabilization, so the distinction is irrelevant here.

###  2.4. Relative trisections

Let $Z_{0}=B^{4}$ and $Z_{k}=\natural_{k}S^{1}\times B^{3}$ and let
$Y_{k}=\partial Z_{k}$. For any integers $r,s$ let $\Sigma_{s,t}$ denote the
compact, oriented surface of genus $s$ and $t$ boundary components. For any
integers $r,s,t$, let $H_{r,s,t}$ denote a compression body cobordism between
$\Sigma_{r,t}$ and $\Sigma_{s,t}$. In particular, if $r=s$, then
$H_{r,s,t}=\Sigma_{r,t}\times[0,1]$ and if $r>s$, then $H_{r,s,t}$ is obtained
from $H_{r,r,t}$ by attaching $r-s$ 3-dimensional 2-handles.

Let $g,p,b$ be nonnegative integers with $g\geq p$ and $b\geq 1$. Following
[GK16, CGPC18], there is a decomposition

$Y_{k}=Y^{-}_{g,k;p,b}\cup Y^{0}_{g,k;p,b}\cup Y^{+}_{g,k;p,b}$

where

$\displaystyle Y^{+}_{g,k,p,b}$ $\displaystyle\cong H_{g,p,b}$ $\displaystyle
Y^{0}_{g,k,p,b}$ $\displaystyle\cong\Sigma_{p,b}\times[0,1]$ $\displaystyle
Y^{-}_{g,k,p,b}$ $\displaystyle\cong H_{g,p,b}$

######  Definition 2.13.

Let $X$ be a compact, connected, oriented 4-manifold with connected boundary
$Y$. A $(g,k_{1},k_{2},k_{3};p,b)$-relative trisection of $X$ is a
decomposition

$X=X_{1}\cup X_{2}\cup X_{3}$

where

  1. (1)

for each $\lambda=1,2,3$, there is a diffeomorphism
$\phi:X_{\lambda}\rightarrow Z_{k_{\lambda}}=\natural_{k_{\lambda}}S^{1}\times
B^{3}$

  2. (2)

for each $\lambda=1,2,3$ and taking indices mod 3, we have

$\displaystyle\phi(X_{\lambda}\cap X_{\lambda-1})$
$\displaystyle=Y^{-}_{g,k_{\lambda},p,b}$ $\displaystyle\phi(X_{\lambda+1}\cap
X_{\lambda})$ $\displaystyle=Y^{+}_{g,k_{\lambda},p,b}$
$\displaystyle\phi(X_{\lambda}\cap\partial X)$
$\displaystyle+Y^{0}_{g,k_{\lambda},p,b}$

Analogously to the closed case, we define $H_{\lambda}=X_{\lambda}\cap
X_{\lambda-1}$, which is a compression body.

As in the closed case, we refer to $\Sigma=X_{1}\cap X_{2}\cap X_{3}$ as the
central surface, the genus of $\Sigma$ as the genus of the trisection, and
$H_{!}\cup H_{2}\cup H_{3}$ as the spine of the trisection.

It follows from the construction (e.g. [CGPC18, Lemma 11]) that a relative
trisection induces an open book decomposition of $\partial X$. Recall that an
open book decomposition of 3-manifold $Y$ is a pair $(B,\pi)$ where $B$ is
fibered link in $Y$ and $\pi:Y\setminus\nu(B)\rightarrow S^{1}$ is the
fibration.

######  Definition 2.14.

The standard trisection $\mathcal{T}_{std}$ of $B^{k}$ for $k\geq 2$ is the
decomposition $B^{k}=X_{1}\cup X_{2}\cup X_{3}$ where
$X_{\lambda}=\pi^{-1}(D^{s}_{\lambda})$, where
$\pi:\mathbb{R}^{k}\rightarrow\mathbb{R}^{2}$ is a coordinate projection, and
$D=D^{s}_{1}\cup D^{s}_{2}\cup D^{s}_{3}$ is the standard trisection of the
2-disk.

######  Lemma 2.15.

Let $B^{k}=X_{1}\cup X_{2}\cup X_{3}$ be the standard trisection of $B^{k}$.
Then $X_{\lambda}\cap X_{\lambda+1}$ is diffeomorphic to $B^{k-1}$; the triple
intersection $X_{1}\cap X_{2}\cap X_{3}$ is diffeomorphic to $B^{k-2}$; and
$X_{\lambda}\cap\partial B^{k}$ is diffeomorphic to $B^{k-1}$.

In particular, the standard trisection of $B^{4}$ is a $(0,0;0,1)$ relative
trisection.

Note that if $X$ is a smooth 4-manifold with boundary with relative trisection
$\mathcal{T}_{X}$ and $Y$ is a closed, smooth 4-manifold with trisection
$\mathcal{T}_{Y}$, the (interior) connected sum of trisections
$\mathcal{T}_{X}\\#\mathcal{T}_{Y}$ of $X\\#Y$ can be defined as in the closed
case (Definition 2.8).

######  Definition 2.16.

Let $X$ be smooth 4-manifold with boundary and let $\mathcal{T}$ be a relative
trisection of $X$. A sector-$\lambda$ interior stabilization of $\mathcal{T}$
is the trisection obtained by taking the connected sum with the unbalanced,
genus 1 trisection $\mathcal{T}_{\lambda}$ of $S^{4}$.

The main structural result for relative trisections is the following.

######  Theorem 2.17 ([GK16]).

Let $X$ be a smooth, oriented 4-manifold with nonempty boundary.

  1. (1)

For any fixed open book on the boundary $\partial X$, there exists a relative
trisection $\mathcal{T}$ of $X$ inducing that open book deomposition.

  2. (2)

Any two trisections of $X$ inducing the same open book on the boundary become
ambient isotopic after a sequence of interior stabilizations.

  3. (3)

the 4-manifold $X$ is determined up to diffeomorphism by the spine of a
relative trisection.

###  2.5. Bridge position for surfaces

Meier and Zupan generalized bridge position for links in a 3-manifold to
bridge position for surfaces in a 4-manifold [MZ17, MZ18]. Meier has recently
extended the theory of bridge trisections to properly embedded surfaces in
4-manifolds with boundary [Mei20]. We will give a brief overview, focusing
only on the relevant case $X\cong B^{4}$.

A properly embedded tangle $\tau$ in 3-manifold with boundary $M$ is trivial
if the arcs of $\tau$ can be simultaneously isotoped to lie in $\partial M$. A
disk-tangle $\mathcal{D}\subset X$ in a 4-manifold is a collection of properly
embedded disks. A disk-tangle is trivial if the disks can be simultaneously
isotoped to lie in the boundary. All of the trivial tangles and disk-tangles
we consider will be of the following form. Let $\Sigma$ be a compact surface
with nonempty boundary and let $Q$ be a collection of points in $\Sigma$. Then

  1. (1)

$Q\times[0,1]$ is a trival tangle in $\Sigma\times[0,1]$, and

  2. (2)

$Q\times[0,1]^{2}$ is a trivial disk-tangle in $\Sigma\times[0,1]^{2}$

Recall that a link $L\subset Y$ is in bridge position with respect to a
Heegaard splitting $Y=H_{1}\cup_{\Sigma}H_{2}$ if each intersection
$\tau_{\lambda}=L\cap H_{\lambda}$ is a trivial tangle. Extending, we say that
a properly embedded, closed surface $\mathcal{K}$ in a closed 4-manifold $X$
is in bridge position with respect to a trisection $\mathcal{T}$ of $X$ if

  1. (1)

$\mathcal{K}$ intersects the central surface $\Sigma$ of the trisection
transversely in $2b$ points,

  2. (2)

$\mathcal{K}$ intersects each handlebody $H_{\lambda}$ along a trivial tangle
$\tau_{\lambda}$.

  3. (3)

$\mathcal{K}$ intersects each 4-dimensional sector $X_{\lambda}$ along a
trivial disk tangle.

The parameter $b$ is the bridge index of the surface $\mathcal{K}$.

Now suppose that $\mathcal{K}$ is a properly embedded surface in $B^{4}$, let
$\mathcal{T}$ be a relative trisection of $B^{4}$, and suppose that
$\partial\mathcal{K}$ is braided with respect to the induced open book
decomposition on $\partial B^{4}$.

######  Definition 2.18.

We say that $\mathcal{K}$ is in relative bridge position with respect to
$\mathcal{T}$ if

  1. (1)

$\mathcal{K}$ intersects the central surface $\Sigma$ transversely in $2b+n$
points,

  2. (2)

$\mathcal{K}$ intersects each handlebody $H_{\lambda}$ along a trivial tangle,
and

  3. (3)

$\mathcal{K}$ intersects each sector $X_{\lambda}$ along a trivial disk-
tangle.

If $n$ is the braid index of $\partial\mathcal{K}$, we say that $b$ is the
bridge index of $\mathcal{K}$.

###  2.6. Bridge perturbation

For motivation, recall that a bridge perturbation of a link $L\subset Y$ in
bridge position is a local modification that increases the bridge index within
the same ambient isotopy class. We can describe the move abstractly and in
coordinates:

  1. (1)

Abstractly Near a bridge point $x\in\Sigma$, we can choose a local
neighborhood $U$ and coordinates $(x,y,)$ on $U$ such that (a) the Heegaard
surface intersects $U$ along the $xy$-plane, and (b) the link $L$ intersects
$U$ along the $z$-axis. Now choose an arc $\delta$ in $U$, such that one
endpoint lies on $L$ and the other on $\Sigma$. This determines a finger-move
isotopy of $L$ through $\Sigma$.

  2. (2)

In coordinates. In the coordinates on $U$ from above, we can isotope $L$ to be
the linear graph $L=(t,0,t)$. By an isotopy, we can further isotope the link
to be the cubic graph $L^{\prime}=(t,0,t^{3}-\epsilon t)$.

We can define an analogous procedure for surfaces in bridge position.
Following [Mei20, Definition 9.9], we first define a finger-perturbation
abstractly.

######  Definition 2.19 ([Mei20]).

Let $\mathcal{K}$ be a surface in relative bridge position with respect to a
relative trisection $\mathcal{T}$. Let $x$ be a bridge point and $U$ a
neighborhood of $X$ such that $\mathcal{K}$ intersects $U$ along a small disk.
The intersection of $\mathcal{K}$ with $U\cap H_{\lambda}$ is an arc
$\tau_{\lambda}$ with one endpoint at $x$. A sector-$\lambda$ finger
perturbation of $\mathcal{K}$ is the surface $\mathcal{K}^{\prime}$ obtained
by a finger-move isotopy along a small arc $\delta$ in $U\cap H_{\lambda+1}$,
with one endpoint on $\tau_{\lambda+1}$ and the other in $\Sigma$.

The finger move increases the bridge index by 1; it increases the number of
components of $\mathcal{K}\cap X_{\lambda+1}$ by one (since the sector
$X_{\lambda+1}$ of the trisection is opposite to the handlebody
$H_{\lambda}$); and it fixes the number of components of $\mathcal{K}\cap
X_{\lambda-1}$ and $\mathcal{K}\cap X_{\lambda}$.

We can also describe this move visually in 3 dimensions by projecting away one
dimension. Let $\mathcal{K}$ be a surface in bridge position with respect to a
trisection $\mathcal{T}$. We can locally choose coordinates $(x,y,z)$ on
$\mathbb{R}^{3}$ near a bridge point $\mathcal{K}\cap\Sigma$ such that

$\mathcal{K}=\\{z=0\\}$

and the trisection is given by pulling back the standard trisection of the
$xy$-plane by the projection $\pi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$.

A bridge perturbation consists of locally replacing $\mathcal{K}$ by

$\mathcal{K}^{\prime}=\\{x=z^{3}-yz\\}$

Intuitively, this is accomplished by pleating the surface $\mathcal{K}$
through the center of the trisection.

Restricted to $\mathcal{K}$, the projection $\pi$ is a homeomorphism.
Restricted to $\mathcal{K}^{\prime}$, however, there is a critical locus that
projects to a semicubical cusp $(3t^{2},-2t^{3})$. Over the interior of the
cusp, the map is 3-to-1; over the exterior it is 1-to-1; at the origin there
is a cusp singularity; and everywhere else is a fold singularity.

###  2.7. Branched covers of surfaces in relative bridge position

######  Theorem 2.20.

Let $X$ be smooth 4-manifolds with boundary, let $\mathcal{K}\subset X$ be a
properly embedded surface, and let $\pi:\widetilde{X}\rightarrow X$ be an
$n$-fold simple branched covering determined by a homomorphism
$\rho:\pi_{1}(X\setminus\mathcal{K})\rightarrow S_{n}$.

  1. (1)

Suppose that $\mathcal{T}$ is a relative trisection of $X$ and $\mathcal{K}$
is in relative bridge position with respect to $\mathcal{T}$. Then
$\widetilde{\mathcal{T}}=\pi^{-1}(\mathcal{T})$ is a relative trisection of
$\widetilde{X}$.

  2. (2)

Suppose that $\mathcal{K}^{\prime}$ isotopic to $\mathcal{K}$ by a
sector-$\lambda$ interior bridge perturbation. Let
$\widetilde{\mathcal{T}},\widetilde{\mathcal{T}}^{\prime}$ be the induced
relative trisections of $\widetilde{X}$. Then
$\widetilde{\mathcal{T}}^{\prime}$ is obtained from $\widetilde{\mathcal{T}}$
by a sector-$(\lambda+1)$ interior stabilization.

###### Proof.

Part (1) is the relative version of branched covering constructions of
trisections, such as [LCMS21, Theorem 4.1], [LCM22, Proposition 3.1], and
[MZ18, Proposition 13].

An interior bridge perturbation increases the bridge index by 1 and the number
of intersection points $\mathcal{K}^{\prime}\cap\Sigma$ by two. Therefore,
since the map $\pi$ is a simple branched covering, the Riemann-Hurwitz formula
implies that the Euler characteristic of the central surface
$\widetilde{\Sigma}$ upstairs decreases by 2 or equivalently the genus
increases by 1. The number of components of $\mathcal{K}^{\prime}\cap
X_{\lambda-1}$ and $\mathcal{K}^{\prime}\cap X_{\lambda}$ stays the same,
while the number of components of $\mathcal{K}^{\prime}\cap X_{\lambda+1}$
increases by 1. Therefore, the Euler characteristic of
$\widetilde{X}_{\lambda-1},\widetilde{X}_{\lambda}$ stay the same; since they
are 1-handlebodies this implies that their smooth topology stays the same.
However, the Euler characteristic of $\widetilde{X}_{\lambda+1}$ decreases by
1. Thus $\widetilde{X}^{\prime}_{\lambda+1}$ is obtained from
$\widetilde{X}_{\lambda+1}$ by attaching a 1-handle. In summary, this exactly
corresponds to the trisection stabilization. ∎

###  2.8. Singular surfaces

The branch loci for symplectic branched coverings
$\pi:(X,\omega)\rightarrow(\mathbb{CP}^{2},\omega_{FS})$ are singular
surfaces, with self-transverse nodes and cusp singularities that are locally a
cone on the right-handed trefoil. We give a brief summary of [LCMS21, Sections
3 and 4], which extends the (bridge) trisection machinery to singular
surfaces.

A singular knotted surface $\mathcal{K}$ in $X$ is the image of a smooth
immersion, away from a finite number of critical points, such that all
multiple points are transverse double points and in a small neighborhood of
each critical value, the surface is diffeomorphic to the cone on a knot in
$S^{3}$. A singular disk tangle $\mathcal{D}$ in $Z$ bounded by $L\subset
Y=\partial D$ is a collection of properly embedded singular disks such that
(a) each component of $\mathcal{D}$ bounded by an unknot is smoothly embedded
and boundary-parallel; (b) for each component of $\mathcal{D}$ with knotted
boundary $K$, there exists an embedded 4-ball $B$ in $Z$ such that $B\cap Y$
is a 3-ball containing $K$ and the intersection of $\mathcal{D}$ with $B$ is a
smooth cone on $K$; (c) two components of $\mathcal{D}$ are disjoint if their
boundary knots are split and intersect transversely, away from the singular
points, if their boundary knots are linked. A singular surface $\mathcal{K}$
is $X$ is in bridge trisected position with respect to a trisection if it is
transverse to the central surface; the intersection
$\tau_{\lambda}=H_{\lambda}\cap\mathcal{K}$ is a boundary-parallel tangle; and
the intersection of $\mathcal{K}$ with each 4-dimensional sector is a singular
disk tangle.

A singular branched covering $\pi:\widetilde{X}\rightarrow X$ with singular
branch locus $\mathcal{K}\subset X$ is a proper, smooth map such that (a) away
from the singular points of $\mathcal{K}$, the map is a branched covering, and
(b) for each singular point $p$ of $\mathcal{K}$, there exists a 4-ball
neighborhood $U$ such that $\pi{{}^{-}1}(U)$ is a collection of
$\widetilde{U}_{!}\cup\dots\cup\widetilde{U}_{k}$ and the restriction of $\pi$
to $\widetilde{U}_{i}$ is either a diffeomorphism or the cone on a branched
covering $S^{3}\rightarrow S^{3}$. A trisection $\mathcal{T}$ of $X$ pulls
back to a trisection of $\widetilde{X}$ under a singular branched covering,
provided that $\widetilde{X}$ is connected [LCMS21, Theorem 4.1].

##  3\. Existence of symplectic 1-cocycles

###  3.1. Standard trisection of $(\mathbb{CP}^{2},\omega_{FS})$

The standard trisection of $\mathbb{CP}^{2}$ is induced by the toric structure
(see citations). In homogeneous coordinates $\\{[z_{1}:z_{2}:z_{3}]\\}$ on
$\mathbb{CP}^{2}$, we have

$\displaystyle Z_{\lambda}$ $\displaystyle:=\\{|z_{\lambda-1}|=1\,\text{ and
}|z_{\lambda}|,|z_{\lambda+1}|\leq 1\\}$ $\displaystyle H_{\lambda}$
$\displaystyle:=\\{|z_{\lambda-1}|=|z_{\lambda}|=1\,\text{ and
}|z_{\lambda+1}|\leq 1\\}$ $\displaystyle\Sigma$
$\displaystyle=\\{|z_{1}|=|z_{2}|=|z_{3}|=1\\}$

The central surface $\Sigma$ is a torus. Since its Euler characteristic is 0,
every harmonic 1-form is nonvanishing. Therefore, symplectic data for this
trisection consists of a triple of 1-forms $(\beta_{1},\beta_{2},\beta_{3})$
on the handlebodies such that their restrictions to $\Sigma$ satisfy

$\beta_{\lambda}\wedge\beta_{\lambda+1}>0$

These 1-forms can be obtained by tracing the Fubini-Study form $\omega_{FS}$
through the the Cech-DeRham isomorphism, as explained in [LC22b, Section 2.1].
With respect to polar coordinates $z_{j}=r_{j}e^{i\theta_{j}}$, this
isomorphism sends $\omega_{FS}$ to the triple
$(2d\theta_{1},2d\theta_{2},2(-d\theta_{1}-\theta_{2}))$.

###  3.2. Transverse bridge position and geometrically transverse surfaces

######  Definition 3.1.

Let $\mathcal{T}$ be a trisection of $X$, let $\mathcal{K}$ be a surface
transverse to the trisection $\mathcal{T}$, and let
$(\beta_{1},\beta_{2}.\beta_{3})$ be a symplectic 1-cocycle for $\mathcal{T}$.

  1. (1)

$\mathcal{K}$ has complex bridge points if $\mathcal{K}\cap\Sigma$ is disjoint
from the zeros of $\beta$ and for each point of $\mathcal{K}\pitchfork\Sigma$,
there exists an integrable complex structure $J$ such that each
$\widetilde{\beta}_{\lambda}$ defines a $J$-holomorphic foliation on
$H_{\lambda}$ and $\mathcal{K}$ is locally $J$-holomorphic.

  2. (2)

$\mathcal{K}$ is geometrically transverse if it has complex bridge points and
$\widetilde{\beta}_{\lambda}(\tau^{\prime}_{\lambda})>0$ everywhere.

If $\mathcal{K}$ is in bridge position and geometrically transverse, we say it
is in transverse bridge position.

In other words, a surface is geometrically transverse if it has complex bridge
points and each tangle $\tau_{\lambda}$ is positively transverse to the
foliation of $H_{\lambda}$ defined by $\widetilde{\beta}_{\lambda}$.

######  Lemma 3.2.

Let $\mathcal{K}$ be a symplectic surface (possibly nodal and/or cuspidal)
that is geometrically transverse. There exists a sector-$\lambda$ finger
perturbation of $\mathcal{K}$ to $\mathcal{K}^{\prime}$, supported in a
neighborhood of some bridge point, such that $\mathcal{K}^{\prime}$ is also
geometrically transverse.

###### Proof.

Without loss of generality, we assume $\lambda=1$ and the bridge point is a
positive intersection between $\mathcal{K}$ and $\Sigma$. We can locally
assume that $\beta_{1}=dy$ and $\beta_{2}=-dx$ and $\beta_{3}=dx-dy$ and the
tangles $\tau_{1},\tau_{2},\tau_{3}$ are incident to the bridge point
$b=(0,0)$. For some small $\epsilon>0$, add a positive bridge point $b_{+}$ at
$(-2\epsilon,-\epsilon)$ and a negative bridge point $b_{-}$ at
$(-\epsilon,\epsilon)$. Split $\tau_{1}$ into two oriented arcs, one ending at
$b_{+}$ and the other connecting $b_{-}$ to $b$; we can assume these satisfy
$\beta_{1}(\tau^{\prime}_{1})=dy(\tau^{\prime}_{1})>0$. Then we can add
oriented arcs of $\tau_{2}$ and $\tau_{3}$ from $b_{-}$ to $b_{+}$, which
satisfy the positivity condition. Compare with Figure 2(b). ∎

###  3.3. Existence of branch locus

Auroux, building on techniques introduced by Donaldson, proved that every
closed symplectic 4-manifold admits a branched covering over
$(\mathbb{CP}^{2},\omega_{FS})$ [Aur00]. This was strengthened by Auroux-
Katzarkov to show that the branch curve in $\mathbb{CP}^{2}$ is
‘quasiholomorphic’ [AK00]. Intuitively, this means the surface intersects a
generic pencil on $\mathbb{CP}^{2}$ akin to a genuine holomorphic curve. In
[LCMS21], this essential geometric property was called a ’symplectic braided
surface’. However, precise definitions are irrelevant for the current
application. The important point is that the branch locus can be algebraically
encoded, up to isotopy, by words in the braid group $B_{n}$, where $n$ is the
degree of the branch locus in $\mathbb{CP}^{2}$.

######  Theorem 3.3 ([AK00]).

Every closed symplectic 4-manifold admits a symplectic singular branched
covering over $\mathbb{CP}^{2}$, such that the branch locus in
$\mathbb{CP}^{2}$ is a (singular) symplectic braided surface.

######  Theorem 3.4 ([LC23]).

Let $\mathcal{K}$ be a singular, symplectic braided surface in
$\mathbb{CP}^{2}$ with $A_{k}$-singularities. There exists a smooth isotopy of
$\mathcal{K}$ such that it is in transverse bridge position with respect to
the standard trisection of $\mathbb{CP}^{2}$.

######  Remark 3.5.

Great care was taken in [LCMS21] to ensure that the branch locus in
$\mathbb{CP}^{2}$ was isotoped symplectically, which preserves the symplectic
structure obtained by pulling back. However, since we are only interested in
the diffeomorphism type and will construct a potentially new symplectic form
$\widetilde{\omega}$ anyway, we just need to ensure that the branch locus is
smoothly isotopic to the given curve.

Combining these two results, we have the following corollary.

######  Corollary 3.6.

Let $(X,\omega)$ be a closed symplectic 4-manifold. Then there exists a nodal,
cuspidal surface $F$ in transverse bridge position with respect to the
standard trisection of $\mathbb{CP}^{2}$ such that $X$ is the branched cover
of $\mathbb{CP}^{2}$ over $F$.

For completeness, we review the construction here. Surfaces in
$\mathbb{CP}^{2}$ can be encoded by torus diagrams on the central surface
$\Sigma$ of the standard trisection. If $\mathcal{K}$ is in general position
with respect to the trisection, it intersects $\Sigma$ in a finite number of
points and each solid torus $H_{\lambda}$ along a tangle $\tau_{\lambda}$.
Generically, this tangle misses the core $B_{\lambda}$ of $H_{\lambda}$ and
there is a projection $H_{\lambda}\smallsetminus
B_{\lambda}\cong\Sigma\times(0,1]\rightarrow\Sigma$. The tangles
$\tau_{1},\tau_{2},\tau_{3}$ project onto collections of arcs that we label
$\mathcal{A},\mathcal{B},\mathcal{C}$, respectively. See [LC20, Section 2.2].

Let $\Sigma=T^{2}$ be the central surface of the trisection of
$\mathbb{CP}^{2}$. We have coordinates $(x,y)$ for $x,y\in[0,1]$ such that the
foliations are

$\beta_{1}=dy\qquad\beta_{2}=-dx\qquad\beta_{3}=dx-dy$

We can extend these foliations across the handlebodies. Let $\alpha_{1}=(x,0)$
and $\alpha_{2}=(0,x)$ and $\alpha_{3}=(-x,-x)$ for $x\in[0,1]$; these curves
bound disks in $H_{1}$ and $H_{2}$ and $H_{3}$, respectively, that are leaves
of the foliations $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ and
$\mathcal{F}_{3}$, respectively. Let $p=(0,0)=\alpha_{1}\cap\alpha_{2}$ be a
fixed basepoint.

A key result of [AK00] is that the branch locus $\mathcal{R}$ is encoded
algebraically by a braid factorization of the full twist. Specifically, if
$\mathcal{R}$ is a degree $d$ surface in $\mathbb{CP}^{2}$, it is determined
by a factorization

$\Delta^{2}_{d}=(g_{1}\sigma^{i}_{1}g^{-1}_{1})\cdots(g_{n}\sigma^{i_{n}}_{1}g^{-1}_{n})$

where $\sigma_{1}$ is the first half twist in the Artin braid group and
$i_{j}\in\\{-2,1,2,3\\}$. Reconstructing $\mathcal{R}$ from this factorization
is described in [AK00, LC23].

A torus diagram for $\mathcal{R}$ in transverse bridge position in constructed
in [LC23, Proposition 4.6]. It is determined by stacking local models for each
conjugate half-twist $g_{j}\sigma^{i_{j}}_{1}g^{-1}_{j}$, as in Figure 2(a).
Crossings in the conjugating braid $g_{i},g^{-1}_{i}$ can be removed by a
mini-stabilization (Figure 2(b)). The only modification we require here is to
dictate that each $\mathcal{B}$ arc in the local model of a half-twist
intersects the curve $\alpha_{2}$ transversely once. In each sector, we obtain
a transverse link $R_{\lambda}=\mathcal{R}\cap Y_{\lambda}$. Each component of
$R_{\lambda}$ is either an (a) unknot, (b) Hopf link or (c) right-handed
trefoil. The surface $\mathcal{R}$ is obtained by capping off each component
of $R_{\lambda}$ with either (a) trivial disk, (b) pair of transversely
intersecting trivial disks, or (c) a cone on a right-handed trefoil.

2pt $g_{i}$ at 220 396 $g^{-1}_{i}$ at 220 82 $\dots$ at 250 440 $\dots$ at
250 230 $\dots$ at 250 30 $\alpha_{2}$ at 40 40

(a) Local model of a torus diagram of a half-twist

(b) A crossing in the conjugating braid $g_{i}$ can be removed by a mini
stabilization while preserving geometric transversality

Figure 2. Component pieces of a torus diagram for the branch locus
$\mathcal{R}$

###  3.4. Pulling back symplectic 1-cocycles

A symplectic 1-cocycle
$\\{\widetilde{\beta}_{1},\widetilde{\beta}_{2},\widetilde{\beta}_{3})$ on a
trisection $\mathcal{T}$ of $X$ and a branch locus $\mathcal{K}$ in singular,
transverse bridge position pulls back under the branched covering

$\pi:(\widehat{X},\widehat{\mathcal{T}})\rightarrow(X,\mathcal{T})$

  1. (1)

Away from the branch locus $\tau$, the map $\pi$ is a local diffeomorphism so
$\pi^{*}(\widetilde{\beta}_{\lambda})$ remains a closed, nonvanishing 1-form.
Similarly, along $\Sigma$ we can pull back the restrictions
$\\{\beta_{\lambda}\\}$.

  2. (2)

Along the branch locus $\tau_{\lambda}$, we have that
$\widetilde{\beta}_{\lambda}(\tau^{\prime}_{\lambda})>0$ by assumption, since
$\mathcal{K}$ is in transverse bridge position. We can therefore locally
choose coordinates $(x,y,z)$ such that $\widetilde{\beta}_{\lambda}=dz$ and
$\tau=(0,0,t)$ for $t\in(-\epsilon,\epsilon)$. The branched covering map is
therefore given by

$\pi:(x,y,z)\mapsto(x^{2}-y^{2},2xy,z)$

and in particular $\pi^{*}(dz)=dz$ and the pullback of
$\widetilde{\beta}_{\lambda}$ is a closed, nonvanishing 1-form.

  3. (3)

Finally, we check near the bridge points. By assumption, we can choose a local
neighborhood and identify it with $\mathbb{C}^{2}$ with coordinates
$(z_{j}=x_{j}+iy_{j})$ such that

$\displaystyle\mathcal{K}$ $\displaystyle=\\{z_{2}=0\\}$ $\displaystyle\Sigma$
$\displaystyle=\\{z_{2}=\epsilon\overline{z_{1}}\\}$

where $\epsilon\in\\{\pm 1\\}$ is the sign of the intersection of
$\mathcal{K}$ with $\Sigma$. Set

$\displaystyle t$ $\displaystyle=x_{1}-x_{2}$ $\displaystyle s$
$\displaystyle=-y_{1}-y_{2}$ $\displaystyle\widetilde{\beta}_{1}$
$\displaystyle=dy_{1}-dy_{2}$ $\displaystyle\widetilde{\beta}_{2}$
$\displaystyle=-dx_{1}-dx_{2}$

Then the trisection decomposition is given by

$\displaystyle H_{1}$ $\displaystyle=\\{s\geq 0,t=0\\}$ $\displaystyle Z_{1}$
$\displaystyle=\text{max}(-s,t)\leq 0\\}$ $\displaystyle H_{2}$
$\displaystyle=\\{t\geq 0,-t-s=0\\}$ $\displaystyle Z_{2}$
$\displaystyle=\text{max}(-t,-t-s)\leq 0\\}$ $\displaystyle H_{3}$
$\displaystyle=\\{-t-s\geq 0,s=0\\}$ $\displaystyle Z_{3}$
$\displaystyle=\text{max}\\{(t+s,s)\leq 9\\}$

and

$dt\wedge\widetilde{\beta}_{1}-ds\wedge\widetilde{\beta}_{2}=2\left(dx_{1}\wedge
dy_{1}+dx_{2}\wedge dy_{2}\right)=2\omega_{std}$

The branched covering is locally given by the map
$\pi:(z_{1},z_{2})\rightarrow(z_{1},z_{2}^{2})$. The pulled-back trisection
$\pi^{*}(\mathcal{T})$ is determined by the functions

$\displaystyle t$ $\displaystyle=x_{1}+\epsilon(-x_{2}^{2}+y_{2}^{2})$
$\displaystyle s$ $\displaystyle=-y_{1}-\epsilon(2x_{2}y_{2})$

and the 1-forms pull back as

$\displaystyle\widetilde{\beta}_{1}$ $\displaystyle=dy_{1}-2\epsilon
d(x_{2}y_{2})$ $\displaystyle\widetilde{\beta}_{2}$
$\displaystyle=-dx_{1}+2\epsilon d(y_{2}^{2}-x_{2}^{2})$

Restricting this forms to $\Sigma=\\{t=s=0\\}$, we have that

$\displaystyle\beta_{1}=\widetilde{\beta}_{1}|_{\Sigma}$
$\displaystyle=-4\epsilon d(x_{2}y_{2})$
$\displaystyle\beta_{2}=\widetilde{\beta}_{2}|_{\Sigma}$
$\displaystyle=-4\epsilon d(y_{2}^{2}-x_{2}^{2})$

Using $x_{2},y_{2}$ as coordinates along $\Sigma$ we see that

$\beta_{1}\wedge\beta_{2}=16\epsilon^{2}(x_{2}^{2}+y_{2}^{2})dx_{2}\wedge
dy_{2}\geq 0$

and

$\widetilde{\beta}_{1}=\beta_{1}+d(s\cdot\epsilon)\qquad\qquad\widetilde{\beta}_{2}=\beta_{2}+d(t\cdot\epsilon)$

Consequently, the form $\beta_{1}$ has a hyperbolic zero at the branch locus.
Moreover, we can set $f=\epsilon\in\\{\pm 1\\}$ near the branch points in
$\pi^{*}(\Sigma)$, according to the sign of the intersection
$\mathcal{K}\pitchfork\Sigma$. Since $\Sigma$ is nullhomologous, as it bounds
the handlebody $H_{\lambda}$, the algebraic count of intersections is 0.

###  3.5. Branched covers over transverse knots

The final piece in proving Theorem 1.9 is to check that the symplectic
1-cocycle data on $\pi^{*}(\mathcal{T})$ determines tight contact structures
on the three strongly concave boundary components.

Recall that a $n$-fold simple branched covering of $(Y,K)$ is defined by a map

$\rho:\pi_{1}(Y\setminus K)\rightarrow S_{n}$

where each meridian in $\pi_{1}(Y\setminus K)$ is mapped to a transposition in
$S_{n}$. The kernel of $\rho$ corresponds to a covering space
$\widehat{Y}\setminus\widehat{K}$ of $Y\setminus K$. If $K$ is transverse to a
contact structure $(Y,\xi)$, then (after a perturbation along the ramification
locus $\widehat{K}\subset\widehat{Y}$) the contact structure pulls back to a
contact structure $(\widehat{Y},\widehat{\xi})$ that is unique up to ambient
isotopy. A local model is given in [Gei08, Theorem 7.5.4], for example.

######  Example 3.7.

Let $H$ denote the Hopf link and $T$ the right-handed trefoil.

  1. (1)

The link group of $H$ is $\pi_{1}(S^{3}\setminus T)=\langle
x,y|xy=yx\rangle=\pi_{1}(T^{2})$. Therefore, any homormorphism $\rho$ must
send $x,y$ to commuting transpositions

  2. (2)

The knot group of $T$ is $\pi_{1}(S^{3}\setminus T)\cong\langle
x,y|xyx=yxy\rangle$. The homomorphsim $\rho$ to $S_{3}$ defined by
$\rho(x)=(12)$ and $\rho(y)=(23)$ defines an irregular 3-fold simple branched
covering from $S^{3}$ to itself.

######  Proposition 3.8.

Let $(B^{3},\xi_{std})$ denote the 3-ball with the standard tight contact
structure and convex boundary. Let $U$ denote a transverse unknot with $sl=-1$
and let $T$ denote a transverse, right-handed trefoil with $sl=1$.

  1. (1)

the 2-fold contact branched cover of $(B^{3},\xi_{std})$ over $U$ is
$(S^{2}\times[0,1],\xi_{std})$, where $\xi_{std}$ is the unique tight contact
structure.

  2. (2)

the irregular 3-fold simple branched cover of $(B^{3},\xi_{std})$ over $T$ is
contactomorphic to $(B^{3},\xi_{std})$;

  3. (3)

Let $L\subset(S^{3},\xi_{std})$ be a transverse link consists of the split
union of maximal unknots; Hopf links (with both linking numbers) of maximal
self-linking unknots; and maximal self-linking right-handed trefoils. Let
$(Y,\xi)$ be the contact branched covering over $L$ determined by a map
$\rho:\pi_{1}(S^{3}\setminus L)\rightarrow S_{n}$. Then $(Y,\xi)$ is
contactomorphic to $(S^{3}\\#_{k}S^{1}\times S^{2},\xi_{std})$ for some $k\geq
0$, where $\xi_{std}$ is the unique tight contact structure.

###### Proof.

Consider the unit ball $B$ in $\mathbb{C}^{2}$. This is a Stein domain and
therefore its preimage under the branched covering $\pi:(z,w)\mapsto(z,w^{2})$
is also Stein; in particular, the contact structure induced on the boundary is
tight. The ramification locus $\\{w=0\\}$ intersects $S^{3}$ along a $sl=-1$
transverse unknot. Take a $B^{3}$-neighborhood of this knot with convex
boundary; the preimage under the branched covering is homeomorphic to
$S^{2}\times[0,1]$ and the restriction of the contact structure is therefore
tight.

Part (2) follows similarly using the cuspidal branched covering

$\pi:(x,y)\mapsto(x,y^{3}+xy)$

whose branch locus in the codomain intersets $S^{3}$ along a right-handed
trefoil.

Finally, note that we can choose $B^{3}$-neighborhoods of each split link
component. Then $Y$ is union of $n$ copies of a punctured $(S^{3},\xi_{std})$
and components contactomorphic to $(S^{2}\times[0,1],\xi_{std})$ or
$(B^{3},\xi_{std})$, glued along convex 2-spheres with tight neighborhoods.
This gluing preserves tightness, so since each component is tight, the global
contact structure $(Y,\xi)$ is also tight. ∎

###  3.6. First Chern class

Let $\pi:(X,\omega)\rightarrow(\mathbb{CP}^{2},\omega_{FS})$ be a symplectic
branched covering as constructed in this paper. We can interpret the first
Chern class of $(X,\omega)$ geometrically in terms of the Euler class of the
foliations $\mathcal{F}_{\lambda}=\text{ker}(\widetilde{\beta}_{\lambda})$ on
$H_{\lambda}$ determined by a symplectic 1-cocycle
$\\{\widetilde{\beta}_{\lambda}\\}$. The discussion follows [LC22a, Section
3], where it was remarked that compact leaves minimize the (generalized)
Thurston norm.

Let $B$ denote the zeros of $\beta_{\lambda}$ on $\Sigma=\partial
H_{\lambda}$. There is an intersection pairing

$\langle,\rangle:H_{1}(H_{\lambda},B)\times H_{2}(H_{\lambda},\Sigma\setminus
B)\rightarrow\mathbb{Z}$

Choose a vector field $v_{\lambda}$ directing the kernel of $\beta_{\lambda}$
on $\Sigma$ and extend it to a section of
$\text{ker}(\widetilde{\beta}_{\lambda})$ over $H_{\lambda}$. Generically,
this section will vanish along a smooth 1-complex representing a well-defined
Euler class $e(\mathcal{F}_{\lambda})\in H_{1}(H_{\lambda},B)$ of the
foliation. If $L$ is a compact leaf of the foliation $\mathcal{F}_{\lambda}$,
then evaluating $e(\mathcal{F}_{\lambda})$ on $L$ gives its Euler
characteristic.

######  Proposition 3.9.

The triple $(e(\mathcal{F}_{1}),e(\mathcal{F}_{2}),e(\mathcal{F}_{3}))$ is a
1-cocyle that represents the first Chern class of $(X,\omega)$.

###### Proof.

To ensure that we obtain a 1-cocycle, we can assume that the vector fields
satisfy the cocycle condition $v_{1}+v_{2}+v_{3}=0$.

The first Chern class of $(X,\omega)$ is given by the formula

$c_{1}(X,\omega)=\pi^{*}(c_{1}(\mathbb{CP}^{2}))-PD(\mathcal{R})$

where $\mathcal{R}\subset X$ is the (smooth) ramification locus of $\pi$. By
construction, since the symplectic 1-cocycle is constructed via a branched
covering over the standard trisection of $\mathbb{CP}^{2}$, every regular leaf
$L$ of $\mathcal{F}_{\lambda}$ is compact and is the $k$-fold branched cover
of a disk leaf in $H_{\lambda}=S^{1}\times D^{2}$. Therefore, if the
proposition holds with respect to the standard trisection of
$\mathbb{CP}^{2}$, then it will hold in general by the Riemann-Hurwitz
formula.

The first Chern class of $(\mathbb{CP}^{2},\omega_{FS})$ has degree 3. This
can be represented by a triple $(C_{1},C_{2},C_{3})$, where $C_{\lambda}$ is a
core of the handlebody $H_{\lambda}\cong S^{1}\times D^{2}$. Since each
compact leaf $L$ is a 2-disk, we see that $\langle
C_{\lambda},L\rangle=\chi(L)=1$. ∎

###  3.7. Realizing geometrically transverse surfaces

######  Proposition 3.10.

Let $(X,\omega)$ be a symplectic 4-manifold with rational symplectic form. The
branch locus $\mathcal{R}$ in the construction can be chosen such that every
homology class $A\in H_{2}(X;\mathbb{Z})$ with positive symplectic area is
represented by a surface that is geometrically transverse to the trisection of
$X$.

###### Proof.

There exists a surjective map

$\pi_{1}(\widetilde{H}_{1},p_{0})\rightarrow
H_{1}(\widetilde{H}_{1})\rightarrow H_{2}(X)$

The first is given by the abelianization, while the second is due to the fact
that $H_{1}(\widetilde{H}_{1})\cong
H_{2}(\widetilde{H}_{1},\partial\widetilde{H}_{1})$ and there is a handle
decomposition of $X$ such that compressing disks in $\widetilde{H}_{1}$ are
precisely the cores of the 2-handles.

Let $A\in H_{2}(X;\mathbb{Z})$ be a class of positive symplectic area and let
$\zeta$ be a loop based at $p$ such that
$[\zeta]\in\pi_{1}(\widetilde{H}_{1},p)$ is a lift of $A$. By assumption,
$\zeta$ is nullhomologous in $Y_{1},Y_{3}$ and therefore bounds slice surfaces
in $Z_{1},Z_{3}$ whose union is a surface representing $A$.

As shown in the following subsection, there is a decomposition

$\zeta=l_{1}^{n}\mu_{1}\rho_{1}\cdots\mu_{k}\rho_{k}$

where

  1. (1)

$l_{1}$ is loop in $\widetilde{H}_{1}$ transverse to the foliation determined
by $\beta_{1}$

  2. (2)

each loop $\mu_{i}$ is tangent to the leaf of $\mathcal{F}_{1}$ through the
basepoint $p$,

  3. (3)

each $\rho_{i}$ is represented by a loop on $\Sigma$ that is homotopic in
$\widetilde{H}_{2}$ to a loop in the leaf of $\mathcal{F}_{2}$ through $p$ and
that is contractible in $\widetilde{H}_{3}$.

In a neighborhood of $p$, the defining 1-forms $\beta_{1},\beta_{2},\beta_{3}$
exact with local primitives $g_{1},g_{2},$ and $g_{3}=-g_{1}-g_{2}$. Pick $2k$
points $p_{1},\dots,p_{2k}$ is a small neighborhood of the basepoint $p$ such
that

$\displaystyle g_{1}(p_{i})>g_{1}(p_{i-1})$ for $i$ odd $\displaystyle
g_{2}(p_{i})>g_{2}(p_{i-1})$ for $i$ even $\displaystyle
g_{3}(p_{i})>g_{3}(p_{i-1})$ for $i$ even

Note that since the points $\\{p_{i}\\}$ are in an arbitrary neighborhood of
$p$, there is a unique identification between homotopy classes of loops based
at $p$ and homotopy classes of paths from $p_{j}$ to $p_{j^{\prime}}$ for any
$j,j^{\prime}$.

To build the surface, connect $p_{2i-1}$ to $p_{2i}$ by arcs of $\tau_{2}$ and
$\tau_{3}$ homotopic to $\rho_{i}$ and connect $p_{2i}$ to $p_{2i+1}$ be an
arc homotopic to $\mu_{i+1}$. Finally, connect $p_{2k}$ to $p_{1}$ by a loop
homotopic to $l_{1}^{n}\mu_{1}$. If $n>0$, then each arc of $\tau_{\lambda}$
can be assumed geometrically transverse to the foliation
$\mathcal{F}_{\lambda}$.

From Stokes’s Theorem, it follows that

$\int_{\mathcal{K}}\omega=\sum_{\lambda=1}^{3}\int_{\tau_{\lambda}}\beta_{\lambda}$

Since all of the component arcs of $\tau_{2},\tau_{3}$ are perturbations of
flat paths, their contribution is $O(\epsilon)$, whereas the contribution of
$\tau_{1}$ to his is approximately $n\int_{l_{1}}\beta_{1}$. Therefore, the
surface has positive symplectic area if and only if $n>0$. ∎

###  3.8. Fundamental group of $H_{1}\smallsetminus\tau_{1}$

. Since the branch locus $\mathcal{R}$ is in bridge position, the tangle
$(H_{1},\tau_{1})$ is trivial and the arcs of $\tau_{1}$ can be simultaneously
isotoped into the boundary $\partial H_{1}=T^{2}$. Consequently, the
3-manifold $H_{1}\smallsetminus\nu(\tau_{1})$ is a genus $b+1$ handlebody,
where $b$ is the bridge index, and

$\pi_{1}(H_{1}\smallsetminus\tau_{1},p)=\langle
l,\mu_{1},\dots,\mu_{b}\rangle$

where $l$ is the longitude generating $\pi_{1}(H_{1},p)$ and the $\mu$’s are
meridians of the arcs of $\tau_{1}$. However, we require a different
presentation of the fundamental group adapted to the foliations
$\mathcal{F}_{\lambda}=\text{ker}(\beta_{\lambda})$.

######  Definition 3.11.

Let $Y$ be a compact 3-manifold with (smooth) foliation $\mathcal{F}$. We say
that a path is flat if it can be homotoped, rel boundary, to lie in a leaf of
$\mathcal{F}$.

Consider the cover $\overline{H}_{1}=D^{2}\times\mathbb{R}$ associated to the
longitude $l$ and let $p_{0}$ denote the lift of $p$ with
$\mathbb{R}$-coordinate 0. The tangle $(\overline{H}_{1},\overline{\tau}_{1})$
consists of an infinite number of arcs and
$\pi_{1}(\overline{H_{1}}\smallsetminus\overline{\tau}_{1},p_{0})$ is freely
generated by meridians of these arcs. Define the following subgroups:

$\displaystyle M_{0}\coloneqq$ the subgroup generated by meridians that can be
represented by loops in the disk $D\times\\{0\\}$. $\displaystyle
M_{r}\coloneqq$ the subgroup generated by meridians that can be represented by
loops in the cylinder $D\times[0,r]$ or $D\times[r,0]$. $\displaystyle
M\coloneqq$ the subgroup generated by meridians.

Note that each of these groups projects to a subgroup of
$\pi_{1}(H_{1}\smallsetminus\tau_{1},p)$ and by abuse of notation we let it
denote both groups. Note that if $r>0$, then $lM_{r}l^{-1}$ is contained in
$M_{r+1}$, and if $r<0$ then $l^{-1}M_{r}l$ is contained in $M_{r-1}$.

By construction, we have a torus diagram for the branch locus $\mathcal{R}$
such that $\tau_{1}$ projects onto $T^{2}$ as a collection $\mathcal{A}$ of
disjoint arcs. This diagram also pulls back to a diagram on
$S^{1}\times\mathbb{R}$ for $\overline{\tau}_{1}$.

######  Lemma 3.12.

The subgroup $M_{r}$ is generated by a collection of meridians, one for each
embedded arc in the torus diagram in the region $S^{1}\times[0,r]$ or
$S^{1}\times[r,0]$

###### Proof.

Immediate application of the Wirtinger presentation, where the tangle is
inside the solid torus and we view it from the outside. ∎

######  Lemma 3.13.

If there are no bridge points in the cylinder $S^{1}\times[s,r]$, then
$M_{s}=M_{r}$.

###### Proof.

All of the arcs are embedded and end at bridge points. Thus the number of arcs
can only increase at a bridge point. ∎

######  Proposition 3.14.

There exist a subgroup $R\subset\pi_{1}(\Sigma\smallsetminus\overline{b},p)$
such that

  1. (1)

$M_{r}\subset M_{s}\ast R$ if $0<s<r$ or $r<s<0$

  2. (2)

$M\subset M_{0}\ast R$

  3. (3)

$R$ is invariant under conjugation by the longitude $l$

  4. (4)

every loop in $R$ contracts in $H_{3}\smallsetminus\tau_{3}$

  5. (5)

every loop in $R$ maps to a flat loop in $H_{2}\smallsetminus\tau_{2}$

###### Proof.

We will prove part (1) by induction; then (2) clearly follows from (1). The
properties of $R$ will come out of the construction.

For simplicity, we will assume that $0<s<r$; the negative case is essentially
identical. The group $M_{r}$ changes at a discrete collection of values of
$r$, corresponding to two cases: (a) a crossing in the torus diagram, or (b) a
local model of a half-twist.

Case (a): crossings. Suppose that the crossing happens at level $r$. In the
local picture 2(b), we have three $\tau_{1}$-arcs and two bridge points. We
can choose loops $a,b$ in $\Sigma$ that descend to the Wirtinger meridians of
the two arcs ending at the basepoints, so that $M_{r+\epsilon}$ is obtained
from $M_{r-\epsilon}$ by adding the generator $b$. Note that $a,b$ map to the
same meridians of $\tau_{2}$ in $H_{2}$ and $\tau_{3}$ in $H_{3}$; thus
$ab^{-1}$ contracts in $H_{2}\smallsetminus\tau_{2}$ and in
$H_{3}\smallsetminus\tau_{3}$. Include $\rho=ab^{-1}$ in $R$; clearly it
satisfies properties (3)-(5) and $b=a\rho^{-1}$.

Case (b): half-twist. In the local model, we have four bridge points. Choose
four loops $a,b,c,d$ in $\Sigma$ based at $p$ as in Figure 3(b); we can assume
they agree outside this local picture. Pushing these loops into $H_{\lambda}$,
they become meridians of $\tau_{\lambda}$. In $H_{1}$, they become the
Wirtinger meridians of the four incident $\tau_{1}$ arcs.

First, we claim that each of the four loops maps to a flat meridian in
$H_{2}$. Note that $a=c$ and $b=d$ in $H_{2}$. Then each $\mathcal{B}$-arc in
the projection intersects $\alpha_{2}$ transversely in a point. We can then
represent these by a path that travels from $p$ along $\alpha_{2}$, then makes
a small meridian loop, and returns to $p$ along $\alpha_{2}$.

Secondly, we claim that the following loops, depending on the value of $k$,
contract in $H_{3}\smallsetminus\tau_{3}$. See Figure 3(a). These are
precisely the loops corresponding to the arcs of $\tau_{3}$.

$k$ | $\rho_{1}$ | $\rho_{2}$
---|---|---
1 | $ad$ | $cb$
2 | $a^{-1}dab$ | $cb^{-1}ab$
-2 | $ab^{-1}a^{-1}cab$ | $b^{-1}dbaba^{-1}$
3 | $a^{-1}dab^{-1}ab$ | $cb^{-1}a^{-1}bab$

In each case, we have that

$c=\rho_{1}c_{0}\qquad d=\rho_{2}d_{0}$

where $c_{0},d_{0}$ lie in $M_{s}$ and $\rho_{1},\rho_{2}$ are loops that
bound in $H_{3}\smallsetminus\tau_{3}$ and are flat loops in $H_{2}$.

We now let $R$ consist of all $l$-conjugates of all the relations at each
half-twist. Thus, it satisfies property (3) by definition. Since each relation
vanishes in $H_{3}\smallsetminus\tau_{3}$, so does every conjugate and
therefore every element in $R$. Finally we can homotope the longitude to lie
in the leaf of $H_{2}$ bounded by $\alpha_{2}$. This implies that all
$l$-conjugates of flat loops in $H_{2}$ are also flat. ∎

(a) Four local models for $k=1,3,2,-2$ (counterclockwise, starting in bottom
left)

2pt $a$ at 105 135 $b$ at 110 250 $d$ at 225 270 $c$ at 240 160

(b) Paths in $\widetilde{\Sigma}$ that descend to meridians of the tangles
$\tau_{1},\tau_{2},\tau_{3}$. All oriented counterclockwise.

Figure 3. Local fundamental group calculation

######  Proposition 3.15.

Every element $\gamma\in\pi_{1}(H_{1}\smallsetminus\tau_{1},p)$ can be
factored as

$\gamma=l^{p}\gamma_{0}$

where $p$ is an integer and $\gamma_{0}\in M_{0}\ast R$.

###### Proof.

Clearly, $\gamma$ can be factored into a product of the longitude and the
meridians of $\tau_{1}$. For any meridian $x$, we have that

$xl=l(l^{-1}xl)=l\overline{x}$

where $\overline{x}=(l^{-1}xl)$ is a different meridian. Consequently, we can
move all longitudes to the front of the factorization. We are then left with a
product of meridians $\gamma_{0}$, which lies in $M_{0}\ast R$ by Proposition
3.14. ∎

Let $\pi_{1}(\widetilde{H}_{1},\overline{p})$ denote the set of homotopy
classes of paths in $\widetilde{H}_{1}$ connecting any two points
$p_{i},p_{j}$ in the preimage of $p$. It consists precisely of all lifts of
all paths in $\pi_{1}(H_{1}\smallsetminus\tau_{1},p)$. Let $\widetilde{M}_{0}$
and $\widetilde{R}$ denote the set of all lifts of $M_{0}$ and $R$.

######  Proposition 3.16.

Every element $\gamma\in\pi_{1}(\widetilde{H}_{1},p_{1})$ can be factored in
$\pi_{1}(\widetilde{H}_{1},\overline{p})$ into the form

$\gamma=l_{1}^{p}\gamma_{0}$

where $\gamma_{0}$ is a sequence of paths in $\widetilde{M}_{0}$ and
$\widetilde{R}$. and $l_{1}$ is the lift of $l$ based at $p_{1}$.

###### Proof.

Every loop in the branched cover projects to a loop in
$H_{1}\smallsetminus\tau_{1}$ based at $p$. We can use Proposition 3.15 to
factor it. Then each longitude $l$ lifts to $l_{1}$, the lift based at
$p_{1}$, and each element of $M_{0},R$ lifts to a path in $\widetilde{M}_{0}$
or $\widetilde{R}$. ∎

##  4\. Holomorphic trisections

######  Definition 4.1.

Let $(X,J)$ be an almost-complex 4-manifold. A holomorphic trisection of
$(X,J)$ is a trisection such that

  1. (1)

$J$ is integrable in an open neighborhood of the spine of $\mathcal{T}$,

  2. (2)

there exists a neighborhood $U_{\lambda}$ of $H_{\lambda}$ and a holmorphic
function $g_{\lambda}$ on $U_{\lambda}$ such that
$H_{\lambda}=\\{Re(g_{\lambda})=0\\}$ and $Z_{\lambda}$ is locally defined as
$\\{\text{max}(Re(g_{1}),-Re(g_{2}))\leq 0\\}$; and

  3. (3)

the triple $(g_{1},g_{2},g_{3})$ satisfy the cocycle relation

$Re(g_{1})+Re(g_{2})+Re(g_{3})=0$

on $U_{1}\cap U_{2}\cap U_{3}$, which is a neighborhood of the central surface
$\Sigma$.

######  Example 4.2.

  1. (1)

The standard trisection of $\mathbb{CP}^{2}$ induces a holomorphic trisection,
where $g_{\lambda}=\log z_{\lambda}$ for homogeneous coordinates
$[z_{1}:z_{2}:z_{3}]$ on $\mathbb{CP}^{2}$.

  2. (2)

Zupan shows that the curves

$C_{d}=\\{z_{1}^{d-1}z_{2}+z_{2}^{d-1}z_{3}+z_{3}^{d-1}z_{1}=0\\}$

are in bridge position with respect to the standard trisection of
$\mathbb{CP}^{2}$. Therefore, the projective surfaces

$V_{d}\\{z_{4}^{d}+z_{1}^{d-1}z_{2}+z_{2}^{d-1}z_{3}+z_{3}^{d-1}z_{1}=0\\}\subset\mathbb{CP}^{3}$

admit holomorphic trisections, obtained by pulling back the standard
trisection of $\mathbb{CP}^{2}$ by the projection
$V_{d}\rightarrow\mathbb{CP}^{2}$.

######  Definition 4.3.

A Kähler structure on a holomorphic trisection is a Kähler form

$\omega=i\partial\overline{\partial}\phi$

where $\phi$ is a Kähler potential such that

  1. (1)

$\phi$ is also a defining function for $\partial\nu(H_{1}\cup H_{2}\cup
H_{3})$ making each boundary component strictly pseudoconcave with respect to
the outward-normal coorientation.

  2. (2)

the induced contact structures on the boundary components are tight

######  Example 4.4.

On $\mathbb{CP}^{2}$, consider the Kähler potential

$\phi_{N}=\log(\frac{1}{N}|\mathbf{z}|^{2}+|\mathbf{z}|^{2N})$

for $N\gg 0$. As $N\rightarrow\infty$, the 0-level sets converge to the
standard trisection of $\mathbb{CP}^{2}$.

By following the proof of Theorem 1.9, combined with Theorem 4.8 proved below,
we obtain the following.

######  Theorem 4.5.

Let $(X,\omega)$ be a closed symplectic 4-manifold with rational symplectic
form. There exists a holomorphic trisection $\mathcal{T}$ on $X$ with a Kähler
structure $\omega^{\prime}$ cohomologous to $\omega$.

###### Proof.

By Theorem 4.8, the branch locus can be assumed holomorphic in some
neighborhood of the spine. Then, for $N$ sufficiently large, the Kähler
potential in Example 4.4 defines a Kähler form and the superlevel set
$\phi_{n}^{-1}([0,\infty))$ is a neighborhood of the spine in which the branch
locus is holomorphic. Pulling this back gives a Kähler structure on the
holomorphic trisection of the branched cover. ∎

Note that the form $\pi^{*}(\omega_{FS})$ is degenerate along the ramification
locus in $X$, but this can be eliminated by a $C^{\infty}$-small perturbation
of the Kähler potential $\pi^{*}(\phi)$.

###  4.1. Holomorphic ribbon

Let $K\subset\mathbb{C}^{n}$ be a subset. The rational hull of $K$ is the set

$\widehat{K}_{\mathbb{Q}}:=\\{z\in\mathbb{C}^{n}:|f(z)|\leq\text{max}_{w\in
K}|f(w)|\text{ for all meromorphic functions $f$ on $\mathbb{C}$}\\}$

A compact set $K$ is rationally convex if $\widehat{K}_{\mathbb{Q}}=K$.

######  Lemma 4.6.

Let $K$ be a piecewise smooth 1-complex in $\mathbb{C}^{2}$. Then $K$ is
rationally convex and admits a Stein neighborhood basis.

###### Proof.

Let $z$ be a point in the complement of $K$. Consider the ‘projection from a
point’ map $\mathbb{CP}^{2}$ $\setminus z\rightarrow\mathbb{CP}^{1}$, which
sends each point $y\in\mathbb{CP}^{2}$ $\setminus z$ to the unique line
through $z$ and $y$. The image of $K$ is a piecewise-smooth 1-complex. In
particular, it has measure 0. Let $L$ be the projective line corresponding to
some point in the complement and let $L^{\prime}$ be some $C^{\infty}$-close
projective line that misses $z$. If $f$ is the defining equation for
$L^{\prime}$, then $1/f$ is a rational function that has a pole along
$L^{\prime}$ and if $L^{\prime}$ is sufficiently close to $L$, then $1/|f(z)|$
is greater than the maximum modulus of $1/f$ along $K$. ∎

Consequently, we can find a strictly plurisubharmonic function defining a
tubular neighborhood of $\mathcal{K}$. Within this neighborhood, we can find a
holomorphic function whose intersection with the spine of the trisection is
arbitrarily close to $\mathcal{K}$.

The following is a standard fact in complex analysis, which for example can be
proven by Hörmander’s weighted $L^{2}$ spaces [H9̈0] (see also [Don96, Lemma
28]).

######  Lemma 4.7.

Let $\phi:\Omega\rightarrow\mathbb{R}$ be a smooth plurisubharmonic function
on a domain $\Omega\subset\mathbb{C}^{2}$ and let
$\Delta_{r}=\phi^{-1}((-\infty,r])$ denote a sublevel set of $\phi$. For each
pair $r<s$, there exists a constant $K=K(r,s)$ such that if $f$ is any smooth,
complex-valued function on $\Delta_{s}$, there exists a holomorphic function
$\widetilde{f}$ on the interior of $\Delta_{r}$ such that

$||f-\widetilde{f}||_{C^{1}(\Delta_{r})}\leq
K\left(||\overline{\partial}f||_{C^{0}(\Delta_{s})}+||\nabla(\overline{\partial}f)||_{C^{0}(\Delta_{s})}\right)$

With this fact in hand, we can approximate the spine of the surface by a
holomorphic curve.

######  Theorem 4.8.

Let $\mathcal{K}$ be a surface that is geometrically transverse with respect
to the standard trisection of $\mathbb{CP}^{2}$. There exists a neighborhood
$U$ of the spine $\mathcal{S}$, an isotopic surface $\mathcal{K}^{\prime}$
whose restriction to $U$ is holomorphic, and such that
$\mathcal{K}^{\prime}\cap\mathcal{S}$ is $C^{1}$-close to
$\mathcal{K}\cap\mathcal{S}$.

###### Proof.

By assumption, the surface $\mathcal{K}$ is holomorphic in some neighborhood
of the bridge points $\\{b_{j}\\}=\mathcal{K}\pitchfork\Sigma$. Let $U_{j}$
denote a fixed open neighborhood of $b_{j}$ on which $\mathcal{K}$ is
holomorphic and defined as the vanishing locus of $\widetilde{B}_{j}$. Let
$\tau_{\lambda,j}$ be an arc of $\mathcal{K}\cap H_{\lambda}$. We can choose a
holomorphic bidisk neighbohood $\Delta_{\lambda,j}$ of $\tau_{\lambda,j}$ such
that $\\{\Delta_{\lambda,j}\\}$ and $\\{U_{j}\\}$ give an open cover of
$\mathcal{K}\cap\mathcal{S}$. By shrinking the open sets as necessary, we can
assume that all triple intersections among the open cover are empty. Finally,
we choose a strongly pseudoconvex neighborhood $V$ of $\mathcal{K}$ contained
within this open cover.

Along $H_{\lambda}$, have coordinates $z_{\lambda}$ and
$\theta_{\lambda+1}=\text{Im}\log(z_{\lambda+1})$. There exists real-valued
functions $f_{\lambda,j},g_{\lambda,j}$ such that $\tau_{\lambda,j}$ is cut
out by

$z_{\lambda}=f_{\lambda,j}(\theta_{\lambda+1})+ig_{\lambda,j}(\theta_{\lambda+1})$

We can take real analytic approximations $\widetilde{f},\widetilde{g}$ of
these defining functions and complexify. The vanishing locus of

$z_{\lambda}=\widetilde{f}_{\lambda,j}+i\widetilde{g}_{\lambda,j}$

is a holomorphic curve in some neighborhood of $\tau_{\lambda,j}$. In
particular, $\widetilde{f},\widetilde{g}$ are defined on some smaller bidisk
$\Delta^{\prime}_{\lambda,j}\subset\Delta_{\lambda,j}$, but by the Oka-Weil
theorem we can arbitrarily approximate $\widetilde{f},\widetilde{g}$ by
holomorphic functions on $\Delta_{\lambda,j}$. Let
$\widetilde{F}_{\lambda,j}=\widetilde{f}_{\lambda,j}+i\widetilde{g}_{\lambda,j}$.
Suppose that $\tau_{\lambda,j}$ is incident to the bridge point $b_{k}$. We
can glue the defining functions $\\{\widetilde{F}_{\lambda,j}\\}$ and
$\\{\widetilde{B}_{k}\\}$ into a defining function $G$ on $V$ as follows.
Choose a fixed partition of unity $\\{\phi_{\lambda,j},\phi_{k}\\}$
subordinate to the open cover and define

$G=\phi_{\lambda,j}\widetilde{F}_{\lambda,j}+\phi_{k}\widetilde{B}_{k}$

Since all triple overlaps in the open cover are empty, on the double overlaps
$\Delta_{\lambda,j}\cap U_{k}$ we have

$\phi=\phi_{\lambda,j}=1-\phi_{k}$

Then

$\displaystyle|\overline{\partial}G|$
$\displaystyle=|\overline{\partial}\phi|\cdot|\widetilde{F}-\widetilde{B}|$
$\displaystyle|\nabla\overline{\partial}G|$
$\displaystyle\leq|\nabla(\overline{\partial}\phi)|\cdot|\widetilde{F}-\widetilde{B}|+|\overline{\partial}\phi|\cdot|\nabla(\widetilde{F}-\widetilde{B})|$

Therefore, as $\widetilde{F}_{\lambda,j}\rightarrow\widetilde{B}_{k}$ on each
$\Delta_{\lambda,j}\cap U_{k}$, we use Lemma 4.7 to obtain a holomorphic
function $\widetilde{G}$ on $V^{\prime}\subset V$ whose vanishing locus
$C^{1}$-approximates $\mathcal{K}\cap\mathcal{S}$. We can then perturb
$\mathcal{K}$ to agree with the vanishing locus of $\widetilde{G}$, which
yields the required surface. ∎

######  Corollary 4.9.

Let $(X,\mathcal{T})$ be a holomorphic trisection constructed as a branched
cover of $(\mathbb{CP}^{2},\mathcal{T}_{std})$ and let $\mathcal{K}$ be a
surface that is geometrically transvese with respect to $\mathcal{T}$. Then
$\mathcal{K}$ can be $C^{1}$-approximated by a surface $\mathcal{K}$ that is
holomorphic in an open neighborhood of the spine of $\mathcal{T}$.

###### Proof.

Let $\pi:X\rightarrow\mathbb{CP}^{2}$ be the branched covering, which is
holomorphic in a neighborhood of the spines. After a $C^{\infty}$-small
perturbation, we can assume that $\pi(\mathcal{K})$ is disjoint from the
branch locus in $\mathbb{CP}^{2}$ in some neighborhood of the spine. The image
is geometrically transverse to the standard trisection of $\mathbb{CP}^{2}$.
By the previous theorem, we can approximate it by a holomorphic ribbon, then
pull this back to a holomorphic ribbon in $X$. ∎

##  5\. Hodge Theory

Since we can assume that $(X,\omega)$ admits a integrable complex structure
over the 2-skeleton, we can investigate complex geometric objects.

###  5.1. Picard group

The first important object is the Picard group of holomorphic line bundles.

######  Definition 5.1.

Let $(X,J)$ be an almost-complex 4-manifold with holomorphic trisection
$\mathcal{T}$. The Picard group of $(X,J,\mathcal{T})$ is the group of
holomorphic line bundles over the spine of $\mathcal{T}$, up to biholomorphic
equivalence, that admit smooth trivializations along $\partial\mathcal{S}$.
The group structure is given by tensor product.

One immediate question regarding the Picard group concerns the Hodge structure
induced on $H^{2}_{DR}(X)$. The first Chern class gives a map

$c_{1}:Pic(X,J,\mathcal{T})\rightarrow H^{2}(X;\mathbb{Z})\rightarrow
H^{2}(X;\mathbb{C})$

For Kähler surfaces, the image is precisely the integral $(1,1)$-classes.
However, unlike the Kähler case, we can modify the construction and assume
that every class in $H^{2}(X;\mathbb{Z})$ can be represented by a holomorphic
line bundle.

###### Proof of Theorem 1.7.

Start with some class $\alpha\in H^{2}(X;\mathbb{Z})$ such that
$\int_{X}\omega\wedge\alpha>0$. Then by Proposition 3.10, the Poincare-dual of
$\alpha$ can be represented by a surface that is geometrically transverse to
the holomorphic trisection of $(X,\omega)$. Applying Corollary 4.9, this
surface can be $C^{1}$-approximated by a holomorphic curve in some
neighborhood $\nu(\mathcal{S})^{\prime}$ of the spine. This is an effective
divisor, which immediately determines a holomorphic line bundle whose first
Chern class is Poincare-dual to the divisor.

To prove the theorem, note that $H^{2}(X;\mathbb{Z})$ is finitely-generated
and we can assume it is generated by cohomology classes with positive
symplectic area. Therefore, we can realize a generating set as holomorphic
line bundles, then composing by tensor product realize a holomorphic line
bundle with any given first Chern class. ∎

###  5.2. Smooth Hodge decomposition

Let $(X,\omega,J)$ be the symplectic structure constructed in the main
theorem. Then $\mathcal{S}$ is a compact Riemannian 4-manifold with boundary.
Let $d$ be the exterior derivative on $k$-forms and $d^{*}$ is adjoint with
respect to the metric. Define

$\mathcal{D}=(d+d^{*})\qquad\Delta=\mathcal{D}^{2}=dd^{*}+d^{*}d$

Define the space of harmonic forms

$\mathcal{H}^{k}_{\Delta}=\\{\alpha\in\Omega^{k}(X):\Delta\alpha=0\\}$

and harmonic fields

$\mathcal{H}^{k}_{\mathcal{D}}=\\{\alpha\in\Omega^{k}(X):(d+d^{*})\alpha=0\\}$

Note that harmonic fields are $d$\- and $d^{*}$-closed, but harmonic forms are
not necessarily $d$\- and $d^{*}$-closed on a compact manifold with boundary.
Moreover, in the case of nonempty boundary, the spaces of harmonic forms and
fields are in general larger than DeRham cohomology groups of the manifold
itself.

Since $X$ is almost-complex, we get splittings
$\Omega^{k}(X)=\bigoplus_{p+q=k}\Omega^{p,q}(X)$ of $k$-forms into their
$(p,q)$-components. We also have operators

$\overline{\partial}:\Omega^{p,q}(X)\rightarrow\Omega^{p,q+1}(X)\qquad\partial:\Omega^{p,q}(X)\rightarrow\Omega^{p+1,q}(X)$

and their metric adjoints $\overline{\partial}^{*},\partial^{*}$ and
Laplacians

$\Delta_{\overline{\partial}}=(\overline{\partial}+\overline{\partial}^{*})^{2}\qquad\Delta_{\partial}=(\partial+\partial^{*})^{2}$

Let $U$ be a connected open set on which $J$ is integrable. Then we have a
fiberwise equality

$\Delta=2\Delta_{\overline{\partial}}=2\Delta_{\partial}$

The latter two Laplacians clearly preserve $(p,q)$-bigrading. Consequently, we
get decompositions

$\mathcal{H}^{k}_{\Delta}=\bigoplus_{p+q=k}\mathcal{H}^{p,q}_{\Delta}$

over $U$. Complex conjugation gives an isomorphisms

$\mathcal{H}^{p,q}_{\Delta}\cong\mathcal{H}^{q,p}_{\Delta}$

and the Hodge star gives isomorphisms

$\mathcal{H}^{p,q}_{\Delta}\cong\mathcal{H}^{n-p,n-q}_{\Delta}$

Wedging with the Kähler form $\omega$ induces maps

$L_{\Delta}:\mathcal{H}^{p,q}_{\Delta}\rightarrow\mathcal{H}^{p+1,q+1}_{\Delta}$

which has a metric adjoint $\Lambda_{\Delta}$. The Kähler identifies imply
that $\oplus\mathcal{H}^{p,q}_{\Delta}$ is a
$\mathfrak{sl}(2,\mathbb{C})$-modules.

However, we do not necessarily get a decomposition of
$\mathcal{H}^{k}_{\mathcal{D}}$ into $(p,q)$-summands. We let $H^{p,q}_{d}(X)$
denote the vector space of $d$-closed $(p,q)$-forms modulo $d$-exact
$(p,q)$-forms. On this subspace, we can partial recover the classical Hodge-
Riemann bilinear relations.

######  Theorem 5.2 (Hodge-Riemann bilinear relations).

Let $X$ be a $n$-dimensional complex manifold with strictly pseudoconcave
boundary. Let $\rho$ be a strictly plurisubharmonic function defining the
boundary. Suppose that $X$ admits a Kähler form with Kähler potential $\rho$
near $\partial X$. Then there is a nondegenerate pairing

$H^{p,q}_{d}(X)\times H^{q,p}_{d}(X)\rightarrow\mathbb{C}$

for $p+q\leq n-1$.

###### Proof.

Since the Kähler form is compatible with the boundary, we will show that we
can replace $\omega$ with some $\omega^{\prime}$ that vanishes along $\partial
X$. Then the pairing $\langle\alpha,\beta\rangle$ is given by composing the
wedge product

$H^{p,q}_{d}\times H^{q,p}_{d}\rightarrow H^{p+q,p+q}_{d}\subset
H^{2(p+q)}_{DR}(X)$

with the pairing

$H^{2(p+q)}_{DR}(X)\times H^{2(n-p-q)}_{DR}(X,\partial X)\rightarrow
H^{2n}_{DR}(X,\partial X)\cong\mathbb{C}$

In particular,

$\langle\alpha,\beta\rangle=\int_{X}\alpha\wedge\beta\wedge(\omega^{\prime})^{n-p-q}$

which is well-defined since $\omega^{\prime}\in H^{2}_{DR}(X,\partial X)$.

Let $\rho$ be the defining function and let $f$ be a function such that

$f^{\prime}(0)=f^{\prime\prime}(0)=0$

and $f^{\prime}(t)>0$ and $f^{\prime\prime}(t)\geq 0$ for $t>0$ and $f(t)=t$
for $t\geq\delta>0$. Then

$\omega^{\prime}=i\partial\overline{\partial}f(\rho)$

is a 2-form such that $\omega^{\prime}|_{\partial X}=0$ but
$(\omega^{\prime})^{n}>0$ on the interior of $X$. ∎

######  Corollary 5.3.

Let $\alpha\in H^{p,q}_{\overline{\partial}}(X)$ be a global holomorphic
$(p,0)$-form for $p\leq n-2$. Then $\alpha$ is $d$-closed.

In particular, if $n\geq 2$ then $X$ admits no nonconstant holomorphic
functions.

###### Proof.

We have that

$\int_{X}|d\alpha|^{2}dvol=\int_{X}d\alpha\wedge
d\overline{\alpha}\wedge\omega^{n-p-1}=\int_{\partial X}\alpha\wedge
d\overline{\alpha}\wedge\omega^{n-p-1}=0$

when $n-p\geq 2$. This implies that $|d\alpha|=0$ pointwise, hence
$d\alpha=0$. ∎

###  5.3. Dolbeault cohomology

By construction, the spine $\nu(\mathcal{S})$ is a Kähler surface with
pseudoconcave boundary. Consequently, we have a Frölicher spectral sequence

$H^{\bullet,\bullet}_{\overline{\partial}}(\nu(\mathcal{S}))\Rightarrow
H^{\bullet}_{d}(\nu(\mathcal{S}))$

and for each $0\leq k\leq 2n$ obtain the bound

$\sum_{p+q=k}\text{dim}H^{p,q}_{\overline{\partial}}(\nu(\mathcal{S}))\geq\text{dim}H^{k}_{d}(\nu(\mathcal{S}))$

which is weaker than the Hodge decomposition for a closed Kähler surface.

Let $E$ be a holomorphic vector bundle over $\nu(\mathcal{S})$. One can then
define the Dolbeault cohomology groups
$H^{p,q}_{\overline{\partial}}(\nu(S),E)$. Moreover, since the boundary of
$\nu(\mathcal{S})$ is strictly pseudoconcave, we can apply the classical
Andreotti-Grauert theorem.

######  Theorem 5.4 (Andreotti-Grauert [AG62]).

Let $E$ be a holomorphic vector bundle over $\nu(\mathcal{S})$. The rank of
$H^{p,0}(\nu(\mathcal{S}),E)$ is finite for all $p$.

However, $q=1$ is the critical dimension and there is no finiteness result.
With suitable $\overline{\partial}$-Neumann boundary conditions along
pseudoconcave boundary, Epstein proved finite-dimensionality for
$\overline{\partial}$-harmonic sections of holomorphic bundles for all $q$
[Eps07a, Eps07b, Eps08].

Given a complex surface $X=X_{+}\cup X_{-}$ decomposed along a real
hypersurface $Y$ such that $X_{+}$ has pseudoconvex boundary and $X_{-}$ has
pseudoconcave boundary, Andreotti and Hill proved a Meyer-Vietoris sequence
relating the Dolbeault cohomologies of $X,X_{+},X_{-}$ and the Kohn-Rossi
cohomology of the CR structure induced on $Y$ [AH72]. The failure of a
comparable sequence here may shed light on non-Kähler symplectic structures.

###  5.4. Harmonic spinors

In order to get finite-dimensional spaces of solutions, we take advantage of
harmonic spinors.

Let $E$ be a holomorphic vector bundle over a Kähler surface $X$ and let
$\mathcal{A}^{p,q}(E)$ be the space of $(p,q)$-forms on $X$ valued in $E$.
Given the Kähler metric and complex structure, we have the Dolbeault operator
$\overline{\partial}$ and its adjoint $\overline{\partial}^{*}$

$\overline{\partial}:\mathcal{A}^{p,q}(E)\rightarrow\mathcal{A}^{p,q+1}(E)\qquad\overline{\partial}^{*}:\mathcal{A}^{p,q}(E)\rightarrow\mathcal{A}^{p,q-1}(E)$

The combined operator
$\mathcal{D}=\sqrt{2}(\overline{\partial}+\overline{\partial}^{*})$ is a
Spin-$\mathbb{C}$ Dirac operator

$\mathcal{D}:\mathcal{A}^{p,even/odd}(E)\rightarrow\mathcal{A}^{p,odd/even}(E)$

For fixed $p$, the index of the operator is the Euler characteristic of the
Dolbeault cohomology valued in $E$:

$\text{ind}(\mathcal{D})=\chi(E)=\text{dim}H^{p,0}(E)-\text{dim}H^{p,1}(E)+\text{dim}H^{p,2}(E)$

Generalizing this construction, now suppose $(X,\omega,J)$ is merely an
almost-Kähler 4-manifold and $E$ is complex vector bundle with Hermitian
connection $A$. We can define Dolbeault operators

$\overline{\partial}_{A}:\mathcal{A}^{p,q}(E)\rightarrow\mathcal{A}^{p,q+1}(E)\qquad\overline{\partial}^{*}_{A}:\mathcal{A}^{p,q}(E)\rightarrow\mathcal{A}^{p,q-1}(E)$

as before. In addition, we have bundles of positive and negative spinors

$S_{+}=\mathcal{A}^{p,0}(E)\oplus\mathcal{A}^{p,2}(E)\qquad
S_{-}=\mathcal{A}^{p,1}(E)$

and the sum
$\mathcal{D}_{A}=\sqrt{2}(\overline{\partial}_{A}+\overline{\partial}^{*}_{A})$
is a Dirac operator

$\mathcal{D}_{A}:S_{+}\oplus S_{-}\rightarrow S_{-}\oplus S_{+}$

that exchanges positive and negative spinors. In particular, the Dirac
operator splits into the sum of two operators
$\mathcal{D}^{\pm}_{A}:S_{\pm}\rightarrow S_{\mp}$. The Dirac operator
$\mathcal{D}_{A}$ is a Fredholm operator with index

$\text{ind}(\mathcal{D}_{A})=\text{dim ker }\mathcal{D}^{+}_{A}-\text{dim
coker }\mathcal{D}^{+}_{A}$

By the Atiyah-Singer index theorem, we have that

$\text{ind}(\mathcal{D}_{A})=\chi(E)=\int_{X}ch(E)td(X)$

where $ch(E)$ is the Chern character of $E$ and $td(X)$ is the Todd class of
$X$.

On a general almost-Kähler manifold, the linear subspace $\text{ker
}\mathcal{D}^{+}_{A}$ no longer splits into $(p,0)$\- and $(p,2)$-graded
subspaces. However, suppose that $J$ is integrable over some subset $U$. The
2-form $\omega$ acts fiberwise by Clifford multiplication and satisfies

$\mathcal{D}_{A}(\omega\cdot\psi)=\left(\sqrt{2}(\overline{\partial}+\overline{\partial}^{*})\omega\right)\cdot\psi+\omega\cdot\mathcal{D}_{A}\psi=\omega\cdot\mathcal{D}_{A}$

fiberwise over $U$. In particular, the kernel of $\mathcal{D}_{A}$ is
invariant under the action of $\omega$ by Clifford multiplication. Since
$\mathcal{A}^{p,0}(E)$ and $\mathcal{A}^{p,2}(E)$ are the $(\pm
2i)$-eigenspaces for $\omega$ acting by Clifford multiplication, we get a
locally-constant decomposition

$\text{ker}(\mathcal{D}^{+}_{A})=\text{ker}(\mathcal{D}^{+}_{A})^{(p,0)}\oplus\text{ker}(\mathcal{D}^{+}_{A})^{(p,2)}$

over $U$. Given an element
$(\psi_{0},\psi_{2})\in\text{ker}(\mathcal{D}^{+}_{A})$, we get holomorphic
sections

$\psi_{0}\in\mathcal{A}^{p,0}(E|_{U})\qquad\ast\psi_{2}\in\mathcal{A}^{2-p,0}(E^{*}\otimes
K|_{U})$

where $\ast:\mathcal{A}^{p,q}(E)\rightarrow\mathcal{A}^{2-p,2-q}(E^{*}\otimes
K)$ is the Hodge star operation.

######  Definition 5.5.

Let $U$ be an connected open subset of an almost-Kähler 4-manifold
$(X,\omega,J)$, let $E$ be a complex vector bundle over $X$ that is
holomorphic over $U$, and let $A$ be a Hermitian connection on $E$. Define the
subspaces

$\displaystyle H^{p,0}_{U}(E,A)$
$\displaystyle=\text{ker}(\mathcal{D}^{+}_{A})^{p,0}$ $\displaystyle
H^{p,1}_{U}(E,A)$ $\displaystyle=\text{coker}(\mathcal{D}^{+}_{A})$
$\displaystyle H^{p,2}_{U}(E,A)$
$\displaystyle=\text{ker}(\mathcal{D}^{+}_{A})^{p,2}$

where the splitting of positive spinors is the eigenspace decomposition of the
Kähler form $\omega$ acting by Clifford multiplication.

The Betti numbers of $E$ over $U$ with respect to $A$ are

$\displaystyle h^{p,0}_{U}(E,A)$ $\displaystyle=\text{dim
ker}(\mathcal{D}^{+}_{A})^{p,0}$ $\displaystyle h^{p,1}_{U}(E,A)$
$\displaystyle=\text{dim coker}(\mathcal{D}^{+}_{A})$ $\displaystyle
h^{p,2}_{U}(E,A)$ $\displaystyle=\text{dim ker}(\mathcal{D}^{+}_{A})^{p,2}$

######  Proposition 5.6.

There are Serre duality isomorphisms

$H^{p,q}_{U}(E,A)\cong H^{2-p,2-q}_{U}(E^{*}\otimes K,-A+C)$

and equality of Betti numbers

$h^{p,q}_{U}(E,A)=h^{2-p,2-q}_{U}(E^{*}\otimes K,-A+C)$

where $C$ is the Chern connection on the canonical bundle $K$.

######  Definition 5.7.

A partial holomorphic section of $E$ over $U$ is an element of
$H^{p,0}|_{U}(E,A)$.

######  Lemma 5.8.

Let $L$ be a line bundle on $(X,\omega,J)$ that is holomorphic over $U$ and
let $\psi\in H^{0,0}|_{U}(L,A)$ be a partial holomorphic section. The
vanishing locus

$D=\psi^{-1}(0)$

is a $J$-holomorphic curve in $U$ (with boundary in $\partial U$) and in
particular, is $\omega$-symplectic in $U$.

An effective partial divisor is the vanishing locus $\psi^{-1}(0)$ of a
partial holomorphic section.

###  5.5. Partial sections in symplectic 4-manifolds

Within this section, we restrict to $(X,J,\mathcal{T})$ obtained as branched
covers of $\mathbb{CP}^{2}$. While the Hodge-Riemann bilinear relations
(Theorem 5.2) only hold in subcritical dimensions, we can recover the Hodge
index theorem on the Picard group by using harmonic spinors.

Given a holomorphic line bundle $L\in\text{Pic}(X,J,\mathcal{T})$, since it is
smoothly trivial along $\partial\mathcal{S}$, we can extend it over all of $X$
as a trivial line bundle over each Weinstein sector $Z_{\lambda}$. After
choosing a Hermitian connection $A$ on this line bundle, we can extract
cohomological information from the harmonic spinors and partial holomorphic
sections described in the previous subsection.

We begin with a lemma.

######  Lemma 5.9.

Let $L\in\text{Pic}(X,J,\mathcal{T})$ be a holomorphic line bundle and
$\psi\in H^{0,0}_{U}(L,A)$ a partial section. If $D=\psi^{-1}(0)$ is smooth,
then after a $C^{\infty}$-small perturbation of the boundary, the surface $D$
intersects each component of $\partial\mathcal{S}$ along a link transverse to
the contact structure.

###### Proof.

By a $C^{\infty}$-small perturbation, we can modify the defining function
$\rho_{\lambda}$ on the boundary so that
$(\rho_{\lambda},\psi):X\rightarrow\mathbb{R}\oplus\mathbb{C}$ is transverse
to 0. The link $\Lambda_{\lambda}=D\cap\widehat{Y}_{\lambda}$ is the regular
level set of this map and it represents $PD(c_{1}(L))$ in $H_{1}(Y)$; since
$L$ is smoothly trivial over $\widehat{Y}_{\lambda}$, this link is
nullhomologous. Secondly, the contact structure is exactly the field of
complex tangencies along $\widehat{Y}_{\lambda}$. By a $C^{2}$-small
perturbation of the boundary, we can assume it intersects $D$ transversely
along a smooth 1-manifold. Positivity of intersections of complex lines in
$\mathbb{C}^{2}$ implies that the link is positively transverse to the contact
structure. ∎

######  Proposition 5.10 (Vanishing).

Let $L\in\text{Pic}(X,J,\mathcal{T})$ with $c_{1}(L)$ nonzero and suppose that
$\omega\cdot L\leq 0$. Then $h^{0,0}_{U}(L,A)=0$ for any $A$.

###### Proof.

Suppose that the Betti number is nonzero, which means we can find a nontrivial
partial section $\psi$ with corresponding effective partial divisor
$D=\psi^{-1}(0)$. We can split $D$ into its irreducible components
$D=D_{1}\cup\dots\cup D_{n}$. Each is smooth away from a compact set of
singular points. As in the previous lemma, we can perturb the boundary so that
$D_{i}$ intersects the boundary along a transverse link. Any singular points
in the interior can be smoothed so that $D_{i}$ is a symplectic surface with
transverse boundary.

The boundary $\partial D=\partial D_{1}\cup\dots\partial D_{n}$ is a
nullhomologous transverse link in each $\widehat{Y}_{\lambda}$. We can cap it
off in $Z_{\lambda}$ with a properly embedded surface $F_{\lambda}$. By
Stokes’s Theorem

$\int_{F_{\lambda}}\omega=\int_{\Lambda_{\lambda}}d^{\mathbb{C}}\rho_{\lambda}\geq
0$

with equality if and only if $\Lambda_{\lambda}$ is empty. Here
$\rho_{\lambda}$ is the plurisubharmonic function defining the boundary and
$d^{\mathbb{C}}\rho_{\lambda}$ is the induced contact form, which is a
primitive for $\omega$.

Consequently, we obtain a closed surface $\overline{D}=D\cup F_{1}\cup
F_{2}\cup F_{3}$ in $X$ representing $L$ and with positive symplectic area,
which is a contradiction. ∎

Via the Riemann-Roch theorem for surfaces, if $L$ is a holomorphic line
bundle, we have

$\chi(L)=\frac{1}{2}L\cdot(L-K)+\chi_{h}(X)$

where $K$ is the canonical class of $(X,J)$ and
$\chi_{h}(X)=\frac{1}{12}(c_{1}^{2}+c_{2})$ is its holomorphic Euler
characteristic.

######  Theorem 5.11.

Let $L\subset\text{Pic}(X,J,L)$ be a holomorphic line bundle.

  1. (1)

If $\chi(L)>0$, then either $[L]$ or $[K-L]$ in $H_{2}(X;\mathbb{Z})$ can be
represented by an effective partial divisor.

  2. (2)

If $\chi(L)>0$ and $\omega\cdot L>\omega\cdot K$, then $[L]$ can be
represented by an effective partial divisor.

  3. (3)

If $L^{2}>0$ and $\omega\cdot L>0$, then $mL$ can be represented by a partial
effective divisor for positive $m$ sufficiently large.

###### Proof.

Part (1) follows since by the definition of Euler characteristic:

$\chi(L)=h^{0}(L)-h^{1}(L)+h^{2}(L)$

Part (2) follows from Part (1), because $K-L$ will have negative symplectic
area, hence the symplectic surface must represent $[L]$. Part (3) follows from
parts (1) and (2) because

$\chi(mL)=\frac{1}{2}m^{2}L^{2}+O(m)$

which is positive for $m$ sufficiently large. Moreover,
$\omega\cdot(mL)=m(\omega\cdot L)>\omega\cdot K$ for $m\gg 0$ as well. ∎

######  Lemma 5.12.

Let $L\in\text{Pic}(X,J,\mathcal{T})$ and let $H$ denote the class of
$\omega$. There exist $m,n>0$ such that $mH+nL$ can be represented by a
partial effective divisor

###### Proof.

Consider $H^{\prime}=m^{\prime}H+L$. Then $(H^{\prime})^{2}>0$ and $H\cdot
H^{\prime}=m^{\prime}H^{2}+H\cdot L>0$ for $m^{\prime}>0$ sufficiently large,
so $nH^{\prime}=(nm^{\prime})H+nL$ can be represented by a partial effective
divisor for $n$ sufficiently large. ∎

A version of the Kodaira vanishing theorem for Spin-$\mathbb{C}$ Dirac
operators on symplectic manifolds was proved by Borthwick and Uribe [BU96].

##  6\. BMY Inequality

###  6.1. Miyaoka divisors

Let $X$ be a complex surface of general type, let $\Omega_{X}$ denote its
holomorphic cotangent bundle, let $K_{X}=\text{det}(\Omega_{X})$ be its
canonical bundle, and let $S^{m}\Omega_{x}$ denote the
$m^{\text{th}}$-symmetric power of the (complex) cotangent bundle (which has
complex rank $m+1$). Define

$Q_{\alpha}=S^{m}\Omega_{x}\otimes K^{-m\alpha}$

where $m\gg 0$ is a large integer and $\alpha$ is some rational number such
that $m\alpha$ is integral. The Serre dual of $Q_{\alpha}$ is

$Q^{*}_{\alpha}\otimes K=S^{m}T\otimes K^{1+m\alpha}=S^{m}\Omega\otimes
K^{1-m(1-\alpha)}=Q_{1-\alpha}\otimes K$

Let $\beta=\frac{c_{2}(X)}{c_{1}(X)^{2}}$ (so the Bogomolev-Miyaoka-Yau
inequality states that $\beta\geq\frac{1}{3}$). The Euler characteristic of
$Q$ is

$\chi(Q)=\left(-\frac{1}{6}c_{2}(X)+\frac{1}{6}c_{1}(X)^{2}-\frac{1}{2}\alpha
c_{1}(X)^{2}+\frac{1}{2}\alpha^{2}c_{1}(X)^{2}\right)m^{3}+O(m^{2})$

Therefore, for $m$ sufficiently large, the Euler characteristic is positive if
and only if

$\beta\leq(3\alpha^{2}-3\alpha+1)$

In particular, this forces either $Q_{\alpha}$ or
$Q^{*}_{\alpha}=Q_{1-\alpha}\otimes K$ to admit a holomorphic section.

Now consider $Q_{m,n}=S^{m}\Omega\otimes K^{-n}$. Holomorphic sections of
$Q_{m,n}$ can be identified with effective divisors of the line bundle
$mH-n\pi^{*}(K)$ on $\mathbb{P}(\Omega_{X})$, the projectivization of the
holomorphic cotangent bundle $\Omega_{X}$. Miyaoka proves that no such
effective divisors can exist if $\frac{n}{m}>\frac{c_{2}}{c_{1}^{2}}=\beta$.

The BMY inequality follows since if $\beta<\frac{1}{3}$, we can choose an
$\alpha$ such that

$\beta\leq(3\alpha^{2}-3\alpha+1)\qquad\text{and}\qquad\beta<\alpha,1-\alpha$

The first inequality implies the existence of holomorphic sections, while the
second rules it out. Therefore, we cannot have $\beta<\frac{1}{3}$ and,
equivalently, must have $c_{1}^{2}\leq 3c_{2}$.

Now let $(X,\omega)$ be a closed symplectic 4-manifold and let
$(X,\omega^{\prime},J)$ be the symplectic model from Theorem 1.3. The almost-
complex structure $J$ induces a complex structure on $T^{*}X$, which is
holomorphic over the spine $\mathcal{S}$ of the trisection. We can then form
$\mathbb{P}(T^{*}X)$ as the (complex) projectivization of the cotangent
bundle, which is a $\mathbb{CP}^{1}$-bundle over $X$ that is holomorphic over
$\mathcal{S}(X)$. Moreover, we have line bundles $K$, representing the
canonical class, and $H$, representing the class of the
$\mathbb{CP}^{1}$-fiber, that are holomorphic over $\mathcal{S}(X)$ and
$\mathbb{P}(T^{*}X)|_{\mathcal{S}(X)}$, respectively.

######  Theorem 6.1.

Let $\alpha$ satisfy the inequality

$\frac{c_{2}}{c_{1}^{2}}\leq(3\alpha^{2}-3\alpha+1)$

Then for $m\gg 0$, there exists a holomorphic partial section of $Q_{\alpha}$
and corresponding effective partial divisor $W$ in $\mathbb{P}(\Omega_{X})$
over $\nu(\mathcal{S})$ representing either

$mH-(m\alpha)K\qquad\text{or}\qquad mH-(1-m(1-\alpha))K$

In addition, $W$ is a symplectic 4-manifold with three strongly concave
boundary components.

###### Proof.

The existence of a holomorphic partial section follows from the fact that the
Euler characteristic of the complex bundle $Q_{\alpha}$ is positive for
$\beta,\alpha$ satisfying the inequality. Therefore, either $Q_{\alpha}$ or
its Serre dual $Q_{1-\alpha}\otimes K$ admit a partial holomorphic section
$\psi$.

Viewing $\psi$ as a holomorphic section of a line bundle over
$\mathbb{P}(\Omega_{X})$, the vanishing locus of $\psi$ is an effective
divisor $W_{\psi}$. The composition of the inclusion
$\iota:W_{\psi}\hookrightarrow\mathbb{P}(\Omega_{X})$ with the bundle
projection $\pi:\mathbb{P}(\Omega_{X})\rightarrow X$ is a holomorphic branched
covering. Therefore, the pulled back 2-form $(\pi\circ i)^{*}(\omega)$ can be
perturbed to a symplectic form on $\omega$. In addition, the plurisubharmonic
defining functions on $\partial\mathcal{S}$ pull back to plurisubharmonic
defining functions on $W_{\psi}$. ∎

###  6.2. Stability of the cotangent bundle

For the most part, the proofs of Bogomolev (see [Rei78] or [BHPVdV04] for
example) and Miyaoka’s inequalities follow from computations with Chern
numbers and intersection theory, which carry over immediately to the
symplectic case. However, a key aspect for Bogomolev and for Miyaoka is that
the holomorphic cotangent bundle $\Omega_{X}$ is stable. In this case, this
means that if $L$ is a holomorphic sub-line bundle of $\Omega_{X}$, then the
dimension of $H^{0}(mL)$ grows at most linearly in $m$. A critical part of the
argument is that a holomorphic section $\psi$ of $L$ is a global holomorphic
1-form, hence by Hodge theory it must be closed. This makes it possible to
argue as in the classical Castelnuovo-de Franchis theorem that there is some
holomorphic map $\pi:X\rightarrow C$ to a curve such that
$L=\pi^{*}(L^{\prime})$ for some line bundle on $C$, hence the space of
sections can grow at most linearly. However, it is not necessarily true that a
global holomorphic 1-form on a Kähler surface with strongly pseudoconcave
boundary must be closed with respect to the DeRham exterior derivative.

Moreover, the Kobayashi-Hitchen correspondence between stable holomorphic
bundles and Hermitian Yang-Mills metrics breaks down on Kähler surfaces with
boundary. Donaldson [Don92] and Li-Zhang [LZ05] solved the Dirichlet problem
for Hermitian Yang-Mills metrics. In particular, given any real number
$\lambda$, any holomorphic bundle $E$, and any Hermitian metric $h$ for $E$
along the boundary of a Kähler manifold, there is a Hermitian metric $H$
extending $h$ whose curvature satisfies

$2i\Lambda F_{H}=\lambda\text{Id}$

where $\Lambda$ is the adjoint operator to wedging with the Kähler form. For
example, one can find always find a metric on the holomorphic cotangent bundle
whose curvature form satisfies the weaker Bogomolov-Gieseker inequality
$c_{2}-4c_{1}^{2}\geq 0$ over the interior. However, the true Chern number
computation includes the boundary term, given by the Chern-Simons-Dirac
functional of the connection along the boundary. The key obstruction arises
from the fact that one can prescribe the Hermitian metric along the boundary,
but one cannot prescribe its curvature along the boundary under the Hermitian
Yang-Mills flow.

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