# Cohomological splitting over rationally connected bases

Shaoyun Bai  Columbia University, 2990 Broadway, New York, NY 10027, USA
<EMAIL_ADDRESS>,  Daniel Pomerleano  University of Massachusetts, Boston,
100 William T, Morrissey Blvd, Boston, MA 02125, USA
<EMAIL_ADDRESS>and  Guangbo Xu  Department of Mathematics, Rutgers
University, Hill Center–Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ
08854-8019, USA<EMAIL_ADDRESS>
###### Abstract.

We prove a cohomological splitting result for Hamiltonian fibrations over
enumeratively rationally connected symplectic manifolds. As a key application,
we prove that the cohomology of a smooth, projective family over a smooth
stably rational projective variety splits additively over any field. The main
ingredient in our arguments is the theory of Fukaya–Parker–Ono (FOP)
perturbations developed by the first and third author, which allows one to
define integer-valued Gromov–Witten type invariants.

D.P. is supported by NSF DMS-2306204.

G.X. is supported by NSF DMS-2345030.

###### Contents

  1. 1 Introduction
  2. 2 Integer-valued Gromov–Witten invariants
  3. 3 Seidel maps
  4. 4 Cohomological splitting
  5. 5 Proof of Theorem 4.1
  6. 6 A fibration which does not split
  7. A Pseudocycles in characteristic $p$

##  1\. Introduction

###  1.1. Statement of result

Let $\pi_{P}:P\to B$ be a smooth, projective morphism from a smooth complex
projective variety $P$ to a smooth complex projective base $B$. Fix a ground
field $\mathbb{k}$ and consider the associated Leray spectral sequence

$\displaystyle
E_{2}^{p,q}:=H^{p}(B;\mathbf{R}^{q}\pi_{P}(\mathbb{k}))=>H^{p}(P;\mathbb{k}).$
(1.1)

A celebrated result of Blanchard and Deligne [Bla56, Del68] proved that this
spectral sequence degenerates when $\mathbb{k}$ is any field of characteristic
zero. The purpose of this article is to exhibit settings where the spectral
sequence degenerates when $\mathbb{k}$ is a field of finite characteristic
using tools from enumerative geometry: moduli spaces of stable maps and
Gromov–Witten invariants.

Recall that an algebraic variety is called rationally connected if every two
points in this variety are connected by a rational curve. Rationally connected
varieties are central objects in birational geometry, see [Kol96] for an
introduction. In particular, all smooth Fano varieties over an algebraically
closed field of characteristic $0$ are rationally connected by a famous
theorem of Kollár–Miyaoka–Mori [KMM92] and Campana [Cam92].

Our approach to studying the degeneration of the Leray spectral sequence is
via the symplectic topology of Hamiltonian fibrations, i.e. smooth fiber
bundles of symplectic manifolds whose structure group is reduced to the group
of Hamiltonian symplectomorphisms of the fiber. As such, we introduce the
following symplectic version of rational connectedness.

######  Definition 1.1.

Let $(B,\omega_{B})$ be a closed symplectic manifold. It is called
_symplectically or enumeratively rationally connected_ if there exists
$k\in{\mathbb{Z}}_{\geq 0}$ such that it admits a genus $0$ Gromov–Witten
invariant

$\mathrm{GW}_{0,k+2}^{B,A}([{\rm pt}],[{\rm pt}],a_{1},\cdots,a_{k})\neq 0,$
(1.2)

where $[{\rm pt}]\in H^{\rm top}(B;{\mathbb{Q}})$ is the Poincaré dual of the
point class, $a_{i}\in H^{*}(B;{\mathbb{Q}})$, and $A\in
H_{2}(B;{\mathbb{Z}})$ is a curve class.

This definition is motivated by a conjecture of Kollár [Kol98, Conjecture
4.2.7], which conjectures that for any pair of symplectically deformation
equivalent complex smooth projective varieties, one is rationally connected if
and only if the other is so. This conjecture is resolved in full generality
for $3$-folds by the work of Voisin and Tian [Voi08, Tia12], with significant
progress for $4$-folds [Tia15]. All of these proofs are based on exhibiting a
nonvanishing (possibly descendant) Gromov–Witten invariant of the form (1.2).
So, we use Definition 1.1 as our notion for rational connectedness in the
symplectic category.

To state our first result, we focus on monotone symplectic manifolds, i.e.,
when the first Chern class is positively proportional to the cohomology class
represented by the symplectic form, known as the symplectic counterparts of
Fano varieties. When $(B,\omega_{B})$ is a monotone symplectic manifold, it is
well-known that we can choose such $\mathrm{GW}_{0,k+2}^{B,A}([{\rm pt}],[{\rm
pt}],a_{1},\cdots,a_{k})$ to be an integer, and $a_{i}\in
H^{*}(B;{\mathbb{Z}})$ (cf. [RT95, MS04]).

######  Theorem A.

Let $(B,\omega_{B})$ be a compact, monotone, and enumeratively rationally
connected symplectic manifold with a nonzero Gromov–Witten invariant
$\mathrm{GW}_{0,k+2}^{B,A}([{\rm pt}],[{\rm pt}],a_{1},\cdots,a_{k})$. If
$(M,\omega_{M})$ is a compact symplectic manifold and

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{P}}$$\textstyle{B}$

is a Hamiltonian fibration, then for any coefficient field $\mathbb{k}$ with
positive characteristics $p$ which does not divides
$\mathrm{GW}_{0,k+2}^{B,A}([{\rm pt}],[{\rm pt}],a_{1},\cdots,a_{k})$, there
holds

$H^{*}(P;\mathbb{k})\cong
H^{*}(B;\mathbb{k})\otimes_{\mathbb{k}}H^{*}(M;\mathbb{k})$ (1.3)

as $\mathbb{k}$-vector spaces. If $\mathbb{k}={\mathbb{Q}}$, the additive
splitting (1.3) always holds under the assumption
$\mathrm{GW}_{0,k+2}^{B,A}([{\rm pt}],[{\rm pt}],a_{1},\cdots,a_{k})\neq 0$.

Returning to our original algebro-geometric situation, Theorem A implies the
following result.

######  Corollary 1.2.

Let $\pi_{P}:P\to B$ be a smooth projective family over base $B$ which is Fano
and enumeratively rationally connected. Then for any coefficient field
${\mathbb{k}}$, $H^{*}(P;{\mathbb{k}})\cong
H^{*}(B;{\mathbb{k}})\otimes_{\mathbb{k}}H^{*}(M;{\mathbb{k}})$ provided that
there is a genus zero primary Gromov–Witten invariant of $B$ with at least two
point insertions which is not divisible by ${\rm char}({\mathbb{k}})$.

###### Proof.

By the theorem of Kollár–Miyaoka–Mori [KMM92] and Campana [Cam92],
$\pi_{1}(B)=0$. On the other hand, as $P$ is projective, there is a Kähler
form $\omega_{P}\in\Omega^{2}(P)$; as $\pi_{P}$ is a holomorphic submersion,
each fiber is a complex, hence symplectic submanifold. Then by [MS98, Lemma
6.2] (or see [BGS88, Theorem 1.7] for Kähler fibrations), the map
$\pi_{P}:P\to B$ admits the structure of symplectic fibration. Then by [LM03,
Theorem 1.1], the symplectic fibration reduces to a Hamiltonian fibration.
Then Theorem A implies the splitting result. ∎

Note that we do not impose any positivity condition of the fiber
$(M,\omega_{M})$. At this level of generality, Theorem A was only known for
$B=\mathbb{CP}^{1}$ by Abouzaid–McLean–Smith [AMS21, Theorem 1.1] (see the
work of the first and third author [BX22b] for an alternative proof), and $B$
being a product of Grassmannians by a result of the first and second author
[BP24, Lemma 2.8]. To compare these previous results with Theorem A, note that
for $\mathbb{CP}^{1}$, the Gromov–Witten invariant
$\mathrm{GW}_{0,2}^{\mathbb{CP}^{1},[\mathbb{CP}^{1}]}([{\rm pt}],[{\rm
pt}])=1$ because itself is the unique line connecting two points; for the
Grassmannians, as discussed in [Hu19, Example 3.10], there is a nonvanishing
Gromov–Witten invariant $\mathrm{GW}_{0,3}^{B,A}([{\rm pt}],[{\rm pt}],a)=1$,
so $\eqref{eqn:coh-splitting}$ holds for any field $\mathbb{k}$ when $B$ is
either $\mathbb{CP}^{1}$ or a Grassmannian directly from Theorem A.

######  Remark 1.3.

In Section 6 below, we provide an example of a projective fibration over a
smooth Fano variety whose cohomology does not split in characteristic $2$.

The monotone condition on $B$ can be relaxed as long as the nonzero GW
invariant is realized by a transverse moduli space (see Theorem 4.1). This
seemingly technical extension has a noteworthy consequence. To state it, we
recall the following definition:

######  Definition 1.4.

A complex projective variety $B$ of dimension $m$ is stably rational if
$B\times\mathbb{CP}^{r}$ is birational to $\mathbb{CP}^{m+r}$.

It is not difficult to see that stably rational varieties are rationally
connected [Har01]. By combining the above methods with certain topological
arguments from [LM03, BP24], we are able to show:

######  Theorem B.

Suppose $B$ is a smooth stably rational projective variety. Then for any
Hamiltonian fibration $P\to B$ (in particular, any smooth algebraic family
$P\to B$ with $P$ projective) and any coefficient field ${\mathbb{k}}$, there
is an isomorphism of graded ${\mathbb{k}}$-vector spaces

$H^{*}(P;{\mathbb{k}})\cong
H^{*}(B;{\mathbb{k}})\otimes_{\mathbb{k}}H^{*}(M;{\mathbb{k}}).$

######  Remark 1.5.

Historically, the study of cohomological splitting of Hamiltonian fibrations
over $\mathbb{CP}^{1}$ was initiated by Lalonde–McDuff–Polterovich [LMP99] and
McDuff [McD00a] using holomorphic curve methods, building on Seidel’s work on
Seidel representations [Sei97]. For general symplectic fibers, Lalonde–McDuff
[LM03] used various “soft” topological arguments to establish the
cohomological splitting over ${\mathbb{Q}}$ for Hamiltonian fibrations over a
large class of manifolds, and proposed the relation between cohomological
splitting and enumeratively rational connectedness [LM03, Section 4.2]. They
established cohomological splitting over ${\mathbb{Q}}$ for any Hamiltonian
fibration over blowups of $\mathbb{CP}^{n}$. The Lalonde–McDuff proposal was
partially realized by Hyvrier [Hyv12], who proved the cohomological splitting
with ${\mathbb{Q}}$ coefficients for Hamiltonian fibrations over
symplectically rationally connected base assuming the fibers are semi-
positive. Theorem A and Theorem B are the first general results along this
line of inquiries with allowance of torsion classes in cohomology.

###  1.2. Discussion of the proof

Because we work over a field, we know that our result about the splitting of
cohomology (which is equivalent to the splitting of homology) follows from the
degeneration of the Leray–Serre spectral sequence of the fibration, which, in
turn, is implied by the injection of the inclusion map

$\iota_{*}:H_{*}(M;\mathbb{k})\to H_{*}(P;\mathbb{k}).$

We prove this injectivity via a certain non-vanishing result of
${\mathbb{k}}$-valued Gromov–Witten invariants of the total space. The
Hamiltonian fibration structure induces a closed extension of the fiberwise
symplectic form to the total space $P$, which becomes a symplectic form after
adding a multiple of the pullback of the symplectic form on the base $B$. When
${\rm char}({\mathbb{k}})>0$, the ${\mathbb{k}}$-valued Gromov–Witten
invariants can be obtained via the integral Gromov–Witten pseudocycle defined
in [BX22b] following the idea of Fukaya–Ono [FO97]. Based on this, Theorem A
follows from the following nonvanishing result. To illustrate the idea of the
proof without getting into the details we assume that $k=0$ so the base $B$
admits a nonvanishing integral Gromov–Witten invariant ${\rm
GW}_{0,2}^{B,A}({\rm pt},{\rm pt})\neq 0$.

######  Theorem 1.6.

(See a more general version as Theorem 5.2) Under the same assumptions as in
Theorem A while assuming $k=0$, for any nonzero homology class $\beta_{0}\in
H_{*}(M;{\mathbb{k}})$, there exists a homology class $\beta_{\infty}\in
H_{*}(M;{\mathbb{k}})$ and a curve class $\tilde{A}\in H_{2}(P;{\mathbb{Z}})$
such that

$\mathrm{GW}^{P,{\mathbb{k}};\tilde{A}}_{0,2}\Big{(}{\rm
PD}(\iota_{*}(\beta_{0})),{\rm PD}(\iota_{*}(\beta_{\infty}))\Big{)}\neq 0,$
(1.4)

where the above Gromov–Witten invariant is the ${\mathbb{k}}$-valued version
induced from the integral Gromov–Witten invariants.

We provide a heuristic explanation of this result and indicate the technical
difficulties. Very ideally, the moduli space of two-pointed stable genus $0$
maps to $B$ representing the second homology class $A\in
H_{2}(B;{\mathbb{Z}})$

$\overline{\mathcal{M}}_{0,2}(B,A)$

is smooth and compact, with the evaluation map

${\rm ev}:\overline{\mathcal{M}}_{0,2}(B,A)\to B\times B$

a local diffeomorphism of finite degrees. In this case, for any pair of
distinct points $p_{1},p_{2}\in B$, the Gromov–Witten invariant is a signed
sum of the finitely many rational curves in
$\overline{\mathcal{M}}_{0,2}(B,A)$ which pass through $p_{1}$ and $p_{2}$. If
$\tilde{A}\in H_{2}(P;{\mathbb{Z}})$ is a lift of $A$, then we see that the
composition with $\pi_{P}$ defines a map between the moduli spaces of stable
rational curves

$\overline{\mathcal{M}}_{0,2}(P,\tilde{A})\to\overline{\mathcal{M}}_{0,2}(B,A).$

The preimage of an element ${\rm
ev}^{-1}(\\{p_{1}\\}\times\\{p_{2}\\})\in\overline{\mathcal{M}}_{0,2}(B,A)$,
if can be represented by a smooth rational curve $C$ in $B$, can be identified
with the moduli space of stable genus $0$ holomorphic “sections” of the
fibration over the projective line, $P|_{C}$. Then, in this ideal situation,
we can appeal to the nonvanishing result of Gromov–Witten section invariants
of fibrations over $\mathbb{CP}^{1}$ [Sei97, McD00a] to find the desired
invariant, at least for the ${\mathbb{Q}}$-valued version.

The technical difficulties of making the above sketch feasible in general are
of a familiar kind: to compactify the moduli space of rational curves using
stable maps, one needs to introduce maps with nodal domains; also, the moduli
spaces are “stacky” in nature, which means that the nontrivial automorphisms
of stable maps forbid one to go beyond characteristic zero when defining
enumerative invariants following the traditional approach to Gromov–Witten
theory.

We resolve these obstacles using tools from symplectic topology, both
classical and modern. In the monotone setting as Theorem A, one uses the
classical regularization method written up systematically by McDuff–Salamon
[MS04] to find a generic almost complex structure on $B$ compatible with
$\omega_{B}$ to ensure that only smooth curves in
$\overline{\mathcal{M}}{}_{0,2}(B,A)$ contributes to the relevant
Gromov–Witten invariant. In the situation of Theorem B where $B$ is not
necessarily monotone, we reduce the consideration to the case that $B$ is a
blowup of $\mathbb{CP}^{n}$, which has a nonzero Gromov–Witten invariant
realized by a single smooth rational curve. In order to allow the field
${\mathbb{k}}$ to have positive characteristics, we appeal to the
Fukaya–Ono–Parker (FOP) perturbation method developed by the first and third
author [BX22b, BX22a] to develop a theory of Seidel elements in finite
characteristics and find the required nonzero Gromov–Witten type invariant.
These two regularization methods are combined fruitfully via variants of the
global Kuranishi charts of Abouzaid–McLean–Smith [AMS21, AMS23].

Finally, we note that the approach to proving cohomological splitting in
finite characteristic from [AMS21] goes through Morava $K$-theories. It is an
interesting open problem to see whether one can prove Theorem A using
Gromov–Witten Morava $K$-theory invariants. We expect that to make progress
along these lines, one needs to revisit the Baas–Sullivan approach [Baa70] to
generalized cohomology theory to establish a dictionary between ordinary
Gromov–Witten invariants and these generalized Gromov–Witten invariants. There
are other motivations to investigate this question: via chromatic lifting
techniques, one may go beyond additive splitting of ordinary cohomology. We
note that [BP24, Lemma 2.8] implies that for any smooth projective family
$P\to B$ over a product of Grassmannians and any complex oriented cohomology
theory $\mathbb{E}^{*}$, we have:

$\displaystyle\mathbb{E}^{*}(P)\cong\mathbb{E}^{*}(M)\otimes_{\mathbb{E}^{*}(pt)}\mathbb{E}^{*}(B).$
(1.5)

It would be interesting to know whether a similar splitting held in the stably
rational case.

###  1.3. Outline of the paper

In Section 2 we review the definition of integer-valued Gromov–Witten
invariants given in [BX22b] via the Fukaya–Ono–Parker (FOP) perturbation
method, as well as the reduction to fields of finite characteristic. In
Section 3 we define the ${\mathbb{k}}$-valued graph Gromov–Witten invariants
for Hamiltonian fibrations over $S^{2}$ and the corresponding Seidel map using
the FOP perturbation method. In Section 4 we state the main technical result
(Theorem 4.1), which implies both Theorem A and Theorem B. In Section 5 we
prove Theorem 4.1. In Section 6, we provide an example of a projective
fibration over a Fano manifold whose cohomology does not split in
characteristic $2$.

### Acknowledgement

We would like to thank Sasha Kuznetsov for some help with Theorem 6.1, Jørgen
Vold Rennemo for a helpful email exchange concerning Remark 6.4, and Ziquan
Zhuang for a discussion on weak factorizations. D.P. would like to thank Paul
Seidel for an inspiring discussion at the start of this project.

##  2\. Integer-valued Gromov–Witten invariants

In this section we review the genus zero integer-valued Gromov–Witten
invariants constructed in [BX22b]. The ordinary GW invariants of a general
symplectic manifold fail to be integers because of the orbifold nature of the
moduli spaces of stable maps and because the orbifold parts are rather “large”
in the moduli space. The integer-valued GW invariants originated from the
insight of Fukaya–Ono [FO97], where they observed the possibility of using a
particular kind of single-valued perturbation on Kuranishi structures on the
moduli spaces of stable maps. A crucial point is that the moduli spaces carry
(stable) complex structures. On the technical level the idea was further
explored by B. Parker [Par13] who clarified crucial properties needed for the
special kind of perturbations Fukaya–Ono proposed. Eventually the first and
the third author [BX22b] polished the ideas of Fukaya–Ono and B. Parker and
rigorously constructed the ${\mathbb{Z}}$-valued GW invariants in genus zero.

###  2.1. Derived orbifold charts and FOP pseudocycle

A derived orbifold chart (D-chart for short) is a triple

${\mathcal{C}}=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$

where ${\mathcal{U}}$ is a smooth orbifold, ${\mathcal{E}}\to{\mathcal{U}}$ is
an orbifold vector bundle (known as the obstruction bundle), and
${\mathcal{S}}:{\mathcal{U}}\to{\mathcal{E}}$ is a continuous section such
that ${\mathcal{S}}^{-1}(0)$ is compact. A morphism of D-charts from
${\mathcal{C}}_{1}$ to ${\mathcal{C}}_{2}$ consists of a map
$\iota_{21}:{\mathcal{U}}_{1}\to{\mathcal{U}}_{2}$ and a bundle map
$\widehat{\iota}_{21}:{\mathcal{E}}_{1}\to{\mathcal{E}}_{2}$ covering
$\iota_{21}$ such that

$\widehat{\iota}_{21}\circ{\mathcal{S}}_{1}={\mathcal{S}}_{2}\circ\iota_{21}.$

There are the following typical types of morphisms.

  1. (1)

Shrinking/open embedding induced by shrinking ${\mathcal{U}}$ to an open
neighborhood of ${\mathcal{S}}^{-1}(0)$ and restricting ${\mathcal{E}}$
together with ${\mathcal{S}}$.

  2. (2)

Stabilization by a vector bundle. Namely, there is a vector bundle
$\pi_{\mathcal{F}}:{\mathcal{F}}\to{\mathcal{U}}_{1}$ such that
${\mathcal{U}}_{2}$ is the total space of ${\mathcal{F}}$,
${\mathcal{E}}_{2}=\pi_{\mathcal{F}}^{*}{\mathcal{E}}_{1}\oplus\pi_{\mathcal{F}}^{*}{\mathcal{F}}$,
and
${\mathcal{S}}_{2}=\pi_{\mathcal{F}}^{*}{\mathcal{S}}_{1}\oplus\tau_{\mathcal{F}}$
where $\tau_{\mathcal{F}}$ is the tautological section; moreover $\iota_{21}$
is the inclusion of the zero section and $\widehat{\iota}_{21}$ is the
embedding of ${\mathcal{E}}_{1}$ into the first summand of
${\mathcal{E}}_{2}$.

Moreover, a strict cobordism from ${\mathcal{C}}_{1}$ to ${\mathcal{C}}_{2}$
is a D-chart with boundary
$\tilde{\mathcal{C}}=(\tilde{\mathcal{U}},\tilde{\mathcal{E}},\tilde{\mathcal{S}})$
together with identifications

$\partial\tilde{\mathcal{C}}=(\partial\tilde{\mathcal{U}},\tilde{\mathcal{E}}|_{\partial\tilde{\mathcal{U}}},\tilde{\mathcal{S}}|_{\tilde{\mathcal{U}}})\cong{\mathcal{C}}_{1}\sqcup{\mathcal{C}}_{2}.$

######  Definition 2.1.

We say that two D-charts ${\mathcal{C}}_{1}$ and ${\mathcal{C}}_{2}$ are
equivalent if ${\mathcal{C}}_{1}$ can be connected to ${\mathcal{C}}_{2}$ by a
zig-zag of open embeddings or stabilizations; ${\mathcal{C}}_{1}$ and
${\mathcal{C}}_{2}$ are called cobordant if they are equivalent to a strictly
cobordant pairs.111The relation induced by the shrinking morphisms is a
special case of the cobordism relation. We keep the shrinking morphisms
explicit because they naturally arise from transversality arguments in the
geometric construction of D-charts for moduli spaces of $J$-holomorphic
curves. By [Par21, Proposition 5.1], the cobordism relation defines an
equivalence relation among D-charts.

We expect that the zero locus ${\mathcal{S}}^{-1}(0)$ to carry a fundamental
class which is invariant under cobordism relation. This is the case in
rational coefficients and the fundamental class is roughly the Poincaré dual
of the Euler class. Equivalently, a fundamental cycle can be obtained by
taking the zero locus of a multi-valued transverse perturbation of
${\mathcal{S}}$. Fukaya–Ono’s idea to obtain a fundamental class with integer
coefficients is to carefully choose single-valued perturbations satisfying a
more refined version of the transversality condition, although the topological
nature of such a fundamental class is not completely clear.

To carry out the Fukaya–Ono–Parker (FOP) perturbation scheme, one needs a
version of complex structures on orbifolds and the obstruction bundles. A
normal complex structure on an effective orbifold ${\mathcal{U}}$ associates
to each orbifold chart $(U,\Gamma)$ and each subgroup $H\subset\Gamma$ an
$H$-invariant complex structure on the normal bundle $NU^{H}\to U^{H}$ of the
$H$-fixed point set $U^{H}\subset U$ such that orbifold coordinate changes
respect these complex structures. If ${\mathcal{E}}\to{\mathcal{U}}$ is an
orbifold vector bundle, a normal complex structure on ${\mathcal{E}}$
associates to each bundle chart $(U,E,\Gamma)$ (where $(U,\Gamma)$ is an
orbifold chart and $E\to U$ is a $\Gamma$-equivariant vector bundle) and each
subgroup $H\subset\Gamma$ an $H$-invariant complex structure on the subbundle
$\check{E}^{H}\subset E|_{U^{H}}$ (which is the direct sum of nontrivial
irreducible $H$-representations contained in $E|_{U^{H}}$) such that the
bundle coordinate changes respect these complex structures. A normal complex
structure on a D-chart
${\mathcal{C}}=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ consists of a
normal complex structure on ${\mathcal{U}}$ and a normal complex structure on
${\mathcal{E}}$. The notion of derived cobordism can be defined for normally
complex D-charts if we require that stabilizations are via complex vector
bundles.

######  Theorem 2.2.

[BX22b] Given any normally complex and effective orbifold ${\mathcal{U}}$ and
a normally complex orbifold vector bundle ${\mathcal{E}}\to{\mathcal{U}}$,
there is a class of (single-valued) smooth sections, called FOP transverse
sections which satisfy the following conditions.

  1. (1)

The FOP transversality condition is local; moreover, over the isotropy-free
part ${\mathcal{U}}_{\rm free}\subset{\mathcal{U}}$ (which is a manifold),
being FOP transverse is equivalent to being transverse in the classical sense.

  2. (2)

The condition that a smooth section ${\mathcal{S}}$ is FOP transverse only
depends on the behavior of ${\mathcal{S}}$ near ${\mathcal{S}}^{-1}(0)$. In
particular, any smooth section ${\mathcal{S}}$ is FOP transverse away from
${\mathcal{S}}^{-1}(0)$.

  3. (3)

Given a continuous norm on ${\mathcal{E}}$, for any continuous section
${\mathcal{S}}_{0}:{\mathcal{U}}\to{\mathcal{E}}$ and any $\delta>0$, there
exists an FOP transverse section ${\mathcal{S}}:{\mathcal{U}}\to{\mathcal{E}}$
such that

$\|{\mathcal{S}}_{0}-{\mathcal{S}}\|_{C^{0}}\leq\delta.$

  4. (4)

(CUDV property) Given any closed subset $C\subset{\mathcal{U}}$ and a smooth
section ${\mathcal{S}}_{0}:O\to{\mathcal{E}}$ defined over an open
neighborhood $O$ of $C$, if ${\mathcal{S}}_{0}$ is FOP transverse near $C$,
then there exists an FOP transverse section
${\mathcal{S}}:{\mathcal{U}}\to{\mathcal{E}}$ which agrees with
${\mathcal{S}}_{0}$ near $C$.

  5. (5)

(Stabilization property) Suppose
$\pi_{\mathcal{F}}:{\mathcal{F}}\to{\mathcal{U}}$ is an orbifold vector bundle
and $\tau_{\mathcal{F}}:{\mathcal{F}}\to\pi_{\mathcal{F}}^{*}{\mathcal{F}}$ is
the tautological section. If ${\mathcal{S}}:{\mathcal{U}}\to{\mathcal{E}}$ is
an FOP transverse section, then the section

$\pi_{\mathcal{F}}^{*}{\mathcal{S}}\oplus\tau_{\mathcal{F}}:{\mathcal{F}}\to\pi_{\mathcal{F}}^{*}{\mathcal{E}}\oplus\pi_{\mathcal{F}}^{*}{\mathcal{F}}$

is a also an FOP transverse section.

  6. (6)

If ${\mathcal{Z}}\subset{\mathcal{U}}$ is a closed and proper sub-orbifold
whose normal bundle is an ordinary vector bundle (which implies
${\mathcal{Z}}$ is also normally complex), and if
${\mathcal{S}}:{\mathcal{Z}}\to{\mathcal{E}}$ is an FOP transverse section,
then there exists an FOP transverse extension of ${\mathcal{S}}$ to
${\mathcal{U}}$.

  7. (7)

For any FOP transverse section ${\mathcal{S}}:{\mathcal{U}}\to{\mathcal{E}}$,
the isotropy-free part of the zero locus

${\mathcal{S}}^{-1}(0)_{\rm free}:={\mathcal{S}}^{-1}(0)\cap{\mathcal{U}}_{\rm
free}$

is a transverse intersection in the classical sense. Moreover, its boundary

$\overline{{\mathcal{S}}^{-1}(0)_{\rm
free}}\setminus{\mathcal{S}}^{-1}(0)_{\rm
free}\subset{\mathcal{U}}\setminus{\mathcal{U}}_{\rm free}$

is the union of images of smooth maps from manifolds of dimension at most
${\rm dim}{\mathcal{U}}-{\rm rank}{\mathcal{E}}-2$.

Now if we are given a normally complex effective derived orbifold chart
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$, we can choose an FOP transverse
section ${\mathcal{S}}^{\prime}$ which agrees with ${\mathcal{S}}$ outside a
compact neighborhood of ${\mathcal{S}}^{-1}(0)$. Then
$({\mathcal{S}}^{\prime})^{-1}(0)$ is compact and the isotropy-free part

$({\mathcal{S}}^{\prime})^{-1}(0)_{\rm free}$

is a pseudocycle 222The notion of pseudocyle in orbifolds is defined in
[BX22b]. of dimension ${\rm dim}{\mathcal{U}}-{\rm rank}{\mathcal{E}}$. Hence
if ${\mathcal{C}}$ is oriented, this pseudocylce represents an integral
homology class in ${\mathcal{U}}$. Moreover, given any two FOP transverse
perturbations ${\mathcal{S}}_{1}^{\prime},{\mathcal{S}}_{2}^{\prime}$, the two
pseudocycles $(S_{1}^{\prime})^{-1}(0)_{\rm free}$ and
$(S_{2}^{\prime})^{-1}(0)_{\rm free}$ are cobordant. Therefore the integral
homology class, which we call the FOP Euler class

$\chi^{\rm FOP}({\mathcal{C}})\in H_{*}({\mathcal{U}};{\mathbb{Z}}),$

is well-defined.

One typically needs to push forward the homology class into another space. To
connect with classical cobordism theory, we restrict to the case when the
derived orbifold charts are stably complex, meaning that the virtual vector
bundle $T{\mathcal{U}}-{\mathcal{E}}$ has a stable complex structure (see
[BX22b, Definition 6.11, 6.14]). In particular, if $T{\mathcal{U}}$ and
${\mathcal{E}}$ are both complex vector bundles, then
${\mathcal{C}}=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ is automatically
stably complex and normally complex. In the special case when ${\mathcal{U}}$
is a manifold and ${\mathcal{S}}:{\mathcal{U}}\to{\mathcal{E}}$ is transverse,
then a stable complex structure on
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ induces a stable complex
structure on the manifold ${\mathcal{S}}^{-1}(0)$. On the other hand, if a
derived orbifold chart ${\mathcal{C}}$ is stably complex, then via a
stabilization, it is equivalent to a normally complex one.

For any topological space $X$, the stably complex derived orbifold bordism
group

$\Omega^{{\rm der},{\mathbb{C}}}_{k}(X)$

is the abelian group generated by quadruples
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},f)$ where
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ is an stably complex D-chart of
virtual dimension ${\rm dim}{\mathcal{U}}-{\rm rank}{\mathcal{E}}=k$ and
$f:{\mathcal{U}}\to X$ is a continuous map, modulo the equivalence relation
induced from stabilization by complex vector bundles and cobordism respecting
the stable complex structures. The assignment $\Omega^{{\rm
der},{\mathbb{C}}}_{*}(-)$ actually defines a generalized homology theory.
Then the pushforward of the FOP pseudocycle (and its homology class) induces a
natural transformation of generalized homology theories

$\Omega^{{\rm der},{\mathbb{C}}}_{*}(-)\to H_{*}(-;{\mathbb{Z}}).$

In fact, when the target space is a manifold, this natural transformation
factors through pseudocycles. As we do not know if pseudocycles up to
cobordism is a homology theory or not, we only consider the naive properties.
Namely, for any manifold $X$, one has a group homomorphism

$\Omega_{k}^{{\rm deg},{\mathbb{C}}}(X)\to{\mathcal{H}}_{k}(X)$

where ${\mathcal{H}}_{k}(X)$ is the abelian group of $k$-dimensional oriented
pseudocycles up to cobordism (see Appendix).

###  2.2. AMS charts and the integer-valued GW invariants

Let $(X,\omega)$ be a compact symplectic manifold and $J$ be an
$\omega$-compatible almost complex structure. Given $k\geq 0$ and $A\in
H_{2}(X;{\mathbb{Z}})$, consider the moduli space of stable $J$-holomorphic
spheres with $k$ marked points in class $A$, denoted by

$\overline{\mathcal{M}}_{0,k}(X,J,A).$

There is a continuous evaluation map

${\rm ev}:\overline{\mathcal{M}}_{0,k}(X,J,A)\to X^{k}.$

A D-chart lift of $(\overline{\mathcal{M}}_{0,k}(X,J,A),{\rm ev})$ is a tuple

$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\psi,\widetilde{\rm ev})$

where $({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ is a D-chart, $\psi$ is an
isomorphism of orbispaces between ${\mathcal{S}}^{-1}(0)$ and
$\overline{\mathcal{M}}_{0,k}(X,J,A)$, and $\widetilde{\rm
ev}:{\mathcal{U}}\to X^{k}$ is an extension of the evaluation map to
${\mathcal{U}}$. In [AMS21, AMS23] (see also [HS22]) a class of global
Kuranishi charts (see Definition 3.2) on
$\overline{\mathcal{M}}{}_{0,k}(X,J,A)$ were constructed, inducing a class of
derived orbifold charts lifts (which we call AMS D-charts). Moreover, for such
a chart, one can make sure that both $T{\mathcal{U}}$ and $T{\mathcal{E}}$ are
complex vector bundles. In particular, the AMS D-charts are stably complex.
The construction of an AMS chart depends on various choices but the
equivalence class does not. Hence the quadruple
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\tilde{\rm ev})$ induces an
element of $\Omega_{i(A)}^{{\rm der},{\mathbb{C}}}(X^{k})$ where $i(A)$ is the
expected dimension of $\overline{\mathcal{M}}{}_{0,k}(X,J,A)$. Moreover, if
$(\omega,J)$ can be deformed to a pair $(\omega^{\prime},J^{\prime})$ via a
smooth path $(\omega_{t},J_{t})$, then the resulting AMS charts are derived
cobordant. Therefore the corresponding element, denoted by

${\rm GW}^{\mathbb{Z}}_{0,k}(X,A)^{\rm vir}\in\Omega_{i(A)}^{{\rm
der},{\mathbb{C}}}(X^{k}),$

only depends on the deformation class of $\omega$.

####  2.2.1. Reduce to characteristic $p$

For the purpose of this article, we would like to reduce the integral cycle
and invariants to characteristic $p$. Let ${\mathbb{k}}$ be a field. For any
$n$-dimensional compact oriented manifold $Y$, there holds the Poincaré
duality

$H_{k}(Y;{\mathbb{k}})\cong{\rm Hom}(H^{k}(Y;{\mathbb{k}}),{\mathbb{k}}).$

Now the cobordism class of the integral Gromov–Witten pseudocycle induces a
well-defined integral homology class

${\rm GW}_{0,k}^{\mathbb{Z}}(X,A)^{\rm vir}\in H_{i(A)}(X^{k};{\mathbb{Z}}).$

By the discussion in Appendix A, the associated integral pseudocycle also
defines a pseudocycle with characteristic $p$ coefficients, which induces a
homology class with $\mathbb{k}$-coefficient

${\rm GW}_{0,k}^{\mathbb{k}}(X,A)^{\rm vir}\in H_{i(A)}(X^{k};{\mathbb{k}}).$

Hence we can define the correlator in ${\mathbb{k}}$-coefficients as

${\rm GW}^{\mathbb{k}}_{0,k}(\alpha_{1},\ldots,\alpha_{k})={\rm
GW}^{\mathbb{k}}_{0,k}(X,A)^{\rm
vir}\cap(\alpha_{1}\times\cdots\times\alpha_{k}),\
\forall\alpha_{1},\ldots,\alpha_{k}\in H^{*}(X;{\mathbb{k}}).$

######  Remark 2.3.

  1. (1)

When ${\rm char}({\mathbb{k}})=p$, even when all $\alpha_{i}$ come from
integral cohomology, the above ${\mathbb{k}}$-valued GW invariants are not
necessarily the mod $p$ reductions of the corresponding ${\mathbb{Z}}$-valued
GW invariants, as $\alpha_{i}$ might be a $p$-torsion class in integral
cohomology. In this situation, the correlator over ${\mathbb{Z}}$ has to
vanish.

  2. (2)

When ${\mathbb{k}}={\mathbb{Q}}$, the above invariants do not necessarily
match the ordinary GW invariants. They do match when the symplectic manifold
$(X,\omega)$ is semi-positive.

##  3\. Seidel maps

In this section we define a version of Seidel representation in finite
characteristics. First recall the original construction. Given a loop of
Hamiltonian diffeomorphisms $\phi=(\phi_{t})_{t\in S^{1}}$ on a compact
symplectic manifold $(M,\omega_{M})$, one can construct a Hamiltonian
fibration

$\pi_{\phi}:\tilde{M}_{\phi}\to S^{2}$

with fibers being $M$ and clutching function along the equator being
$\phi_{t}$. Endowing $S^{2}$ with the standard complex structure $J_{S^{2}}$,
we can choose an almost complex structure $\tilde{J}_{\phi}$ on
$\tilde{M}_{\phi}$ such that the projection $\pi_{\phi}$ is pseudo-
holomorphic, the vertical tangent spaces are invariant under the action of
$\tilde{J}_{\phi}$, and the fiberwise almost complex structure is tamed by the
fiberwise symplectic form. One can consider the moduli space of
$\tilde{J}_{\phi}$-holomorphic graphs, namely sections

$\tilde{u}:S^{2}\to\tilde{M}_{\phi}$

such that

$\tilde{J}_{\phi}\circ d\tilde{u}=d\tilde{u}\circ J_{S^{2}}.$

Let $\pi_{2}^{\rm graph}(\tilde{M}_{\phi})$ be the set of homotopy classes of
sections of $\tilde{M}_{\phi}$, i.e., elements in $\pi_{2}(\tilde{M}_{\phi})$
whose composition with $(\pi_{\phi})_{*}$ represent the generator of
$\pi_{2}(S^{2})$. For each $\tilde{A}\in\pi_{2}^{\rm
graph}(\tilde{M}_{\phi})$, let

$\overline{\mathcal{M}}{}_{0,2}^{\rm graph}(\tilde{J}_{\phi},\tilde{A})$

be the moduli space of stable $\tilde{J}_{\phi}$-holomorphic graphs of class
$\tilde{A}$ with two marked points that are fixed to be
$z_{0}=0,z_{\infty}=\infty$ in the domain $S^{2}$. The way to compactify the
graph moduli space is to view the bubbles as holomorphic spheres in the
fibers. Then there are evaluation maps

${\rm ev}:={\rm ev}_{0}\times{\rm
ev}_{\infty}:\overline{\mathcal{M}}{}_{0,2}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A})\to\tilde{M}_{\phi}|_{z_{0}}\times\tilde{M}_{\phi}|_{z_{\infty}}\cong
M\times M.$

If we assume that $(M,\omega_{M})$ satisfies the $\mathbf{W}^{+}$ condition, a
condition slightly stronger than semipositivity (cf. [Sei97]), then one can
choose $\tilde{J}$ such that ${\rm ev}$ is a pseudocycle. Notice that the
isomorphisms

$H^{*}(\tilde{M}_{\phi}|_{z_{0}};{\mathbb{Z}})\cong H^{*}(M;{\mathbb{Z}})\cong
H^{*}(\tilde{M}_{\phi}|_{z_{\infty}};{\mathbb{Z}})$

are canonical. Then by using intersection numbers between transverse
pseudocycles, one can define the graph Gromov–Witten invariants

$\widetilde{\rm
GW}{}_{0,2}^{\tilde{M}_{\phi},\tilde{A}}(\alpha_{0},\alpha_{\infty})\in{\mathbb{Z}},\
\alpha_{0},\alpha_{\infty}\in H^{*}(M;{\mathbb{Z}})$

which is independent of the choice of almost complex structures and
pseudocycle representatives: see the original construction [Sei97].

If we drop Seidel’s ${\bf W}^{+}$ condition on $M$, one can construct a graph
virtual cycle with rational coefficients (cf. [McD00b]), leading to
${\mathbb{Q}}$-valued graph Gromov–Witten invariants. These GW invariants can
be made into a generating series which leads to invertible elements in quantum
cohomology of $M$, and they can be shown to define a group homomorphism

$\pi_{1}(\mathrm{Ham}(M,\omega_{M}))\to QH^{*}(M)^{\times},$

known as the Seidel representation.

In this paper, we will consider a variant of the graph moduli space. Suppose
${\bf z}=(z_{1},\ldots,z_{k})$ is a $k$-tuple of distinct points of
$S^{2}\setminus\\{0,\infty\\}$. One can consider the moduli space

${\mathcal{M}}{}_{0,2,{\bf z}}^{\rm graph}(\tilde{J}_{\phi},\tilde{A})$

of $\tilde{J}_{\phi}$-holomorphic graphs $u:S^{2}\to\tilde{M}_{\phi}$ with
fixed marked $k+2$ marked points, ${\bf z}$ together with $z_{0}=0$ and
$z_{\infty}=\infty$. Its compactification, denoted by
$\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A})$, is not exactly identical to
$\overline{\mathcal{M}}{}_{0,2}^{\rm graph}(\tilde{J}_{\phi},\tilde{A})$.
There is still the evaluation map

${\rm ev}:={\rm ev}_{0}\times{\rm
ev}_{\infty}:\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A})\to\tilde{M}_{\phi}|_{z_{0}}\times\tilde{M}_{\phi}|_{z_{\infty}}\cong
M\times M$

which pushes forward an integral virtual fundamental cycle that we will
construct shortly.

###  3.1. Global Kuranishi charts for graph moduli spaces

To define the Seidel representation for general symplectic fibers in arbitrary
field coefficient, we would like to adapt the Abouzaid–McLean–Smith
construction of global Kuranishi charts to the graph moduli spaces.

######  Proposition 3.1.

There is a derived orbifold chart
${\mathcal{C}}=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ with stable
complex structure and a submersion

$\tilde{{\rm ev}}:{\mathcal{U}}\to M_{0}\times M_{\infty}.$

such that $({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\tilde{{\rm ev}})$ is a
D-chart lift of $(\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A}),{\rm ev})$.

Our construction resembles that of Abouzaid–McLean–Smith [AMS23]. That is, we
first construct a global Kuranishi chart (defined below) and then reduce to a
derived orbifold chart.

######  Definition 3.2.

Let ${\mathcal{M}}$ be a topological space.

  1. (1)

A smooth (resp. topological) global Kuranishi chart for ${\mathcal{M}}$ is a
tuple

$K=(G,V,E,S,\Psi)$

where $G$ is a compact Lie group, $V$ is a smooth (resp. topological)
$G$-manifold with at most finite isotropy groups, $E\to V$ is a smooth (resp.
topological) $G$-equivariant vector bundle, $S:V\to E$ is a continuous
$G$-equivariant section, and $\Psi:S^{-1}(0)/G\to{\mathcal{M}}$ is a
homeomorphism.

  2. (2)

An almost complex structure on a smooth global Kuranishi chart
$K=(G,V,E,S,\Psi)$ consists of a $G$-invariant complex structure on the bundle
$TV\oplus{\mathfrak{g}}$ and a $G$-invariant complex structure on $E$.

  3. (3)

A singular global Kuranishi chart over ${\mathcal{M}}$ is a tuple $K$
satisfying the same conditions for topological global Kuranishi chart as above
except that $V$ is not required to be a topological manifold. $K$ is said to
be regular/smooth over a $G$-invariant open subset $V^{\prime}\subset V$ if
$V^{\prime}$ is a topological/smooth $G$-manifold. $K$ is said to be almost
complex over a $G$-invariant open subset $V^{\prime}\subset V$ if
$(G,V^{\prime},E|_{V^{\prime}},S|_{V^{\prime}},\Psi|_{V^{\prime}})$ is a
smooth global Kuranishi chart with an almost complex structure.

The notion of singular global Kuranishi charts will not be used until the
proof of the main theorem in the next section.

One can see that given a smooth global Kuranishi chart as above, the quotient

${\mathcal{C}}=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\psi):=(V/G,E/G,S/G,\Psi/G)$

is a derived orbifold chart. Moreover, if the given global Kuranishi chart has
an almost complex structure, the resulting derived orbifold chart is stably
complex.

####  3.1.1. Construction of global Kuranishi charts

We describe how to construct an Abouzaid–McLean–Smith style global Kuranishi
chart for the moduli space $\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A})$.

First, we need to modify the topological energy. Upon choosing a Hamiltonian
connection on $\tilde{M}_{\phi}$, one obtains a coupling form
$\tilde{\omega}_{\phi}\in\Omega^{2}(\tilde{M}_{\phi})$ which is closed and
whose fiberwise restriction is the symplectic form of $M$. The cohomology
class of $\tilde{\omega}_{\phi}$ in $\tilde{M}_{\phi}$ may not be rational. We
find a rational approximation, i.e., a closed 2-form
$\tilde{\Omega}_{\phi}\in\Omega^{2}(\tilde{M}_{\phi})$ which is sufficiently
close to $\tilde{\omega}_{\phi}$ and whose cohomology class is rational. One
can guarantee that the fiberwise restrictions of the almost complex structure
$\tilde{J}_{\phi}$ is still tamed by the fiberwise restrictions of
$\tilde{\Omega}_{\phi}$. Then after an integral rescaling, one can assume that
the cohomology class of $\tilde{\Omega}_{\phi}$ is integral.

Next, consider the moduli space of stable genus 0 holomorphic sections of the
trivial fibration $\mathbb{CP}^{d}\times\mathbb{CP}^{1}\to\mathbb{CP}^{1}$
whose projection to the $\mathbb{CP}^{d}$ factor represents $d$ times the
generator of the second homology, and we endow the parametrized domain with
$k+2$ fixed marked points, two of which are identified with $0$ and $\infty$,
and the other $k$ marked points are denoted by ${\bf z}=(z_{1},\ldots,z_{k})$,
being a $k$-tuple of distinct points of $S^{2}\setminus\\{0,\infty\\}$. This
moduli space is denoted by

$\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm graph}(\mathbb{CP}^{d},d).$

Let $\tilde{B}_{d}\subset\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\mathbb{CP}^{d},d)$ be the subset of curves whose images under the
projection to $\mathbb{CP}^{d}$ are not contained in any hyperplane of
$\mathbb{CP}^{d}$. Then $\tilde{B}_{d}$ is a smooth complex manifold with an
action by $G_{d}:=U(d+1)$ via projective linear transformations. There is also
a universal curve

$\tilde{C}_{d}\to\tilde{B}_{d}$

which is $G_{d}$-equivariant. For each $\rho\in\tilde{B}_{d}$, let
$C_{\rho}\subset\tilde{C}_{d}$ be the fiber, which is a prestable genus zero
curve with a parametrized componennt (called the principal component).
Consider smooth maps $u:C_{\rho}\to\tilde{M}_{\phi}$ such that after composing
with $\tilde{M}_{\phi}\to S^{2}$, it contracts all components except that it
identifies the principal components with the base $S^{2}$. To each such map
$u$ there is an associated section

$\overline{\partial}_{\tilde{J}_{\phi}}u\in\Omega^{0,1}(C_{\rho},u^{*}T^{\rm
vert}\tilde{M}_{\phi}).$

We follow [AMS23] to thicken up the moduli space. Let
$\tilde{C}_{d}^{*}\subset\tilde{C}_{d}$ be the complement of nodes and marked
points, which is a smooth $G_{d}$-manifold. A finite-dimensional approximation
scheme on $\tilde{C}_{d}$ is a finite-dimensional complex
$G_{d}$-representation $W$ and a $G_{d}$-equivariant linear map

$\iota:W\to
C^{\infty}_{c}\Big{(}\tilde{C}_{d}^{*}\times\tilde{M}_{\phi},\Omega^{0,1}_{\tilde{C}_{d}^{*}/\tilde{B}_{d}}\otimes
T^{\rm vert}\tilde{M}_{\phi}\Big{)}$

(which, on the contrary to [AMS23], we do not require that the image of
$\iota$ surjects onto the obstruction space). Then for each $e\in W$ and
$\rho\in\tilde{B}_{d}$, one obtains a section

$\iota(e)|_{C_{\rho}}\in
C^{\infty}_{c}(C_{\rho}^{*}\times\tilde{M}_{\phi},\Omega^{0,1}_{C_{\rho}}\otimes
T^{\rm vert}\tilde{M}_{\phi}).$

For any $u:C_{\rho}\to\tilde{M}_{\phi}$, the restriction to the graph of $u$

$\iota(e)(u)\in\Omega^{0,1}(C_{\rho},u^{*}T^{\rm vert}\tilde{M}_{\phi})$

lies in the same space of $\overline{\partial}_{\tilde{J}_{\phi}}(u)$.

Now we define the so-called pre-thickening. For
$d=d(\tilde{A})=\langle\tilde{\Omega}_{\phi},\tilde{A}\rangle\in{\mathbb{Z}}$,
consider triples $(\rho,u,e)$ where $\rho\in\tilde{B}_{d}$,
$u:C_{\rho}\to\tilde{M}_{\phi}$ is a smooth graph, and $e\in W$ satisfying 1)

$\overline{\partial}_{\tilde{J}_{\phi}}u+\iota(e)(u)=0,$

and 2) the homotopy class of the graph $u$ is $\tilde{A}$. Denote by
$\tilde{V}^{\rm pre}$ the moduli space of such triples (notice that there is
no nontrivial identifications of different triples as all domains are
specified as a fiber of $\tilde{C}_{d}$). There are natural maps

$\tilde{S}^{\rm pre}:\tilde{V}^{\rm pre}\to W,\ \tilde{S}^{\rm
pre}(\rho,u,e)=e$

and

$\tilde{\Psi}^{\rm pre}:(\tilde{S}^{\rm
pre})^{-1}(0)\to\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A}),\ (\rho,u,e)\to[C_{\rho},u].$

######  Lemma 3.3.

There exists a finite-dimensional approximation scheme $(W,\iota)$ such that
$\tilde{V}^{\rm pre}$ is a topological manifold near $(\tilde{S}^{\rm
pre})^{-1}(0)$ of dimension $j(\tilde{A})+{\rm dim}\tilde{B}_{d}+{\rm dim}W$
where

$j(\tilde{A})={\rm dim}M+2c_{1}(T^{\rm vert}\tilde{M}_{\phi})(\tilde{A})$

is the expected dimension of the moduli space of holomorphic graphs in the
class $\tilde{A}$.

###### Proof.

[AMS23, Lemma 4.2] guarantees the existence of finite-dimensional
approximation schemes which can eliminate all cokernels of the linearized
operators. ∎

Next we consider the analogue of framings on the graph moduli space. Any
smooth section $u:C_{\rho}\to\tilde{M}_{\phi}$ of homotopy class $\tilde{A}$
pulls back a 2-form $u^{*}\tilde{\Omega}_{\phi}\in\Omega^{2}(C_{\rho})$ whose
integral over each component is an integer. By a Hitchin–Kobayashi
correspondence on $C_{\rho}$, there exists a unique Hermitian holomorphic line
bundle (up to unitary isomorphism) on $C_{\rho}$, denoted by $L_{u}\to
C_{\rho}$ whose curvature form is $-2\pi{\bf i}u^{*}\tilde{\Omega}_{\phi}$.
The total degree of $L_{u}$ is $d=d(\tilde{A})$. A framing over $u$ is a basis

$F=(f_{0},\ldots,f_{d})$

of the space $H^{0}(L_{u})$. For each framing $F$, define a Hermitian matrix
$H(F)$ with entries

$H_{ij}(F):=\int_{C_{\rho}}\langle f_{i},f_{j}\rangle
u^{*}\tilde{\Omega}_{\phi}.$

Each framing (which must be base point free) also defines a holomorphic map

$\rho_{F}:C_{\rho}\to\mathbb{CP}^{1}\times\mathbb{CP}^{d},$

whose projection to the $\mathbb{CP}^{1}$-component contracts all components
except for mapping the principal component isomorphically to
$\mathbb{CP}^{1}$, and the projection to the $\mathbb{CP}^{d}$ component is
given by $\rho_{F}(z)=[f_{0}(z),\ldots,f_{d}(z)]$. In particular, $\rho_{F}$
defines an element in $\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\mathbb{CP}^{d},d)$.

Let $\tilde{V}$ be the space of quadruples $(\rho,u,e,F)$ where
$(\rho,u,e)\in\tilde{V}^{\rm pre}$ and $F$ is a framing over $u$. This is the
base of the AMS chart. Notice that $\tilde{V}\to\tilde{V}^{\rm pre}$ has the
structure of a $G_{d}$-equivariant principal
$G_{d}^{\mathbb{C}}=GL(d+1)$-bundle where $GL(d+1)$ transform the framing
linearly. Moreover

${\rm dim}\tilde{V}=j(\tilde{A})+{\rm dim}W+{\rm dim}\tilde{B}_{d}+2{\rm
dim}G_{d}.$

The obstruction bundle of the AMS chart is the direct sum

$\tilde{E}=W\oplus\pi_{\tilde{V}/\tilde{B}}^{*}T\tilde{B}_{d}\oplus{\mathfrak{g}}_{d}$

where $\pi_{\tilde{V}/\tilde{B}}:\tilde{V}\to\tilde{B}_{d}$ is the forgetful
map $(\rho,u,e,F)\mapsto\rho$, and $W$ and ${\mathfrak{g}}_{d}$ denote the
trivial bundle with fibers $W$ and the Lie algebra ${\mathfrak{g}}_{d}$
respectively. One can see

${\rm dim}\tilde{V}-{\rm rank}\tilde{E}-{\rm dim}G_{d}=j(\tilde{A})={\rm
dim}^{\rm vir}\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A}).$

To define the Kuranishi section $\tilde{S}:\tilde{V}\to\tilde{E}$, we choose a
$G_{d}$-invariant Riemannian metric on $\tilde{B}_{d}$ so that the exponential
map $\exp_{\tilde{B}_{d}}$ identifies an open neighborhood
$\Delta^{+}(\tilde{B}_{d})$ of the diagonal
$\Delta(\tilde{B}_{d})\subset\tilde{B}_{d}\times\tilde{B}_{d}$ with a
neighborhood of the zero section of the tangent bundle $T\tilde{B}_{d}$. By
abuse of notations, shrink $\tilde{V}$ to the open subset of quadruples
$(\rho,u,e,F)$ such that $(\rho,\rho_{F})\in\Delta^{+}(\tilde{B}_{d})$ and
such that the Hermitian matrix $H_{F}$ is invertible. Denote this open subset
still by $\tilde{V}$. Then define

$\displaystyle\tilde{S}:\tilde{V}$ $\displaystyle\to\tilde{E}$
$\displaystyle(\rho,u,e,F)$
$\displaystyle\mapsto\Big{(}e,\exp_{\tilde{B}_{d}}^{-1}(\rho,\rho_{F}),\exp_{H}^{-1}(H(F))\Big{)}.$

Here $\exp_{H}$ is the exponential map of matrices; $\exp_{H}^{-1}(H(F))=0$ is
equivalent to that $F$ is a unitary basis. The AMS global chart for
$\overline{\mathcal{M}}{}_{0,2}^{\rm graph}(\tilde{M}_{\phi},\tilde{A})$ is
the tuple

$\tilde{K}=(G_{d},\tilde{V},\tilde{E},\tilde{S},\tilde{\psi})$ (3.1)

where $\tilde{\psi}$ is the natural map

$\tilde{\psi}:(\tilde{S})^{-1}(0)/G_{d}\to\overline{\mathcal{M}}{}_{0,2,{\bf
z}}^{\rm graph}(\tilde{J}_{\phi},\tilde{A}).$

######  Lemma 3.4.

The map $\tilde{\psi}$ is a homeomorphism.

###### Proof.

This is the same as the case of [AMS23]. We first prove that $\tilde{\psi}$ is
a bijection. Given any point of $\overline{\mathcal{M}}{}_{0,2}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A})$ represented by a stable graph
$u:C\to\tilde{M}_{\phi}$ where $C$ is a smooth or nodal genus zero curve with
a parametrized principal component, consider the pullback bundle $L_{u}\to C$.
Then as $\tilde{J}_{\phi}$ is tamed by $\tilde{\Omega}_{\phi}$, the pullback
form $u^{*}\tilde{\Omega}_{\phi}$ is positive on $C$. Consider any unitary
framing $F$ of $L_{u}$ with respect to the $L^{2}$ inner product induced from
$u^{*}\Omega_{\phi}$. Then $F$ induces a stable map
$\rho_{F}:C\to\mathbb{CP}^{1}\times\mathbb{CP}^{d}$ whose projection to the
second component is of degree $d=d(\tilde{A})$, hence represents a point of
$\tilde{B}_{d}$. One can then identify $u$ with a holomorphic graph
$u_{F}:C_{\rho_{F}}\to\tilde{M}_{\phi}$. Therefore,

$(\rho_{F},u_{F},0,F)\in\tilde{S}^{-1}(0)$

and is sent by $\tilde{\psi}$ to the point represented by the given $u$.
Therefore $\tilde{\psi}$ is surjective. To show that $\tilde{\psi}$ is
injective, suppose $\tilde{\psi}$ sends $(\rho_{1},u_{1},0,F_{1})$ and
$(\rho_{2},u_{2},0,F_{2})$ to the same point of the graph moduli. Then the two
stable graphs $u_{1}:C_{\rho_{1}}\to\tilde{M}_{\phi}$ and
$u_{2}:C_{\rho_{2}}\to\tilde{M}_{\phi}$ are equivalent. Therefore, there
exists a biholomorphic map $\varphi:C_{\rho_{1}}\to C_{\rho_{2}}$ which is the
identity on the principal component such that

$u_{1}=u_{2}\circ\varphi.$

Therefore,
$\varphi^{*}u_{2}^{*}\tilde{\Omega}_{\phi}=u_{1}^{*}\tilde{\Omega}_{\phi}$,
implying that $L_{u_{1}}=\varphi^{*}L_{u_{2}}$ and $\varphi^{*}F_{2}$ is a
unitary framing of $L_{u_{1}}$. Therefore, there is an element $g\in G_{d}$
such that $F_{1}=g\varphi^{*}F_{2}$. This implies that
$(\rho_{1},u_{1},0,F_{1})$ and $(\rho_{2},u_{2},0,F_{2})$ are on the same
$G_{d}$-orbit. Hence $\tilde{\psi}$ is injective. Lastly it is straightforward
to see that $\tilde{\psi}$ is continuous and a homeomorphism. ∎

We also notice that there is a well-defined $G_{d}$-invariant evaluation map

${\rm ev}=({\rm ev}_{0},{\rm ev}_{\infty}):\tilde{V}\to M_{0}\times
M_{\infty},{\rm ev}(\rho,u,e,F)=(u(0),u(\infty))$

which extends the evaluation map on the moduli space.

####  3.1.2. Stable smoothing

Next we need to put a smooth structure on the AMS Kuranishi chart. As the
smoothing procedure will be carefully engineered in the proof of the main
theorem (c.f. the proof of Proposition 5.4), we provide more details here,
despite the fact that for the situation treated in this section it is almost
identical to the original argument of Abouzaid–McLean–Smith [AMS21].

First we recall a few notions related to smoothing. Let $G$ be a compact Lie
group and $Y$ be a topological $G$-manifold. A $G$-smoothing on $Y$ is a
smooth structure on $Y$ such that the $G$-action is smooth. A stable
$G$-smoothing on $Y$ is a $G$-smoothing on the product $Y\times R$ where $R$
is a finite-dimensional representation of $G$ viewed as a $G$-manifold.

Any topological $G$-manifold $Y$ has its $G$-equivariant tangent microbundle
$T_{\mu}Y$ (see [Mil64]). A necessary condition for $Y$ to admit a
$G$-smoothing is the existence of a $G$-equivariant vector bunde lift of
$T_{\mu}Y$, i.e., a $G$-equivariant microbundle isomorphism

$F_{\mu}\to T_{\mu}Y$

where $F\to Y$ is a $G$-equivariant vector bundle and $F_{\mu}$ is the induced
microbundle. The crucial ingredient in the smoothing procedure adopted in
[AMS21] is the stable $G$-smoothing theorem of Lashof [Las79] which says that
once the tangent microbundle $Y$ admits a $G$-equivariant vector bundle lift,
then $Y$ admits a stable $G$-smoothing. In addition, there is a canonical one-
to-one correspondence between stable $G$-isotopy classes of $G$-equivariant
vector bundle lifts of $T_{\mu}Y$ and stable $G$-isotopy classes of stable
$G$-smoothings of $Y$.

The technical result regarding smoothing the AMS chart is stated below.

######  Lemma 3.5.

There exists a stable $G_{d}$-smoothing on $\tilde{V}$, or equivalently, a
$G_{d}$-smoothing on a product $\tilde{V}\times R$ where $R$ is a
$G_{d}$-representation.

One can use the trivial bundle $\underline{\smash{R}}\to\tilde{V}$ to
stabilize the Kuranishi chart so the base of the new chart is smooth. On the
other hand, equivariant continuous vector bundles are always isomorphic to
equivariant smooth vector bundles. Hence the stabilized Kuranishi chart by
$\underline{\smash{R}}$ is smooth.

###### Proof of Lemma 3.5.

The proof is essentially the same as the argument of [AMS21, Section 4]. The
space $\tilde{V}$ is a topological $G_{d}$-manifold. The projection

$\pi_{\tilde{V}/\tilde{B}}:\tilde{V}\to\tilde{B}_{d}$ (3.2)

is a topological submersion (see [AMS21, Definition 4.18]) whose fibers are
canonically smooth manifolds. Moreover, the smooth structures on fibers vary
sufficiently regularly (see [AMS21, Section 4.5] and [AMS21, Corollary 6.29])
so that there is a vertical tangent bundle

$T^{\rm vert}\tilde{V}\to\tilde{V}$

whose restrictions to fibers of $\pi_{\tilde{V}/\tilde{B}}$ are the tangent
bundles of the fibers. Moreover one can construct a $G_{d}$-equivariant vector
bundle lift

$(T^{\rm
vert}\tilde{V}\oplus\pi_{\tilde{V}/\tilde{B}}^{*}T\tilde{B}_{d})_{\mu}\to
T_{\mu}\tilde{V}$ (3.3)

(see [AMS21, Corollary 4.26]). Hence by the theorem of Lashof [Las79], there
is a representation $R$ and a $G_{d}$-smoothing on $\tilde{V}\times R$. ∎

######  Remark 3.6.

We also remark that although the $G_{d}$-equivariant vector bundle lift is
manually constructed, the stable $G_{d}$-isotopy class of the
$G_{d}$-smoothing is canonical. In other words, up to a stabilization by a
$G_{d}$-representation, any two stable smoothings of $\tilde{V}$ built from
the vector bundle lift (3.3) are equivariantly isotopic.

###### Proof of Proposition 3.1.

By Lemma 3.4 and Lemma 3.5, after a stabilization and taking the quotient by
$G_{d}$, the global Kuranishi chart (3.1) gives rise to a derived orbifold
chart of the moduli space $\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{M}_{\phi},\tilde{A})$. As explained in [AMS23, Corollary 4.31],
we see that this derived orbifold chart is stably complex. Finally, using
[AMS21, Lemma 4.5], possibly after a further stabilization, we can make sure
that the extension of the evaluation map to the ambient orbifold of the
derived orbifold chart is a smooth submersion. ∎

####  3.1.3. Invariance

We briefly comment on the independence of the chart
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\tilde{{\rm ev}})$ from
Proposition 3.1 on the choices made in the construction.

The tuple $\tilde{K}$ from (3.1) defines a global Kuranishi chart of
$\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{M}_{\phi},\tilde{A})$. It depends on the following auxiliary
data:

  1. (1)

the integral $2$-form $\tilde{\Omega}_{\phi}$;

  2. (2)

the finite-dimensional approximation scheme $(W,\iota)$;

  3. (3)

the $G_{d}$-invariant Riemannian metric on $\tilde{B}_{d}$ in the definition
of $\tilde{S}$.

To construct the stable smoothing, one needs to choose:

  1. (1)

a vector bundle lift (3.3);

  2. (2)

a $G_{d}$-representation $R$ such that $\tilde{V}\times R$ admits a
$G_{d}$-equivariant smooth structure;

  3. (3)

an identification of the stabilized vector bundle $E$ with a smooth
$G_{d}$-equivariant vector bundle.

By a now-standard doubly framed curve argument (cf. [AMS21, Section 6.10] and
[AMS23, Section 4.9]) and the independence properties of the
$G_{d}$-equivariant smoothing discussed in Remark 3.6, we see that for
different choices, the resulting derived orbifold charts
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}})$ are equivalent. Furthermore,
using [AMS21, Section 6.11], we conclude that the derived orbifold bordism
class over $M\times M$ represented by
$({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\tilde{{\rm ev}})$ is independent
of $\tilde{J}_{\phi}$ and all the data above.

###  3.2. The Seidel map via FOP transverse perturbations

Now we can use the FOP perturbation scheme to construct the Seidel map. For
the graph moduli $\overline{\mathcal{M}}{}_{0,2,{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A})$, let

$\tilde{\mathcal{C}}=(\tilde{\mathcal{U}},\tilde{\mathcal{E}},\tilde{\mathcal{S}},\tilde{\psi})$

be the derived orbifold chart obtained in Proposition 3.1. By choosing an FOP
transverse perturbation of $\tilde{\mathcal{C}}$, via the evaluation map, one
obtains a well-defined homology class in the product $M_{0}\times M_{\infty}$
(with coefficient in the field ${\mathbb{k}}$). By pairing with cohomology
classes one obtains the graph GW invariant

$\widetilde{\rm GW}{}{}_{0,2,{\bf
z}}^{\tilde{M}_{\phi},\tilde{A}}(\alpha_{0},\alpha_{\infty})\in{\mathbb{k}},\
\alpha_{0},\alpha_{\infty}\in H^{*}(M;{\mathbb{k}}).$

We write the Seidel map as an expansion in the Novikov variable. Let the
Novikov field be

$\Lambda=\Big{\\{}\sum_{i=1}^{\infty}a_{i}q^{\lambda_{i}}\ |\
a_{i}\in{\mathbb{k}},\ \lim_{i\to\infty}\lambda_{i}=+\infty\Big{\\}}.$

Let $\tilde{\Omega}_{\phi}\in\Omega^{2}(\tilde{M}_{\phi})$ be the coupling
form associated to the Hamiltonian fibration $\tilde{M}_{\phi}$, whose
cohomology class does not depend on a Hamiltonian connection. Then define a
linear map

$S_{\bf z}(\phi):H^{*}(M;\Lambda)\to H^{*}(M;\Lambda)$

by

$S_{\bf z}(\phi)(\alpha)=\sum_{\tilde{A}\in\pi_{2}^{\rm
graph}(\tilde{M}_{\phi})}q^{\tilde{\Omega}_{\phi}(\tilde{A})}\sum_{\beta,\gamma}\widetilde{\rm
GW}{}_{0,2,{\bf
z}}^{\tilde{M}_{\phi},\tilde{A}}(\alpha,e_{\beta})g^{\beta\gamma}e_{\gamma}$

where $\\{e_{\beta}\\}$ is a basis of $H^{*}(M;{\mathbb{k}})$ and
$g^{\beta\gamma}$ is the inverse of the intersection matrix
$g_{\beta\gamma}=\langle e_{\beta},e_{\gamma}\rangle$.

We expect that the collection of linear transformations $S(\phi)$ (the version
with $\mathbf{z}$ being empty) defines a representation of $\pi_{1}({\rm
Ham}(M,\omega_{M}))$ into the invertibles of quantum cohomology defined by the
FOP perturbaion scheme. However the proof of this statement (which is
essentially equivalent to the associativity of quantum cup product) is beyond
the current understanding of the FOP perturbation method. For the purpose of
this paper we only prove the invertibility of each $S_{\bf z}(\phi)$.

######  Theorem 3.7.

The map $S_{\bf z}(\phi)$ only depends on the homotopy class of $\phi$ and the
number of fixed marked points. Therefore, we can denote it by $S_{k}(\phi)$.

###### Proof.

Except for the discussion in Section 3.1.3, we first need to remove the
dependence on ${\bf z}$. Indeed, any two distinct tuples ${\bf z}$ and ${\bf
z}^{\prime}$ can be connected by a path. One can easily construct a cobordism
between the corresponding derived orbifold charts. On the other hand, a
homotopy of $\phi$ can be extended to a homotopy of geometric data (almost
complex structures etc.) and a cobordism of the derived orbifold charts of the
moduli spaces. As the FOP pseudocycle is cobordism invariant, the resulting
map on cohomology is invariant under homotopy of $\phi$. ∎

######  Theorem 3.8.

The map $S_{k}(\phi)$ is invertible for all $\phi:S^{1}\to{\rm
Ham}(M,\omega_{M})$.

###### Proof.

See Subsection 3.3. ∎

Accordingly, we immediately have

######  Corollary 3.9.

For each $\phi:S^{1}\to{\rm Ham}(M,\omega_{M})$, the bilinear pairing

$(\beta_{0},\beta_{\infty})\mapsto\langle\beta_{0},\beta_{\infty}\rangle_{k,\phi}:=\sum_{\tilde{A}\in\pi_{2}^{\rm
graph}(\tilde{M}_{\phi})}q^{\tilde{\Omega}_{\phi}(\tilde{A})}\widetilde{\rm
GW}{}_{0,2,{\bf
z}}^{\tilde{M}_{\phi},\tilde{A}}(\beta_{0},\beta_{\infty})\in\Lambda$

is nondegenerate. ∎

###  3.3. Proof of Theorem 3.8

We first prove that $S_{k}(1)$ is invertible, where $k\geq 0$ is any
nonnegative integer and $1$ is the constant loop in ${\rm Ham}(M,\omega_{M})$
at the identity Hamiltonian diffeomorphism. The corresponding fibration
$\tilde{M}_{1}$ is the trivial product $M\times S^{2}$. The moduli space of
constant pseudo-holomorphic graphs is cut out transversely, and its induced
map on cohomology is the identity. So, the derived orbifold chart constructed
in Proposition 3.1 is a stabilization of a manifold, thus the FOP pseudocycle
agrees with the pushforward of the fundamental class of the moduli space under
the evaluation map. Hence $S_{k}(1)$ is the identity map plus a higher order
term in terms of the valuation on the Nokikov field $\Lambda$. We do not know
how to prove the vanishing of the higher order term which counts parametrized
holomorphic spheres in $M$. Nonetheless, it follows that $S_{k}(1)$ is
invertible.

####  3.3.1. Floer and Morse flow categories

The difficulty of proving the expected property of the Seidel map

$S_{k}(\phi)\circ S_{l}(\psi)=S_{k+l}(\phi\\#\psi)$

where $\phi\\#\psi$ is the product loop in ${\rm Ham}(M,\omega_{M})$ is that
it is a nontrivial task to identify the contribution of nodal configurations
using the FOP perturbation scheme. To bypass this difficulty, we use Floer and
Morse models; the corresponding moduli spaces can be viewed as certain real
blowups along the codimension two divisors defined by nodal curves.

Introduce the discrete monoid

$\Pi=\\{\omega_{M}(A)\ |\ A\in\pi_{2}(M)\\}\subset{\mathbb{R}}.$

Fix a nondegenerate 1-periodic family of Hamiltonian $H_{t}$ (which is a datum
independent of the Hamiltonian loop $\phi_{t}$). The critical point of the
action functional ${\mathcal{A}}_{H_{t}}$ are equivalence classes of capped
1-periodic orbits, where two cappings are regarded equivalent if their
difference has vanishing symplectic area. Fix an $S^{1}$-family of
$\omega_{M}$-compatible almost complex structures $J_{t}$. The Hamiltonian
Floer flow category of the pair $(H_{t},J_{t})$, denoted by ${\mathbb{F}}^{\rm
Floer}$, has objects being equivalence classes of capped 1-periodic orbits,
while for any pair of objects $(p,q)$, the morphism space is the compactified
moduli space $\overline{\mathcal{M}}{}_{pq}^{\rm Floer}$ of solutions to the
Floer equation for $(H_{t},J_{t})$ connecting $p$ and $q$. There is a natural
free $\Pi$-action on this flow category.

On the other hand, fix a Morse–Smale pair $(f,g)$ on $M$. The Morse flow
category, denoted by $\mathbb{F}^{\rm Morse}$, has objects being pairs $(a,p)$
where $a\in\Pi$ and $p\in{\rm crit}f$, while for any pair $((a,p),(b,q))$, the
morphism space is the moduli space $\overline{\mathcal{M}}{}_{pq}^{\rm Morse}$
of possibly broken unparametrized Morse trajectories connecting $p$ and $q$.
One can choose the pair $(f,g)$ such that each morphism space is an oriented
smooth compact manifold-with-corners such that the composition maps are smooth
embeddings compatible with the orientations (see [Weh12]).

The Morse flow category naturally induces a chain complex (over any
coefficient ring) with differential defined by counting $0$-dimensional
morphism spaces with signs. Let $\Lambda$ be the Novikov field with ground
field ${\mathbb{k}}$. Then the chain complex derived from $\mathbb{F}^{\rm
Morse}$ coincides with

$CM_{*}(f,g)\otimes\Lambda.$

The homology is equal to $H_{*}(M;\Lambda)$. For the purpose of this paper, we
do not need to consider grading.

On the other hand, to define the Floer chain complex from the Floer flow
category $\mathbb{F}^{\rm Floer}$, one must regularize the moduli spaces. In
[BX22a, BX24] the first and third author constructed a class of so-called
Kuranishi lifts of the Floer flow category $\mathbb{F}^{\rm Floer}$ which
admits stable complex structures333In fact there is one technical step called
“outer-collaring” which modifies the original Floer flow category by extending
the boundary and corner strata outward. However, it is not a crucial point for
the constructions in this paper.. By using the FOP perturbation scheme, one
obtains a Floer complex (over any coefficient field ${\mathbb{k}}$), denoted
by

$CF_{*}(H_{t},J_{t};\Lambda).$

The resulting homology, $HF_{*}(H_{t},J_{t})$, as a finite-dimensional
$\Lambda$-vector space, is independent of $(H_{t},J_{t})$. A crucial fact we
need in this paper, proved in the upcoming [BX24], is the following.

######  Theorem 3.10.

[BX24] When ${\mathbb{k}}$ is a field, ${\rm
dim}_{\Lambda}HF_{*}(M;\Lambda)={\rm rank}H_{*}(M;\mathbb{k})$.

Before we define various alternate versions of the Seidel map, we need to
recall the general notion of flow bimodules. If ${\mathbb{F}}$ and
${\mathbb{G}}$ are two flow categories, then a bimodule ${\mathbb{M}}$ over
$({\mathbb{F}},{\mathbb{G}})$ consists of, for each pair $p\in{\rm
Ob}{\mathbb{F}}$ and $q\in{\rm Ob}{\mathbb{G}}$, a stratified space
$M_{pq}^{\mathbb{M}}$ whose boundary and corner strata, roughly speaking, come
from degenerations of morphism spaces of ${\mathbb{F}}$ and ${\mathbb{G}}$.
For example, the union of codimension 1 strata of $M_{pq}^{\mathbb{M}}$ is the
union of products

$\bigcup_{p^{\prime}\in{\rm Ob}{\mathbb{F}}}M^{\mathbb{F}}_{pp^{\prime}}\times
M_{p^{\prime}q}^{\mathbb{M}}\cup\bigcup_{q^{\prime}\in{\rm
Ob}{\mathbb{G}}}M^{\mathbb{M}}_{pq^{\prime}}\times
M_{q^{\prime}q}^{\mathbb{G}}.$

Examples of flow bimodules are moduli spaces of Floer continuation maps or
Piunikhin–Salamon–Schwarz (PSS) maps.

Flow bimodules can be composed (see the definition in [BX24] and the
discussion in the abstract setting [AB24, Section 4]). For example, the
composition of two flow bimodules corresponding to two continuation maps is
the bimodule whose moduli spaces are those of “two-stage” continuations (whose
top strata has one breaking). There is also a notion of homotopy between
bimodules. For example, the above-mentioned composed bimodule corresponding to
concatenations of two continuation maps is homotopic to the bimodule
corresponding to a single continuation map with data coming from “gluing.”

As one expects, a flow bimodule should induce a chain map. In [BX22b], we
defined a chain map induced from the PSS construction

$\Psi^{\rm PSS}:CM_{*}(f,g)\otimes\Lambda\to CF_{*}(H_{t},J_{t};\Lambda)$

and a chain map in the opposite direction (which we call the SSP construction)

$\Psi^{\rm SSP}:CF_{*}(H_{t},J_{t};\Lambda)\to CM_{*}(f,g)\otimes\Lambda.$

In addition, we proved that (cf. [BX22b, Theorem C]) the composition

$\Psi^{\rm SSP}\circ\Psi^{\rm PSS}:CM_{*}(f,g)\otimes{\mathbb{k}}\to
CM_{*}(f,g)\otimes\Lambda$

is invertible (but not for the other composition!). Indeed, this composition
is basically equivalent to the Seidel map $S(1)$. It follows from Theorem 3.10
that on the homology level, for field coefficients, the PSS and SSP maps are
isomorphisms of $\Lambda$-vector spaces. More generally, in the forthcoming
[BX24], we constructed a chain map induced from a continuation map (which we
do not need here) between two choices of Floer data $(H_{t},J_{t})$ and
$(H_{t}^{\prime},J_{t}^{\prime})$.444This result already follows from the
exisiting construction in [AB24, Appendix A] because the FOP perturbation
scheme can be applied to the regularizations of flow bimodule constructed from
the moduli spaces of continuation maps discussed in _loc.cit._ , which induces
the desired chain maps between Floer complexes defined by different data.

####  3.3.2. Alternate Seidel maps

Now consider a bimodule over $(\mathbb{F}^{\rm Morse},{\mathbb{F}}^{\rm
Morse})$, denoted by $\mathbb{M}^{\rm Morse}_{\bf z}(\phi)$, where ${\bf z}$
is a $k$-tuple of distinct marked points on $S^{2}\setminus\\{0,\infty\\}$.
First, for each class $\tilde{A}\in\pi_{2}^{\rm graph}(\tilde{M}_{\phi})$ and
each pair of critical points $y_{0},y_{\infty}$ of the Morse function
$f:M\to{\mathbb{R}}$, consider the moduli space of pearly object, consisting
of a downward gradient ray $\xi_{0}:(-\infty,0]\to M$ starting from $y_{0}$, a
$\tilde{J}_{\phi}$-holomorphic graph $u:S^{2}\to\tilde{M}_{\phi}$ of class
$\tilde{A}$, and a downward gradient ray $\xi_{\infty}:[0,+\infty)\to M$
ending at $y_{\infty}$, such that

$\xi_{0}(0)=u(z_{0}),\ \xi_{\infty}(0)=u(z_{\infty})$

(for which we fix identifications $M_{0}\cong M\cong M_{\infty}$). One can
compactify this moduli space by allowing breaking of gradient rays and
bubbling of holomorphic spheres in the fibers of $\tilde{M}_{\phi}$, while
keeping the fixed marking ${\bf z}$. Denote the moduli space by

$\overline{\mathcal{M}}{}_{y_{0}y_{\infty},{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A}).$

Then to each pair $((a,p),(b,q))$ of objects of $\mathbb{F}^{\rm Morse}$,
define

$M_{(a,p)\ (b,q)\ {\bf z}}^{\rm Morse}(\phi)$

to be the union of $\overline{\mathcal{M}}{}_{pq,{\bf z}}^{\rm
graph}(\tilde{J}_{\phi},\tilde{A})$ for all $\tilde{A}\in\pi_{2}^{\rm
graph}(\tilde{M}_{\phi})$ with $\tilde{\Omega}_{\phi}(\tilde{A})=b-a$. It is
straightforward to see that this provides a bimodule over $(\mathbb{F}^{\rm
Morse},\mathbb{F}^{\rm Morse})$, denoted by $\mathbb{M}_{\bf z}^{\rm
Morse}(\phi)$.

To define a chain map, notice that one has already constructed a derived
orbifold chart for each degree $\tilde{A}$ for the graph moduli space. One can
choose a FOP transverse perturbation for each such derived orbifold chart such
that the evaluation maps restricted to the zero locus (in the free part) is
transverse to (un)stable manifolds of critical point. Hence one obtains (well-
defined) counts

$n_{pq}(\tilde{A})\in{\mathbb{Z}}.$

Sending these integers to the field ${\mathbb{k}}$ and weight them by
$q^{\tilde{\Omega}_{\phi}(\tilde{A})}$, one obtains a chain map

$S_{\bf z}^{\rm Morse}(\phi):CM_{*}(f,g)\otimes\Lambda\to
CM_{*}(f,g)\otimes\Lambda.$

Denote the induced map on homology still by

$S_{\bf z}^{\rm Morse}(\phi):H_{*}(M;\Lambda)\to H_{*}(M;\Lambda).$

######  Lemma 3.11.

On homology level, $S_{\bf z}^{\rm Morse}(\phi)$ is independent of ${\bf z}$
thus only depends on the number of markings $k$. Hence we can denote it by
$S^{\rm Morse}_{k}(\phi)$. Moreover, $S_{k}^{\rm Morse}(\phi)={\rm PD}\circ
S_{k}(\phi)\circ{\rm PD}$.

###### Proof.

The independence on ${\bf z}$ is similar to the proof of Theorem 3.7. For
different choices with the same number of markings, the induced flow bimodule
are homotopic to each other. Therefore, the induced maps on homology agree.
Next we compare the Morse-theoretic Seidel map and the Seidel map constructed
in the previous subsection. In integer coefficients, for any Morse cycle, the
(weighted) union of unstable manifolds defines a pseudocycle (see [Sch99]).
The same argument can be applied to the case of $\mathbb{F}_{p}$ coefficients.
On the other hand, the pairing between homology and cohomology with
${\mathbb{F}}_{p}$ coefficients can be identified with the count of transverse
intersection points between representing $p$-pseudocycles. Hence the identity
holds for $\mathbb{k}=\mathbb{F}_{p}$. For general finite field $\mathbb{k}$
this follows from the universal coefficient theorem. ∎

Now we define a bimodule $\mathbb{M}_{\bf z}^{\rm PSS}(\phi)$ over
$(\mathbb{F}^{\rm Morse},\mathbb{F}^{\rm Floer})$, and another bimodule
$\mathbb{M}_{\bf z}^{\rm SSP}(\phi)$ over $(\mathbb{F}^{\rm
Floer},\mathbb{F}^{\rm Morse})$. When $\phi=1$, they are the same as the PSS
and SSP bimodules considered in [BX22a, BX24]. For a general loop
$\phi:S^{1}\to{\rm Ham}(M,\omega_{M})$, the defining equation (over the
infinite cylinder) just includes the twisting by the loop $\phi$ and the
moduli space only has the additional ingredients of the fixed marking ${\bf
z}$ on the cylinders. Therefore, the construction of an associated Kuranishi
chart lift and the chain map is completely analogous to the special case when
$\phi=1$. Therefore, one obtains linear maps between homologies

$S_{\bf z}^{\rm PSS}(\phi):H_{*}(M;\Lambda)\to HF_{*}(M;\Lambda),\ S_{\bf
z}^{\rm SSP}(\phi):HF_{*}(M;\Lambda)\to H_{*}(M;\Lambda).$

######  Proposition 3.12.

On the homology level, the maps $S_{\bf z}^{\rm PSS}(\phi)$ and $S_{\bf
z}^{\rm SSP}(\phi)$ do not depend on ${\bf z}$ but only the number of markings
$k$. Hence we can denote them by $S_{k}^{\rm PSS}(\phi)$ and $S_{k}^{\rm
SSP}(\phi)$. Moreover,

$S_{k}^{\rm SSP}(\psi)\circ S_{l}^{\rm PSS}(\phi)=S_{k+l}^{\rm
Morse}(\phi\\#\psi),\ \forall\phi,\psi:S^{1}\to{\rm Ham}(M,\omega_{M}).$

###### Proof.

The left hand side can be viewed as the numerical output of composing two flow
bimodules and the equality can be viewed as the homological consequence of a
homotopy of bimodules. The proof, which relies on constructing Kuranishi lifts
and FOP perturbations, is completely analogous to the case of [BX22a] which
proves the equality for $\phi=\psi=1$. ∎

######  Proposition 3.13.

For any $\phi:S^{1}\to{\rm Ham}(M,\omega_{M})$ and $k\geq 0$, the maps
$S_{k}^{\rm PSS}(\phi):H_{*}(M;\Lambda)\to HF_{*}(M;\Lambda)$ and $S_{k}^{\rm
SSP}(\phi):HF_{*}(M;\Lambda)\to H_{*}(M;\Lambda)$ are both invertible.

###### Proof.

Proposition 3.12 implies $S_{k}^{\rm SSP}(\phi^{-1})\circ S^{\rm
PSS}_{l}(\phi)=S_{k+l}(1)$ which is invertible. It implies that the PSS maps
are injective and the SSP maps are surjective. By Theorem 3.10, when the
ground ring $R$ is a field, as $\Lambda$-vector spaces, $HF^{*}(M;\Lambda)$ is
isomorphic to $H^{*}(M;\Lambda)$. Hence the PSS and the SSP maps are both
invertible. ∎

###### Proof of Theorem 3.8.

By composing $S_{k}^{\rm SSP}(\phi)$ with $S^{\rm PSS}(1)$, it follows from
Proposition 3.12 again that $S_{k}^{\rm Morse}(\phi)$ is invertible.
Therefore, by Lemma 3.11, we conclude that $S_{k}^{\rm Morse}(\phi)$ is
invertible. ∎

##  4\. Cohomological splitting

We prove the main theorems of this paper assuming the follwoing technical
statement, whose proof is deferred to Section 5.

######  Theorem 4.1.

Let $(B,\omega_{B})$ be a compact symplectic manifold with a nonzero
Gromov–Witten invariant

${\rm GW}_{0,k+2}^{B,A}([{\rm pt}],[{\rm pt}],a_{1},\ldots,a_{k})\neq 0$

for $a_{1},\ldots,a_{k}\in H^{*}(B;{\mathbb{Z}})$ and a curve class $A\in
H_{2}(B;{\mathbb{Z}})$. Moreover, suppose there exist a compatible almost
complex structure $J_{B}$, points $p_{0},p_{\infty}\in B$, and pseudocycle
representatives

$f_{i}:W_{i}\to B$

of the homology classes ${\rm PD}_{B}(a_{i})$ satisfying the following
conditions.

  1. (1)

Inside the set of simple curves ${\mathcal{M}}{}_{0,k+2}^{\rm
simple}(J_{B},A)\subset\overline{\mathcal{M}}{}_{0,k+2}(J_{B},A)$ there is an
open subset ${\mathcal{M}}_{0,k+2}^{\rm reg}(J_{B},A)$ which is cut out
transversely.

  2. (2)

There holds

${\rm
ev}\Big{(}\overline{\mathcal{M}}{}_{0,k+2}(J_{B},A)\Big{)}\cap\Big{(}\\{p_{0}\\}\times\\{p_{\infty}\\}\times\prod_{i=1}^{k}\overline{f_{i}(W_{i})}\Big{)}\\\
={\rm ev}\Big{(}{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{B},A)\Big{)}\cap\Big{(}\\{p_{0}\\}\times\\{p_{\infty}\\}\times\prod_{i=1}^{k}f_{i}(W_{i})\Big{)}$
(4.1)

and the intersection on the right hand side is transverse.

In particular, the Gromov–Witten invariant ${\rm GW}_{0,k+2}^{B,A}([{\rm
pt}],[{\rm pt}],a_{1},\ldots,a_{k})$ is an integer. Then for any Hamiltonian
fibration $P\to B$ with fiber $(M,\omega_{M})$ and any coefficient field
${\mathbb{k}}$ whose characteristic does not divide this Gromov–Witten
invariant, there is an isomorphism of graded $\mathbb{k}$-vector spaces

$H^{*}(P;{\mathbb{k}})\cong
H^{*}(B;{\mathbb{k}})\otimes_{\mathbb{k}}H^{*}(M;{\mathbb{k}}).$

###### Proof of Theorem A.

If $(B,\omega_{B})$ is monotone, it is standard that for a generic compatible
almost complex structure $J_{B}$, the moduli space
${\mathcal{M}}{}_{0,k+2}^{\rm simple}(J_{B},A)$ is regular, the evaluation map
restricted to ${\mathcal{M}}{}_{0,k+2}^{\rm simple}(J_{B},A)$ is a
pseudocycle, and the Gromov–Witten invariants are defined via counting
transverse intersections of pseudocycles (see [MS04, Chapter 6]). Thus Theorem
A follows from Theorem 4.1. ∎

###  4.1. Proof of Theorem B

Let $B$ be a smooth and stably rational projective variety of complex
dimension $n$, meaning that $B\times\mathbb{CP}^{r}$ is birational to
$\mathbb{CP}^{n+r}$. We first notice that by Lalonde–McDuff’s “surjection
lemma” ([LM03, Lemma 4.1]), it suffices to prove for the case that $B$ is
rational. In this case, there is a rational map
$\mathbb{CP}^{n}\dashedrightarrow B$. By Hironaka’s theorem on resolution of
indeterminacies (see [Hir64, Main Theorem II]), there is another smooth
projective variety $\tilde{B}$ fitting in a commutative diagram

$\textstyle{\tilde{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\textstyle{\mathbb{CP}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$

where $\pi_{1}$ is the compositions of blowups with smooth centers and
$\pi_{2}:\tilde{B}\to B$ is a genuine (birational) morphism. Then $\pi_{1}$
identifies a Zariski open set of $\tilde{B}$ with $\mathbb{CP}^{n}\setminus Z$
where $Z$ is the union of subvarieties of complex codimension at least two.
Let $A\in\pi_{2}(\mathbb{CP}^{n})$ be the line class. As a general line avoids
$Z$, there is a natural lift $A\in\pi_{2}(\tilde{B})$.

######  Lemma 4.2.

Let $\overline{\mathcal{M}}{}_{0,2}(J_{\tilde{B}},A)$ be the moduli space of
genus zero $2$-marked stable maps in class $A$ and ${\rm
ev}:\overline{\mathcal{M}}{}_{0,2}(J_{\tilde{B}},A)\to\tilde{B}^{2}$ be the
evaluation map.

  1. (1)

There is an open subset ${\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{\tilde{B}},A)\subset{\mathcal{M}}{}_{0,2}^{\rm
simple}(J_{\tilde{B}},A)$ which is transverse.

  2. (2)

For a general pair of points $(p,q)\in\tilde{B}$, the preimage ${\rm
ev}^{-1}((p,q))$ is a single point contained in ${\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{\tilde{B}},A)$ and the intersection is transverse.

  3. (3)

${\rm GW}_{0,2}^{\tilde{B},A}([{\rm pt}],[{\rm pt}])=1$.

###### Proof.

This statement is an easy extension of [LM03, Proposition 4.15]. For any pair
of points $(p,q)\in\tilde{B}\times\tilde{B}$, there is a unique line
connecting $\pi_{1}(p)$ and $\pi_{1}(q)$ in $\mathbb{CP}^{n}$ which underlies
a regular stable map from $\mathbb{P}^{1}$. As $\tilde{B}$ and
$\mathbb{CP}^{n}$ are isomorphic over a Zariski open set, when $(p,q)$ is
general, the line is unique and avoids the subset $Z\subset\mathbb{CP}^{n}$,
hence corresponds to a curve in $\tilde{B}$ in class $A$. This provides the
regular open subset ${\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{\tilde{B}},A)\subset{\mathcal{M}}{}_{0,2}^{\rm
simple}(J_{\tilde{B}},A)$. For a fixed general $(p,q)$ this curve is the only
one in $\overline{\mathcal{M}}{}_{0,2}(J_{\tilde{B}},A)$ passing through $p$
and $q$. ∎

######  Corollary 4.3.

For any Hamiltonian fibration $\tilde{P}\to\tilde{B}$ and any coefficient
field ${\mathbb{k}}$, the cohomology $H^{*}(\tilde{P};{\mathbb{k}})$ splits.

###### Proof.

It follows from Lemma 4.2 that we are in a special situation of Theorem 4.1.
As the GW invariant is 1, one concludes the cohomological splitting for any
Hamiltonian fibration over $\tilde{B}$ in all characteristics. ∎

###### Proof of Theorem B.

Now we derive the cohomological splitting property of $B$ from that of
$\tilde{B}$. Remember that there is a birational morphism
$\pi_{2}:\tilde{B}\to B$. Then by [BP24, Lemma 2.6], because any Hamiltonian
fibration over $\tilde{B}$ satisfies cohomological splitting over any field
based on Corollary 4.3, it also holds for $B$. ∎

######  Remark 4.4.

The main theorem of [GHS03] asserts that for a dominant morphism of varieties
$P\to B$, if both $B$ and the general fiber are rationally connected, then $P$
is also rationally connected. On the other hand, the topological arguments in
[LM03] show that for a fiber bundle $M\hookrightarrow P\to B$, if every
Hamiltonian fibration over $B$ and $M$ satisfies cohomological splitting, then
the same holds for $P$. This showcases further potential relation between
rational connectedness and cohomological splitting.

##  5\. Proof of Theorem 4.1

Now we start to prove the main technical result of this paper. First we need
to make a notational modifications related to the topological type of rational
curves. The Hurewicz map

$\pi_{2}(B)\to H_{2}(B;{\mathbb{Z}})$

may not be injective. Therefore, a priori a Gromov–Witten invariant of $B$
with a fixed homological degree may come from different homotopy classes of
holomorphic spheres. It is easy to see that for each homotopy class of
holomorphic sheres, the corresponding Gromov–Witten invariants are well-
defined. Therefore, the non-vanishing condition of the GW invariants in a
fixed homological degree in Theorem A implies that for some homotopy class,
the invariant is nonzero and not divisible by ${\rm char}({\mathbb{k}})$.
Hence without loss of generality, from now on the curve class $A$ labelling
Gromov–Witten invariants of the base $B$ are homotopy classes rather than
homology classes.

We would like to relate Gromov–Witten invariants of the base and Gromov–Witten
invariants of the total space. We first consider the correspondence of between
homotopy classes of spheres.

######  Lemma 5.1.

The natural map $(\pi_{P})_{*}:\pi_{2}(P)\to\pi_{2}(B)$ is surjective.

###### Proof.

Suppose $A\in\pi_{2}(B)$ is represented by a smooth map $u:S^{2}\to B$. The
pullback fibration $u^{*}P\to S^{2}$ is then a Hamiltonian fibration. Let
$\\{\phi_{t}\\}_{t\in S^{1}}$ be clutching function, which is a loop of
Hamiltonian diffeomorphisms. Now consider the long exact sequence

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{2}(u^{*}P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{2}(S^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

The map $\pi_{2}(S^{2})\to\pi_{1}(X)$ sends the generator of $\pi_{2}(S^{2})$
to the homotopy class of the loop $\phi_{t}(x)$ for any point $x$, which is
always trivial (see [MS04, Corollary 9.1.2]). Therefore, the map
$\pi_{2}(u^{*}P)\to\pi_{2}(S^{2})$ is surjective. It follows that the class
$A$ lies in the image of $(\pi_{P})_{*}$. ∎

###  5.1. The non-vanishing result of ${\mathbb{k}}$-valued GW in the total
space

The main object to consider is the ${\mathbb{k}}$-valued Gromov–Witten
invariants of the total space $P$ of the fibration. To start one needs to
choose a symplectic form on $P$. As $P\to B$ is a Hamiltonian fibration, there
exists a Hamiltonian connection and hence a coupling form (see [MS98, Section
6]) $\Omega_{P}^{0}\in\Omega^{2}(P)$ which is a closed extension of the
fiberwise symplectic form. Let $\omega_{B}$ be the symplectic form on $B$. For
sufficiently large $\kappa>0$, the 2-form

$\Omega_{P}^{\kappa}:=\Omega_{P}^{0}+\kappa\pi^{*}\omega_{B}\in\Omega^{2}(P)$

is a symplectic form whose deformation class is independent of $\kappa$ and
the Hamiltonian connection.

Let ${\rm GW}_{0,k}^{P,{\mathbb{k}}}$ denote the ${\mathbb{k}}$-valued
$k$-pointed genus zero GW invariants of the space $(P,\Omega_{P}^{\kappa})$.
More precisely, for $\alpha_{1},\ldots,\alpha_{k}\in H^{*}(P;{\mathbb{k}})$,

${\rm
GW}^{P,{\mathbb{k}}}_{0,k}(\alpha_{1},\ldots,\alpha_{k})=\sum_{\tilde{A}\in\pi_{2}(P)}q^{\Omega_{P}^{\kappa}(\tilde{A})}{\rm
GW}^{P,{\mathbb{k}};\tilde{A}}_{0,k}(\alpha_{1},\ldots,\alpha_{k}).$

This invariant depends on the real number $\kappa$. However, if we fix a
homotopy class

$A\in\pi_{2}(B)$

of the base, then we can consider the contribution

$\sum_{\pi_{*}(\tilde{A})=A}q^{\Omega_{P}^{\kappa}(\tilde{A})}{\rm
GW}_{0,k}^{P,{\mathbb{k}};\tilde{A}}(\alpha_{1},\ldots,\alpha_{k})=q^{\kappa\Omega_{B}(A)}\sum_{\pi_{*}(\tilde{A})=A}q^{\Omega_{P}^{0}(A)}{\rm
GW}_{0,k}^{P,{\mathbb{k}};\tilde{A}}(\alpha_{1},\ldots,\alpha_{k}).$

It is $q^{\kappa\Omega_{B}(A)}$ times of a $\kappa$-invariant term.

######  Theorem 5.2.

Under the same setting as in Theorem 4.1, suppose ${\rm char}({\mathbb{k}})=p$
and

$m={\rm GW}_{0,k+2}^{B,A}([{\rm pt}],[{\rm
pt}],\alpha_{1},\ldots,\alpha_{k})\notin p{\mathbb{Z}}.$

Let

$(\pi_{P}^{*}\alpha_{i})_{\mathbb{k}}\in H^{*}(P;{\mathbb{k}})$

is the image of $\pi_{P}^{*}\alpha_{i}$ under the natural map

$H^{*}(P;{\mathbb{Z}})\to H^{*}(P;{\mathbb{k}}).$

Then for any $\beta_{0}\in H_{*}(M;{\mathbb{k}})$, there exists
$\beta_{\infty}\in H_{*}(M;{\mathbb{k}})$ such that

$\sum_{\pi_{*}(\tilde{A})=A}q^{\Omega_{P}^{0}(\tilde{A})}{\rm
GW}_{0,k+2}^{P,{\mathbb{k}};\tilde{A}}\Big{(}{\rm
PD}_{P}(\iota_{*}(\beta_{0})),{\rm
PD}_{P}(\iota_{*}(\beta_{\infty})),(\pi^{*}\alpha_{1})_{\mathbb{k}},\ldots,(\pi^{*}\alpha_{k})_{\mathbb{k}}\Big{)}\neq
0.$ (5.1)

The finite characteristic case of Theorem 4.1 follows immediately; the
characteristic zero case follows by considering a prime $p$ such that the
homology groups do not have $p$-torsions.

###  5.2. Proof of Theorem 5.2

To prove Theorem 5.2, we translate it into a more specific statement relating
the GW invariants of $P$ with the graph GW invariants of certain Hamiltonian
fibrations over $S^{2}$. By the assumption of Theorem 4.1, one can assume that
the intersection (4.1) consists of $\tilde{m}$ points
$q_{1},\ldots,q_{\tilde{m}}\in{\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{B},A)$
represented by marked smooth spheres

$(u_{l},{\bf z}_{l}=(z_{l,0},z_{l,1},\ldots,z_{l,k},z_{l,\infty})),\
l=1,\ldots,\tilde{m}$

with signs ${\rm sign}(u_{l},{\bf z}_{l})\in\\{\pm 1\\}$. The total sum of
these signed counts is the Gromov–Witten invariant ${\rm
GW}_{0,k+2}^{B,A}([{\rm pt}],[{\rm pt}],a_{1},\ldots,a_{k})$. Each $u_{l}$
pulls back a Hamiltonian fibration

$u_{l}^{*}P\to S^{2},\ l=1,\ldots,\tilde{m}.$

Then there are natural maps

$\pi_{2}^{\rm graph}(u_{l}^{*}P)\to(\pi_{P/B})_{*}^{-1}(A)\subset\pi_{2}(P)$

(which may not be injective). Now consider arbitrary
$\beta_{0},\beta_{\infty}\in H_{*}(M;{\mathbb{k}})$. Denote

$\beta_{0}^{P}=\iota_{*}(\beta_{0}),\beta_{\infty}^{P}=\iota_{*}(\beta_{\infty})\in
H_{*}(P;{\mathbb{k}})$

be the pushforward induced by the inclusion of a fiber.

######  Proposition 5.3.

One has

$\sum_{\pi_{*}(\tilde{A})=A}q^{\Omega_{P}^{\kappa}(\tilde{A})}{\rm
GW}_{0,k+2}^{P,{\mathbb{k}};\tilde{A}}\Big{(}{\rm PD}_{P}(\beta_{0}^{P}),{\rm
PD}_{P}(\beta_{\infty}^{P}),(\pi_{P/B}^{*}\alpha_{1})_{\mathbb{k}},\ldots,(\pi_{P/B}^{*}\alpha_{k})_{\mathbb{k}}\Big{)}\\\
=q^{\kappa\Omega_{B}(A)}\sum_{l=1}^{\tilde{m}}{\rm sign}(u_{l},{\bf
z}_{l})\sum_{\tilde{A}^{\prime}\in\pi_{2}^{\rm
graph}(u_{l}^{*}P)}q^{\tilde{\Omega}_{u_{l}^{*}P}(\tilde{A}^{\prime})}\widetilde{\rm
GW}{}_{0,2,{\bf
z}_{l}}^{u_{l}^{*}P,\tilde{A}^{\prime}}(\beta_{0},\beta_{\infty})\in\Lambda.$
(5.2)

As all $u_{l}$ are in the same homotopy class, the Hamiltonian fibrations
$u_{l}^{*}P\to S^{2}$ can be identified with $\tilde{M}_{\phi}\to S^{2}$ for a
fixed $\phi:S^{1}\to{\rm Ham}(M,\omega_{M})$ and all coupling forms
$\tilde{\Omega}_{u_{l}^{*}P}$ are in the same homology class of a fixed
coupling form $\tilde{\Omega}_{\phi}\in\Omega^{2}(\tilde{M}_{\phi})$.
Therefore the right hand side of (5.2) is equal to

$mq^{\kappa\Omega_{B}(A)}\sum_{\tilde{A}^{\prime}\in\pi_{2}^{\rm
graph}(\tilde{M}_{\phi})}q^{\tilde{\Omega}_{\phi}(\tilde{A}^{\prime})}\widetilde{\rm
GW}{}_{0,2,{\bf
z}}^{\tilde{M}_{\phi},\tilde{A}^{\prime}}(\beta_{0},\beta_{\infty})$

for some fixed marking ${\bf z}$. As $m$ is not divisible by $p$, hence
invertible in ${\mathbb{k}}$, Theorem 5.2 now follows from Proposition 5.3 and
Corollary 3.9. ∎

###  5.3. Proof of Proposition 5.3

The proof of Proposition 5.3 is carried out by carefully constructing a global
Kuranishi chart on the moduli space of stable maps into the total space and
choosing a good FOP transverse perturbation.

Let $J_{B}$ be the almost complex structure given in the assumption of Theorem
4.1. We first choose an almost complex structure $J_{P}$ on the total space
$P$ which is tamed by $\Omega_{P}^{\kappa}$ and such that the projection
$(P,J_{P})\to(B,J_{B})$ is pseudo-holomorphic. For each class
$\tilde{A}\in\pi_{2}(P)$ such that $\pi_{*}(\tilde{A})=A$ consider the moduli
space of stable maps

$\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A}).$

Then there is a natural projection map

$\pi_{P/B}:\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A})\to\overline{\mathcal{M}}{}_{0,k+2}(J_{B},A).$

Denote

$\overline{\mathcal{M}}{}_{0,k+2}^{\rm
sing}(J_{B},A):=\overline{\mathcal{M}}{}_{0,k+2}(J_{B},A)\setminus{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{B},A)$

where we switch the notation ${\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{B},A)={\mathcal{M}}{}_{0,k+2}^{\rm simple}(J_{B},A)$, and

$\displaystyle\ \overline{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{P},\tilde{A})=(\pi_{P/B})^{-1}({\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{B},A)),\ $ $\displaystyle\ \overline{\mathcal{M}}{}_{0,k+2}^{\rm
sing}(J_{P},\tilde{A})=(\pi_{P/B})^{-1}(\overline{\mathcal{M}}{}_{0,k+2}^{\rm
sing}(J_{B},A)).$

For the $\tilde{m}$ specific points $q_{l}=[u_{l},{\bf
z}_{l}]\in{\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{B},A)$, their preimages

$\pi_{P/B}^{-1}(q_{l})\subset\overline{\mathcal{M}}{}_{0,k+2}(J_{P},A)$

is naturally identified with the (disjoint) union of

$\overline{\mathcal{M}}{}_{0,2,{\bf z}_{l}}^{\rm
graph}(J_{P}|_{u_{l}^{*}P},\tilde{A}^{\prime})$

with $\tilde{A}^{\prime}\in\pi_{2}^{\rm graph}(u_{l}^{*}P)$ sent to
$\tilde{A}\in\pi_{2}(P)$. By abuse of notation, denote by

$\overline{\mathcal{M}}{}_{0,2,{\bf z}_{l}}^{\rm
graph}(J_{P}|_{u_{l}^{*}P},\tilde{A})$

this disjoint union.

######  Proposition 5.4.

There exist the following objects.

  1. (1)

A smooth almost complex global Kuranishi chart

$K=(G,V,E,S,\Psi)$

for $\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A})$. Denote the induced
derived orbifold chart by

${\mathcal{C}}=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\psi),$

then it is equivalent to (see Definition 2.1) the derived orbifold chart
induced from an AMS global Kuranishi chart.

  2. (2)

A singular global Kuranishi chart

$K_{\rm vert}=(G,V_{\rm vert},E_{\rm vert},S_{\rm vert},\Psi_{\rm vert})$

which restricts to an almost complex smooth global Kuranishi chart for the
open subset $\overline{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{P},A)\subset\overline{\mathcal{M}}{}_{0,k+2}(J_{P},A)$ with a
$G$-equivariant commutative diagram

$\textstyle{E_{\rm
vert}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\iota}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{\rm
vert}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{S_{\rm
vert}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S}$
(5.3)

where $\iota$ is a $G$-equivariant continuous map injecting onto its image
covered by the bundle homomorphism $\widehat{\iota}$, which restricts to a
$G$-equivariant embedding over $V_{\rm vert}^{\rm reg}$ compatible with the
almost complex structures.

  3. (3)

A $G$-equivariant complex vector bundle $\pi_{\rm hor}:W_{\rm hor}\to V_{\rm
vert}^{\rm vert}$, a $G$-invariant neighborhood $W_{\rm hor}^{\rm reg}\subset
W_{\rm hor}$ of the zero section, and a $G$-equivariant commutative diagram

$\textstyle{\pi_{\rm hor}^{*}E_{\rm vert}\oplus\pi_{\rm hor}^{*}W_{\rm
hor}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\theta}_{\rm
hor}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{\rm
hor}^{\rm
reg}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\rm
hor}}$$\textstyle{V}$ (5.4)

where $\theta_{\rm hor}$ is a homeomorphism extending the embedding of $V_{\rm
vert}^{\rm reg}$ into $V$ and $\widehat{\theta}_{\rm hor}$ is a bundle
isomorphism. Moreover, if we write the $W_{\rm hor}$-component of $S:V\to E$
as $S_{\rm hor}$ and write the tautological section of $\pi_{\rm
hor}^{*}W_{\rm hor}\to W_{\rm hor}^{\rm reg}$ as $\tau_{W_{\rm hor}}$, then

$\widehat{\theta}_{\rm hor}\circ\tau_{W_{\rm hor}}=S_{\rm hor}\circ\theta_{\rm
hor}.$

  4. (4)

A $G$-invariant smooth submersive map

$\tilde{\pi}_{\rm vert}:V_{\rm vert}^{\rm reg}\to{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{B},A)\subset{\mathcal{M}}{}_{0,k+2}^{\rm simple}(J_{B},A)$ (5.5)

which makes the following diagram commutes

$\textstyle{{\mathcal{S}}_{\rm vert}^{-1}(0)\cap V_{\rm vert}^{\rm
reg}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{S}}_{\rm
vert}^{-1}(0)/G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{\rm
vert}^{\rm
reg}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{B},A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}{}_{0,k+2}(J_{B},A).}$

Furthermore, for each point $q_{l}=[u_{l},{\bf
z}_{l}]\in{\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{B},A)$, denote the fiber of
$V^{\rm reg}_{\rm vert}$ by $V_{{\rm vert},q_{l}}^{\rm reg}$, then the
restriction of $K_{\rm vert}$ to $V_{{\rm vert},q_{l}}^{\rm reg}$ is
equivalent to an AMS global Kuranishi chart on the graph moduli space
$\overline{\mathcal{M}}{}_{0,2,{\bf z}_{l}}^{\rm
graph}(\tilde{J}_{P}|_{u_{l}^{*}P},\tilde{A})$.

The proof of Proposition 5.4 is deferred to the next subsection.

Now we can prove Proposition 5.3. We first choose pseudocycle representatives
of the constraints. For each pseudocycle $f_{i}:W_{i}\to B$, denote
$\tilde{W}_{i}=f_{i}^{*}P$ and

$\tilde{f}_{i}:\tilde{W}_{i}\to P$

the induced map. It is easy to see that $\tilde{f}_{i}$ is again a pseudocycle
and represents the class ${\rm PD}_{P}((\pi_{P}^{*}a_{i})_{\mathbb{k}})$. On
the other hand, choose $p$-pseudocycles

$f_{0}:W_{0}\to M,\ {\rm resp.}\ f_{\infty}:W_{\infty}\to M$

representing the classes $\beta_{0}$ resp. $\beta_{\infty}$. The composition
with the inclusions $\iota_{0}:M\to P|_{p_{0}}$ resp. $\iota_{\infty}:M\to
P|_{p_{\infty}}$ we obtain $p$-pseudocycles

$\tilde{f}_{0}=\iota_{0}\circ f_{0}:W_{0}\to P\ {\rm resp.}\
\tilde{f}_{\infty}=\iota_{\infty}\circ f_{\infty}:W_{\infty}\to P.$

Let the induced derived orbifold chart of $K$ be

${\mathcal{C}}=({\mathcal{U}},{\mathcal{E}},{\mathcal{S}},\psi):=(V/G,E/G,S/G,\Psi/G)$

and also denote

${\mathcal{C}}_{\rm vert}^{\rm reg}=({\mathcal{U}}_{\rm
reg},{\mathcal{E}}_{\rm vert},{\mathcal{S}}_{|},\psi_{\rm vert}):=(V_{\rm
vert}^{\rm reg}/G,E_{\rm vert}/G,S_{\rm vert}/G,\Psi_{\rm vert}/G).$

Consider the $\tilde{m}$ specific points $q_{l}=[(u_{l},{\bf z}_{l})]\in
U\subset{\mathcal{M}}{}_{0,k+2}^{\rm simple}(J_{B},A)$ and their fibers

${\mathcal{U}}^{\rm reg}_{{\rm vert},q_{l}}:=\tilde{\pi}_{\rm
vert}^{-1}(q_{l})\subset{\mathcal{U}}_{\rm vert}^{\rm reg}.$

Then the quadruple

${\mathcal{C}}_{{\rm vert},q_{l}}:=({\mathcal{U}}^{\rm reg}_{{\rm
vert},q_{l}},{\mathcal{E}}_{\rm vert},{\mathcal{S}}_{\rm vert})$

is a normally complex derived orbifold chart for the graph moduli
$\overline{\mathcal{M}}{}_{0,2,{\bf z}_{l}}^{\rm
graph}(\tilde{J}_{P}|_{u_{l}^{*}P},\tilde{A})$, where we abuse the notations
by writing ${\mathcal{E}}_{\rm vert}=(E_{\rm vert}/G)|_{{\mathcal{U}}^{\rm
reg}_{\rm vert}(q_{l})}$ and ${\mathcal{S}}_{\rm vert}=(S_{\rm
vert}/G)|_{{\mathcal{U}}^{\rm reg}_{\rm vert}(q_{l})}$. Choose a distance
function on ${\mathcal{U}}$. For any $\delta>0$, let

${\mathcal{U}}_{\rm vert}^{{\rm reg},\delta}(q_{l})\subset{\mathcal{U}}_{{\rm
vert}}^{\rm reg}$

be the $\delta$-neighborhood of ${\mathcal{U}}_{{\rm vert},q_{l}}^{\rm reg}$
inside ${\mathcal{U}}_{\rm vert}^{\rm reg}$.

Then we can choose FOP transverse perturbations on the fibers

${\mathcal{S}}_{{\rm vert},q_{l}}^{\prime}:{\mathcal{U}}_{{\rm
vert},q_{l}}^{\rm reg}\to{\mathcal{E}}_{\rm vert}.$

Claim. One can extend ${\mathcal{S}}_{{\rm vert},q_{l}}^{\prime}$ to an FOP
transverse perturbation

${\mathcal{S}}_{\rm vert}^{\prime}:{\mathcal{U}}_{\rm vert}^{\rm
reg}\to{\mathcal{E}}_{\rm vert}.$

Proof of the claim. Notice that ${\mathcal{M}}_{0,k+2}^{\rm
reg}(J_{B},A)\subset{\mathcal{M}}_{0,k+2}(J_{B},A)$ is a manifold rather than
an orbifold, the normal bundle of ${\mathcal{U}}_{\rm vert}^{\rm reg}(q_{l})$
inside ${\mathcal{U}}_{\rm vert}^{\rm reg}$, which is the pullback of the
tangent space $T_{q_{l}}{\mathcal{M}}_{0,k+2}^{\rm reg}(J_{B},A)$ under the
projection map (5.5), is an ordinary vector bundle. Then by (6) of Theorem
2.2, one can extend the FOP transverse perturbation to an FOP transverse
perturbation

${\mathcal{S}}_{\rm vert}^{\prime}:{\mathcal{U}}_{\rm vert}^{\rm
reg}\to{\mathcal{E}}_{\rm vert}.$

This finishes the proof of this claim.

Next, look at the more thickened chart ${\mathcal{U}}:=V/G$ which contains
${\mathcal{U}}_{\rm vert}$ by the diagram (5.3). As $W_{\rm hor}\to V_{\rm
vert}^{\rm reg}$ is a $G$-equivariant vector bundle, it descends to an
orbifold vector bundle

${\mathcal{W}}_{\rm hor}\to{\mathcal{U}}_{\rm vert}^{\rm reg}.$

By (3) of Proposition 5.4, inside ${\mathcal{U}}$ there is an open subset
identified with ${\mathcal{W}}_{\rm hor}^{\rm reg}\subset{\mathcal{W}}_{\rm
hor}$ which contains ${\mathcal{U}}_{\rm vert}^{\rm reg}$. By choosing an
appropriate metric on the bundle ${\mathcal{W}}_{\rm hor}$, we may assume that
${\mathcal{W}}_{\rm hor}^{\rm reg}$ is the open disk bundle of radius
$2\epsilon$ for some $\epsilon>0$, denoted by ${\mathcal{W}}_{\rm
hor}^{2\epsilon}$.

Claim. There exists an FOP transverse extension of ${\mathcal{S}}_{\rm
vert}^{\prime}$ to the open subset ${\mathcal{W}}_{\rm hor}^{2\epsilon}$,
denoted by ${\mathcal{S}}^{\prime}$, satisfying the following conditions.

  1. (1)

${\mathcal{S}}^{\prime}={\mathcal{S}}$ within ${\mathcal{W}}_{\rm
hor}^{2\epsilon}\setminus{\mathcal{W}}_{\rm hor}^{\epsilon}$.

  2. (2)

$({\mathcal{S}}^{\prime})^{-1}(0)={\mathcal{S}}^{-1}(0)$.

Proof of the claim. Let $x$ be the coordinate on ${\mathcal{U}}_{\rm
vert}^{\rm reg}$ and $y$ be the fiber coordinate on ${\mathcal{W}}_{\rm hor}$.
We can write any extension ${\mathcal{S}}^{\prime}$ as

${\mathcal{S}}^{\prime}(x,y)=\Big{(}{\mathcal{S}}_{\rm
vert}^{\prime}(x,y),{\mathcal{S}}_{\rm hor}^{\prime}(x,y)\Big{)}$

where the first component takes value in ${\mathcal{E}}_{\rm vert}$ and the
second one takes value in ${\mathcal{W}}_{\rm hor}$. The value of
${\mathcal{S}}_{\rm vert}^{\prime}(x,0)$ has been fixed. Withint
${\mathcal{W}}_{\rm hor}^{\epsilon}$, we define

${\mathcal{S}}^{\prime}(x,y)=({\mathcal{S}}_{\rm vert}^{\prime}(x,0),y)$ (5.6)

where $y$ is regarded as the tautological section of ${\mathcal{W}}_{\rm
hor}$. Then by (5) of Theorem 2.2 (stabilization property),
${\mathcal{S}}^{\prime}$ is FOP transverse and obviously
$({\mathcal{S}}^{\prime})^{-}(0)={\mathcal{S}}^{-1}(0)$. To extend to
${\mathcal{W}}_{\rm hor}^{2\epsilon}$, we keep ${\mathcal{S}}_{\rm
hor}^{\prime}$ to be the tautological section and extend ${\mathcal{S}}_{\rm
vert}^{\prime}$ arbitrarily. This will not create any more zeroes and hence
still FOP transverse. End of the Proof of the claim.

Lastly, by the the “CUDV property” of Theorem 2.2, one can find an FOP
transverse perturbation which agrees with ${\mathcal{S}}^{\prime}$ over
$\theta_{\rm hor}(\overline{W_{\rm hor}^{\epsilon}})$. By abuse of notation,
still denote the perturbation by ${\mathcal{S}}^{\prime}$. Notice that by (2)
of Theorem 2.2, one can assume

$\|{\mathcal{S}}-{\mathcal{S}}^{\prime}\|_{C^{0}}\leq\delta$

for any $\delta>0$.

Now we look at the intersection numbers. Consider

$\Big{(}({\mathcal{S}}^{\prime})^{-1}(0)\cap{\mathcal{U}}_{\rm
free}\Big{)}\cap{\rm ev}_{P}^{-1}\Big{(}f_{0}(W_{0})\times
f_{\infty}(W_{\infty})\times\prod_{i=1}^{k}\tilde{f}_{i}(\tilde{W}_{i})\Big{)}$

whose signed count coincides with the GW invariant

${\rm GW}_{0,k+2}^{P,{\mathbb{k}};\tilde{A}}\big{(}{\rm
PD}_{P}(\beta_{0}^{P}),{\rm
PD}_{P}(\beta_{\infty}^{P}),(\pi_{P/B}^{*}\alpha_{1})_{\mathbb{k}},\ldots,(\pi_{P/B}^{*}\alpha_{k})_{\mathbb{k}}\big{)}.$

Also, obviously

$\Big{(}({\mathcal{S}}^{\prime})^{-1}(0)\cap{\mathcal{U}}_{\rm
free}\Big{)}\cap{\rm
ev}_{P}^{-1}\Big{(}\tilde{f}_{0}(\tilde{W}_{0})\times\tilde{f}_{\infty}(\tilde{W}_{\infty})\times\prod_{i=1}^{k}\tilde{f}_{i}(\tilde{W}_{i})\Big{)}\\\
\subset\Big{\\{}x\in{\mathcal{U}}\ |\ \pi_{P/B}({\rm
ev}_{0}(x))=p_{0},\pi_{P/B}({\rm ev}_{\infty}(x))=p_{\infty},\ \pi_{P/B}{\rm
ev}_{i}(x)\in f_{i}(W_{i})\subset B\Big{\\}}$

As inside ${\mathcal{U}}$, the perturbed zero locus
$({\mathcal{S}}^{\prime})^{-1}(0)$ can be arbitrarily close to
${\mathcal{S}}^{-1}(0)$, the above intersection points can be arbitrarily
close to points in
${\mathcal{S}}^{-1}(0)\cong\overline{\mathcal{M}}{}_{0,k+2}(J_{P},A)$ which
projects to the points $q_{l}=[u_{l},{\bf z}_{l}]$. In particular, the above
intersection points are in the open subset

$\bigcup_{l=1}^{\tilde{m}}{\mathcal{W}}_{\rm
hor}^{\epsilon}|_{{\mathcal{U}}_{\rm vert}^{{\rm
reg},\delta}(q_{l})}\subset{\mathcal{W}}_{\rm hor}^{\epsilon}.$

By our construction of ${\mathcal{S}}^{\prime}$, within the disk bundle
${\mathcal{W}}_{\rm hor}^{\epsilon}$ the perturbation is a stabilization of
${\mathcal{S}}_{\rm vert}^{\prime}$, hence

$\Big{(}({\mathcal{S}}^{\prime})^{-1}(0)\cap{\mathcal{U}}_{\rm
free}\Big{)}\cap{\rm
ev}_{P}^{-1}\Big{(}\tilde{f}_{0}(\tilde{W}_{0})\times\tilde{f}_{\infty}(\tilde{W}_{\infty})\times\prod_{i=1}^{k}\tilde{f}_{i}(\tilde{W}_{i})\Big{)}\\\
=\bigsqcup_{l=1}^{\tilde{m}}\Big{(}({\mathcal{S}}_{\rm
vert}^{\prime})^{-1}(0)_{\rm free}\cap{\mathcal{U}}_{{\rm vert},q_{l}}^{\rm
reg}\Big{)}\cap{\rm ev}_{0}^{-1}(\tilde{f}_{0}(\tilde{W}_{0}))\cap{\rm
ev}_{\infty}^{-1}(\tilde{f}_{\infty}(\tilde{W}_{\infty})).$

As the fiber ${\mathcal{U}}_{\rm vert}^{\rm reg}(q_{l})$ is a derived orbifold
chart of the moduli space $\overline{\mathcal{M}}{}_{0,2,{\bf z}_{l}}^{\rm
graph}(J_{P}|_{u_{l}^{*}P},\tilde{A})$ and the restriction of
${\mathcal{S}}_{\rm vert}^{\prime}$ to ${\mathcal{U}}_{\rm vert}^{\rm
reg}(q_{l})$ is an FOP transverse perturbation, the signed count of the above
intersection set is (up to signs) equal to the graph GW invariant

$\widetilde{\rm GW}{}_{0,2,{\bf
z}_{l}}^{u_{l}^{*}P,\tilde{A}}(\beta_{0},\beta_{\infty})\in{\mathbb{k}}.$

The detailed sign verification is given in Subsection 5.5. Summing over all
the points in $\Big{(}({\mathcal{S}}^{\prime})^{-1}(0)\cap{\mathcal{U}}_{\rm
free}\Big{)}\cap{\rm
ev}_{P}^{-1}\Big{(}\tilde{f}_{0}(\tilde{W}_{0})\times\tilde{f}_{\infty}(\tilde{W}_{\infty})\times\prod_{i=1}^{k}\tilde{f}_{i}(\tilde{W}_{i})\Big{)}$,
we obtain the desired statement. ∎

###  5.4. Proof of Proposition 5.4

####  5.4.1. The AMS construction

As before, the first step of construction a global Kuranishi chart is to
approximate the symplectic form $\Omega_{P}^{\kappa}$ by a rational one,
denoted by $\Omega_{P}^{\prime}\in\Omega^{2}(P)$. When $\Omega_{P}^{\prime}$
is sufficiently close to $\Omega_{P}^{\kappa}$, $J_{P}$ is still tamed by
$\Omega_{P}^{\prime}$. After rescaling one may assume $\Omega_{P}^{\prime}$ is
integral. For the curve class $\tilde{A}\in\pi_{2}(P)$, denote

$d(\tilde{A})=\langle\Omega_{P}^{\prime},\tilde{A}\rangle\in{\mathbb{Z}}.$

Second, for each $d>0$, consider the moduli space
$\overline{\mathcal{M}}_{0,k+2}(\mathbb{CP}^{d},d)$ (of unparametrized stable
maps). The group $G_{d}=U(d+1)$ acts on
$\overline{\mathcal{M}}_{0,k+2}(\mathbb{CP}^{d},d)$. There is a
$G_{d}$-invariant subset
$B_{k+2,d}\subset\overline{\mathcal{M}}_{0,k+2}(\mathbb{CP}^{d},d)$ consisting
of configurations whose images are not contained in any hyperplane.
$B_{k+2,d}$ is a smooth complex $G_{d}$-manifold. Then consider the universal
curve

$C_{k+2,d}\to B_{k+2,d}.$

Let $C_{k+2,d}^{*}\subset C_{k+2,d}$ be the subset of points which are not
marked points or nodal points of fibers.

Similar to the AMS construction for the graph moduli, one consider the
following objects.

######  Definition 5.5.

  1. (1)

A finite-dimensional approximation scheme on $C_{k+2,d}$ is a representation
$W$ of $G_{d}$ together with an equivariant linear map

$\iota:W\to C^{\infty}(C_{k+2,d}^{*}\times
P,\Omega^{0,1}_{C_{k+2,d}^{*}/B_{k+2,d}}\otimes TP).$

  2. (2)

Given $\tilde{A}\in\pi_{*}^{-1}(A)$ and a finite-dimensional approximation
scheme $(W,\iota)$ over $C_{k+2,d}$ for $d=d(\tilde{A})$, the pre-thickening
is the set

$V^{\rm pre}=\Big{\\{}(\rho,u,e)\ |\ \rho\in B_{k+2,d},\ u:C_{\rho}\to P,\
e\in W,\ \overline{\partial}_{J_{P}}u+\iota(e)|_{C_{\phi}}=0,\
[u]=\tilde{A}\Big{\\}}.$ (5.7)

######  Definition 5.6.

We say that a finite-dimensional approximation scheme $(W,\iota)$ is regular
over an open subset $U\subset\overline{\mathcal{M}}_{0,k+2}(J_{P},\tilde{A})$
if for each $y\in U$ and each $(\rho,u,0)\in V^{\rm pre}$ sent to $y$ by the
forgetful map, the linearization of the equation
$\overline{\partial}_{J_{P}}u+\iota(e)=0$ is surjective.

Recall that we have a natural projection
$\pi_{P/B}:\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A})\to\overline{\mathcal{M}}{}_{0,k+2}(J_{B},A)$.

######  Lemma 5.7.

One can choose a finite-dimensional approximation scheme $(W^{\rm
vert},\iota^{\rm vert})$ (called the vertical obstruction space) satisfying
the following two conditions.

  1. (1)

For all $e\in W^{\rm vert}$, $\iota^{\rm vert}(e)$ lies in $T^{\rm vert}P$.

  2. (2)

For any compact subset $K\subset\overline{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{B},A)$, $(W^{\rm vert},\iota^{\rm vert})$ is regular near
$\pi_{P/B}^{-1}(K)$.

###### Proof.

This is because downstairs away from multiple covers and nodal curves the
moduli space is already regular. Therefore obstructions only lie in the
vertical direction, so we can choose a sufficiently large finite dimensional
approximation scheme lying in the vertical direction of the tangent bundle of
$P$. ∎

We first use the vertical obstruction space to construct a Kuranishi chart
which is a manifold only away from certain bad locus. We may shrink the open
subset ${\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{B},A)$ given by the assumption of
Theorem 4.1 to a precompact neighborhood of the points
$p_{1},\ldots,p_{\tilde{m}}$. Then we may assume that there is a finite
dimensional approximation scheme $(W_{\rm vert},\iota_{\rm vert})$ as in Lemma
5.7, such that it is regular over the corresponding open subset of
$\overline{\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{P},\tilde{A})$. Let the pre-
thickening obtained from such data be $V_{\rm vert}^{\rm pre}$.

To proceed, we use framings to construct the actual AMS chart. For each triple
$(\rho,u,e)$, the pullback form
$u^{*}\Omega_{P}^{\prime}\in\Omega^{2}(C_{\rho})$ has integral being
$d=d(\tilde{A})$, hence there is a Hermitian holomorphic line bundle $L_{u}\to
C_{\rho}$, unique up to unitary isomorphism, whose curvature form is
$-2\pi{\bf i}u^{*}\Omega_{P}^{\prime}$. A framing on $(\rho,u,e)$ is a basis
of $H^{0}(L_{u})$

$F=(f_{0},\ldots,f_{d}).$

Then define

$V_{\rm vert}:=\Big{\\{}(\rho,u,e,F)\ |\ (\rho,u,e)\in V_{\rm vert}^{\rm
pre},\ F\ {\rm is\ a\ framing}\Big{\\}}$

which has the structure of a $G_{d}$-equivariant principal
$G_{d}^{\mathbb{C}}$-bundle. Define the obstruction bundle $E_{\rm vert}\to V$
as

$E_{\rm vert}=W_{\rm
vert}\oplus\pi_{V/B}^{*}TB_{k+2,d}\oplus{\mathfrak{g}}_{d}.$

Choosing a $G_{d}$-invariant Riemannian metric on $B_{k+2,d}$ and denote the
exponential map by $\exp_{B}$, which identifies a neighborhood
$\Delta^{+}(B_{k+2,d})$ of the diagonal in $B_{k+2,d}\times B_{k+2,d}$ with a
neighborhood of the zero section of the tangent bundle of $B_{k+2,d}$. Shrink
$V$ to the open subset where $(\rho,\rho_{F})\in\Delta^{+}(B_{k+2,d})$ and
where the Hermitian matrix $H_{F}$ with entries

$\int_{C_{\rho}}\langle f_{i},f_{j}\rangle u^{*}\Omega_{P}^{\prime}$

is positive definite. Define the Kuranishi section $S_{\rm vert}:V_{\rm
vert}\to E_{\rm vert}$ as

$S_{\rm
vert}(\rho,u,e,F)=\Big{(}e,\exp_{B}^{-1}(\rho,\rho_{F}),\exp_{H}^{-1}(H_{F})\Big{)}.$
(5.8)

If $S_{\rm vert}(\rho,u,e,F)=0$, we see $u:C_{\rho}\to P$ is a genuine
$J_{P}$-holomorphic map. Then define the $G_{d}$-equivariant map

$\psi_{\rm vert}:S_{\rm
vert}^{-1}(0)/G_{d}\to\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A}),\
\tilde{\psi}(\rho,u,0,F)=[u:C_{\rho}\to P].$

Similar to Lemma 3.4, one has

######  Lemma 5.8.

$\psi_{\rm vert}$ is a homeomorphism. ∎

Therefore one obtains a singular global Kuranishi chart

$K_{\rm vert}=(G_{d},V_{\rm vert},E_{\rm vert},S_{\rm vert},\psi_{\rm vert})$

which is regular over ${\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{P},\tilde{A})$.

One can then choose another finite-dimensional approximation scheme $(W_{\rm
hor},\iota_{\rm hor})$ over $C_{k+2,d}$ (called the horizontal obstruction
space555Although the corresponding sections do not necessarily take value in
the horizontal part of the pulled-back tangent bundle, we hope that this
notion will not cause any confusion.) and denote

$(W,\iota)=(W_{\rm vert}\oplus W_{\rm hor},\iota_{\rm vert}\oplus\iota_{\rm
hor})$

so that it is regular everywhere in the sense of Definition 5.6. Notice that
$W_{\rm hor}$ defines a $G_{d}$-equivariant vector bundle

$\pi_{\rm hor}:W_{\rm hor}\to V_{\rm vert}$

Parallel to the previous discussions on the singular global Kuranishi chart,
now we can construct a global Kuranishi chart for
$\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A})$. Using the finite-
dimensional approximation scheme $(W,\iota)$, the pre-thickening $V^{\rm pre}$
is defined as in (5.7). We obtain a global Kuranishi chart

$K=(G_{d},V,E,S,\psi)$

where $V=\\{(\rho,u,e,F)\ |\ (\rho,u,e)\in V^{\rm pre},\ F\ {\rm is\ a\
framing}\\}$; $E=W\oplus\pi_{V/B}^{*}TB_{k+2,d}\oplus{\mathfrak{g}}_{d}$,
which differs from $E_{\rm vert}$ by the direct summand $W_{\rm hor}$; the
Kuranishi section $S$ is defined using the same formula (5.8); the map $\psi$
is defined similarly to $\psi_{\rm vert}$, which gives rise to a homeomorphism
$S_{\rm
vert}^{-1}(0)/G_{d}\xrightarrow{\sim}\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A})$.

######  Lemma 5.9.

Away from the horizontally singular locus, $V$ is a stabilization of $V_{\rm
vert}$. More precisely, there is a $G_{d}$-invariant open neighborhood $V_{\rm
vert}^{\rm reg}\subset V_{\rm vert}$ of $\psi_{\rm
vert}^{-1}\left(\overline{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{P},\tilde{A})\right)$ which is a topological manifold, a
$G_{d}$-invariant open neighborhood of the zero section of $W_{\rm
hor}|_{V_{\rm vert}^{\rm reg}}$, denoted by $W_{\rm hor}^{\rm reg}$, a
$G_{d}$-equivariant bundle map $\widehat{\theta}_{\rm hor}$ covering a
$G_{d}$-equivariant map $\theta_{\rm hor}$

$\textstyle{\pi_{\rm hor}^{*}E_{\rm vert}\oplus\pi_{\rm hor}^{*}W_{\rm
hor}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\theta}_{\rm
hor}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{\rm
hor}^{\rm
reg}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\rm
hor}}$$\textstyle{V}$ (5.9)

satisfying the following conditions.

  1. (1)

$\theta_{\rm hor}$ is a homeomorphism onto an open subset whose restriction to
the zero section coincides with the embedding $V_{\rm vert}^{\rm
reg}\hookrightarrow V$.

  2. (2)

Let $S_{\rm hor}:V\to W_{\rm hor}$ be the $W_{\rm hor}$-component of the
Kuranishi map $S:V\to E$ and let $\tau_{W_{\rm hor}}:W_{\rm hor}^{\rm
reg}\to\pi_{\rm hor}^{*}W_{\rm hor}$ be the tautological section. Then

$\widehat{\theta}_{\rm hor}\circ\tau_{W_{\rm hor}}=S_{\rm hor}\circ\theta_{\rm
hor}.$

###### Proof.

Consider the forgetful map $V\to B_{k+2,d}$. The fiber of this forgetful map
at $\rho\in B_{k+2,d}$, denoted by $V(\rho)$, is the zero locus of a smooth
Fredholm map from the pseudoholomorphic graph equation over the smooth or
nodal curve $C_{\rho}$. Hence $V(\rho)$ is naturally a smooth manifold. There
is a subset $V_{\rm vert}(\rho)\subset V(\rho)$ which is a submanifold near
the regular locus. The normal bundle can be identified with the bundle $W_{\rm
hor}$.

We can choose a $G_{d}$-invariant family of fiberwise Riemannian metrics on
$V(\rho)$ which varies continuously in $\rho$. This is possible as the
vertical tangent bundle $T^{\rm vert}V$ exists. Then the map $\theta_{\rm
hor}$ can be constructed using the fiberwise exponential map. Indeed, it
follows from the construction and the regularity assumption that for any point
in $v\in V_{\rm vert}^{\rm reg}\subset V$ over $\rho$, the vertical tangent
space of $V$ at this point admits a decomposition $T_{v}^{\rm
vert}V=T_{v}V_{\rm vert}^{\rm reg}\oplus W_{\rm hor}$, where $T_{v}V_{\rm
vert}^{\rm reg}$ is well-defined due to the regularity of the singular global
Kuranishi chart near $v$. Then the fiberwise normal bundle of $V_{\rm
vert}^{\rm reg}\subset V$ is identified with $W_{\rm hor}\to V_{\rm vert}^{\rm
reg}$. So, for any tangent vector lying in $W_{\rm hor}$ with sufficiently
small norm, we can define $\theta_{\rm hor}$ using the exponential map at $v$.
By varying over $\rho$, we see that $\theta_{\rm hor}$ defines a homeomorphism
onto an open subset which agrees with the embedding $V_{\rm
vert}\hookrightarrow V$ along the zero section. As the diagram (5.9) commutes
along $V_{\rm vert}^{\rm reg}$, one can use parallel transports to construct
the bundle map $\widehat{\theta}_{\rm hor}$.

Now by the implicit function theorem, the restriction of $S_{\rm hor}$ to the
fibers of $W_{\rm hor}^{\rm reg}$ is a homeomorphism onto a small disk bundle
of $W_{\rm hor}\subset E$. By reparametrizing the map $\theta_{\rm hor}$, and
accordingly, $\widehat{\theta}_{\rm hor}$, one can ensure that $S_{\rm hor}$
agrees with $\tau_{W_{\rm hor}}$ after being pulled back by $\theta_{\rm
hor}$. ∎

####  5.4.2. Smoothing

We need to construct smooth structures on the base of the Kuranishi chart $V$
which contains a smooth submanifold $V_{\rm vert}^{\rm reg}$. The basic idea
is as follows. First, following the argument of [AMS21], by using the
topological submersion $V_{\rm vert}^{\rm reg}\to B_{k+2,d}$, one can
construct a stable smoothing on $V_{\rm vert}^{\rm reg}$. Then as $V$ is
locally a bundle over $V_{\rm vert}^{\rm reg}$, one obtains a natural smooth
structure of the thickening.

The first stage of this construction is similar to the situation of Lemma 3.5.
The forgetful map

$V_{\rm vert}^{\rm reg}\to B_{k+2,d}$

is a $G_{d}$-equivariant topological submersion with fiberwise smooth
structures varying sufficiently regularly. This allows us to construct a
vector bundle lift of the tangent microbundle $T_{\mu}V_{\rm vert}^{\rm reg}$.
Lashof’s theorem666Note that Lashof’s smoothing theorem only requires finitely
many orbit types rather than the compactness of the ambient topological
manifold. implies the existence of a stable $G_{d}$-smoothing, i.e., a smooth
structure on $V_{\rm vert}^{\rm reg}\times R$ for some $G_{d}$-representation
$R$. One can indeed include $R$ into the vertical obstruction space $W_{\rm
vert}$ such that $R$ is mapped to zero by $\iota_{\rm hor}$. We can then
assume $V_{\rm vert}^{\rm reg}$ is a smooth $G_{d}$-manifold.

We need the smoothing be compatible with the smooth structures on the moduli
space of holomorphic curves in the base $B$. Notice that since the obstruction
space $W_{\rm vert}$ is in the vertical direction of the fibration $P\to B$,
there is a natural $G_{d}$-invariant projection map

$\pi_{P/B}:V_{\rm vert}^{\rm reg}\to{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{B},A)$ (5.10)

where the codomain has a natural smooth structure.

Next, as the bundle $W_{\rm hor}\to V_{\rm vert}^{\rm reg}$ is trivial, it is
automatically a smooth $G_{d}$-equivariant vector bundle. The homeomorphism
$\theta_{\rm hor}$ in (5.9) then induces a $G_{d}$-smooth structure on the
image of $\theta_{\rm hor}$, which is an open subset of $V$.

We would like to extend the smoothing onto the whole of $V$ using the relative
version of Lashof’s smoothing theorem proved in [BX22a, Appendix B]. We first
specify a neighborhood of the singular region where we modify the existing
smoothing. The region is selected properly in order to compare the GW
invariants of the total space $P$ with the GW invariants of the base $B$.

Choose a $G_{d}$-invariant distance function on the base $V$ of the Kuranishi
chart. Choose a $G_{d}$-invariant distance function on $V$. For any
$\epsilon>0$, define

$V^{{\rm sing}+\epsilon}\subset V$

be the open $\epsilon$-neighborhood of
$\Psi^{-1}(\overline{\mathcal{M}}{}_{0,k+2}^{\rm sing}(J_{P},\tilde{A}))$.
Notice that for any $\epsilon>0$, the union

$V^{\epsilon}:={\rm Im}(\theta_{\rm hor})\cup V^{{\rm sing}+\epsilon}$

is a $G_{d}$-invariant open neighborhood of $S^{-1}(0)$ in $V$. One can fix
$\epsilon>0$ such that

$\pi_{P/B}^{k+2}\Big{(}{\rm ev}_{P}\big{(}V^{{\rm
sing}+\epsilon}\big{)}\Big{)}\cap\Big{(}\\{p_{0}\\}\times\\{p_{\infty}\\}\times\prod_{i=1}^{k}f_{i}(W_{i})\Big{)}=\emptyset.$

Then using the relative version of Lashof’s theorem [BX22a, Theorem B.3], one
can construct a stable smoothing on $V^{\epsilon}$ which coincides with the
smooth structure on ${\rm Im}(\theta_{\rm hor})$ taking product with the
corresponding $G_{d}$-representation away from $V^{{\rm sing}+\epsilon}$. By
absorbing the additional representation of $G_{d}$ needed for the stable
smoothing into the obstruction space $W_{\rm vert}$, one may assume that the
stable smoothing is actually a smoothing.

On the other hand, the obstruction bundle can always be identified with a
smooth vector bundle. Hence we obtained a smooth global Kuranishi chart for
$\overline{\mathcal{M}}{}_{0,k+2}(J_{P},\tilde{A})$, denoted by

$K^{\epsilon}=(G_{d},V^{\epsilon},E,S,\Psi).$

####  5.4.3. Wrapping up the proof

With the above preparations, we can finish the proof.

###### Proof of Proposition 5.4.

Fixing an $\epsilon>0$, we can use the global Kuranish chart $K=K^{\epsilon}$.
As discussed in the proof of Proposition 3.1, by choosing all the involved
stabilizations to be complex, the fact that the vertical tangent spaces can be
identified with the index of Cauchy–Riemann operators ensures that we can
build an almost complex $K$. As this chart is built from the finite-
dimensional approximation $(W,\iota)$ after stabilization and shrinking, it
follows from the construction that $K$ is equivalent to an AMS global
Kuranishi chart. This proves (1).

The singular global Kuranishi chart $K_{\rm vert}=(G_{d},V_{\rm vert},E_{\rm
vert},S_{\rm vert},\psi_{\rm vert})$ for
$\overline{\mathcal{M}}{}_{0,k+2}(J_{P},A)$ from (5.4.1) is regular over the
open subset $\overline{\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{P},A)$. Then the
commutativity of (5.3) and (5.4), and the agreement of $S_{\rm hor}$ and
$\tau_{W_{\rm hor}}$, follow from (5.9), and the compatibility of smooth
structures, in particular, the assertion that $\iota:V_{\rm vert}^{\rm reg}\to
V$ is a smooth embedding, follows from the relative smoothing construction in
Section 5.4.2. Because the map $\iota:V_{\rm vert}^{\rm reg}\to V$ covers the
identity map on the complex manifold $B_{k+2,d}$, and for each fiber over
$\rho\in B_{k+2,d}$, a homotopy of the linearized Cauchy–Riemann operator of
an element $v\in V_{\rm vert}^{\rm reg}$ to a complex linear operator can be
extended to such a homotopy viewing $v\in V$, the fact that we can choose
$W_{\rm hor}$ to be a complex vector space implies that the embedding is
compatible with the almost complex structures. This proves (2) and (3).

As for (4), the submersive property follows from a technical lemma of
Abouzaid–McLean–Smith [AMS21, Lemma 4.5]. Indeed, for the projection (5.10),
upon choosing a further stabilization of the chart $V_{\rm vert}^{\rm reg}$ by
the bundle

$\pi_{P/B}^{*}T{\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{B},A)$

which admits a $G_{d}$-invariant smooth structure, we can find a
$G_{d}$-invariant smooth submersive extension of the map $\pi_{P/B}$ to
${\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{B},A)$. The commutativity of the diagram
follows from the construction. The differences between the restriction of
$K_{\rm vert}$ to $V_{{\rm vert},q_{l}}^{\rm reg}$ and the global Kuranishi
chart arising from the proof of Proposition 3.1 are

  * •

the (approximations of) the coupling $2$-forms,

  * •

the finite dimensional approximation scheme which regularizes the moduli
spaces,

  * •

the Riemannian metrics required in the definition of the exponential maps on
the base.

Up to equivalence of global Kuranishi chart, these differences can be removed
again from the doubly framed curve construction (cf. [AMS21, Section 6.10] and
[AMS23, Section 4.9]). Because of the isotopic uniqueness of stable smoothing,
this finishes the proof. ∎

###  5.5. Sign comparison

####  5.5.1. Linearized operators

Here we show that the linearization of the nonlinear Cauchy–Riemann equation
into the total space $P$ is “block upper-triangular.” Consider the fibration
$\pi_{P}:P\to B$. The Hamiltonian connection we have specified provides a
decomposition

$TP\cong T^{\rm vert}P\oplus T^{\rm hor}P$ (5.11)

where $T^{\rm vert}P\subset TP$ is a canonical subbundle and $T^{\rm hor}P$ is
isomorphic to the pullback $\pi_{P}^{*}TB$. Indeed we can choose the almost
complex structure $J_{P}$ to be the form

$J_{P}=J_{P}^{\rm vert}\oplus\pi_{P}^{*}J_{B}.$

To linearize the corresponding $\overline{\partial}$-operator, one needs to
specify a (complex-linear) connection on $TP$ and a way to identify nearby
maps (an exponential map); see details in [MS04, Section 3]. As the almost
complex structure $J_{P}$ respects the splitting (5.11), one can choose a
complex linear connection $\nabla^{TP}$ of the form

$\nabla^{TP}:=\nabla^{T^{\rm vert}P}\oplus\pi_{P}^{*}\nabla^{TB}.$

On the other hand, choose a Riemannian metric on $P$ of the form

$g^{TP}=g^{T^{\rm vert}P}\oplus\pi_{P}^{*}g^{TB}.$

Given any point $x\in P$ and $\xi\in T_{x}P$, write $\xi=\xi^{\rm
vert}+\xi^{\rm hor}$. Then define

$\widetilde{\exp}_{x}\xi=\exp_{\exp_{x}^{\rm vert}(\xi^{\rm
vert})}(\hat{\xi}^{\rm hor})$

where $\exp^{\rm vert}$ is the fiberwise exponential map with respect to the
fiberwise metric $g^{T^{\rm vert}P}$ and the vector $\hat{\xi}^{\rm hor}$ is
the image of $\xi^{\rm hor}$ via the parallel transport along the path
$\exp_{x}^{\rm vert}(t\xi^{\rm vert})$ with respect to the connection
$\pi_{P}^{*}\nabla^{TB}$. $\widetilde{\exp}_{x}$ is obviously a local
diffeomorphism. One can see that both the connection and the exponential map
preserve fibers. The connection and the exponential map induce the
linearization of the Cauchy–Riemann operator. One can see the following lemma
is true.

######  Lemma 5.10.

Let $u:\Sigma\to P$ be a smooth map. Then with respect to the horizontal-
vertical decomposition (5.11), the linearized operator $D_{u}$ is of the form

$D_{u}=\left[\begin{array}[]{cc}D_{u}^{\rm vert}&*\\\ 0&D_{u}^{\rm
hor}\end{array}\right].$ (5.12)

If we only deform in the fiber direction, then it is easy to see that the
$D_{u}^{\rm vert}$ coincides with the linearization of the equation for
pseudoholomorphic graphs. Moreover, we can verify that the horizontal part is
the same as the linearized operator on the base.

######  Lemma 5.11.

Denote $\underline{\smash{u}}:=\pi_{P}\circ u:\Sigma\to B$ and let
$D_{\underline{\smash{u}}}:\Gamma(\underline{\smash{u}}^{*}TB)\to\Omega^{0,1}(\underline{\smash{u}}^{*}TB)$
be the linearization corresponding coming from the $J_{B}$-linear connection
$\nabla^{TB}$. Then one has

$D_{u}^{\rm hor}=\pi_{P}^{*}D_{\underline{\smash{u}}}.$

###### Proof.

Let $\xi^{\rm hor}$ be a vector field along $u$ taking values in $T^{\rm
hor}P$. For $t\in[0,1]$, let $\gamma(z,t)=\exp_{u(z)}(t\xi^{\rm hor})$ be the
family of maps from $\Sigma$ to $P$. Let

$\Phi_{t}:T_{\gamma(z,t)}P\to T_{u(z)}P$

be the parallel transport along the curve $\gamma(z,t)$ with respect to the
chosen connection $\nabla^{TP}$. Then by the definition of the linearized
operator and the properties of the connection (and hence the associated
parallel transport)

$\begin{split}&\ D_{u}^{\rm hor}(\xi^{\rm hor})\\\ ({\rm Definition\ of\
}D_{u})\ =&\ {\rm Proj}_{T^{\rm
hor}P}\left(\left.\frac{d}{dt}\right|_{t=0}\Phi_{t}\left(\overline{\partial}_{J_{P}}\gamma(z,t)\right)\right)\\\
(\Phi_{t}\ {\rm is\ complex\ linear})\ =&\ \left({\rm Proj}_{T^{\rm
hor}P}\left(\left.\frac{d}{dt}\right|_{t=0}\Phi_{t}(d_{z}\gamma(z,t))\right)\right)^{0,1}\\\
({\rm horizontal}-{\rm vertical\ decomposition})\ =&\ \left({\rm Proj}_{T^{\rm
hor}P}\left(\left.\frac{d}{dt}\right|_{t=0}\Phi_{t}(d_{z}^{\rm
hor}\gamma(z,t)+d_{z}^{\rm vert}\gamma(z,t))\right)\right)^{0,1}\\\ (\Phi_{t}\
{\rm preserves\ the\ decomposition})\ =&\
\left(\left.\frac{d}{dt}\right|_{t=0}\Phi_{t}(d_{z}^{\rm
hor}\gamma(z,t))\right)^{0,1}\\\ ({\rm property\ of\ }d_{z}^{\rm hor})\ =&\
\left(\left.\frac{d}{dt}\right|_{t=0}\Phi_{t}\left(d_{z}(\exp_{\underline{\smash{u}}(z)}(t\xi^{\rm
hor}))\right)^{\rm hor}\right)^{0,1}\\\ (\nabla^{T^{\rm
hor}P}=\pi_{P}^{*}\nabla^{TB})\ =&\ D_{\underline{\smash{u}}}(\xi^{\rm
hor}).\end{split}$

Here in the second last line,
$(d_{z}(\exp_{\underline{\smash{u}}(z)}(t\xi^{\rm hor})))^{\rm hor}$ is the
horizontal lift. ∎

####  5.5.2. Sign verification

Now we turn to the sign verification. Abbreviate
${\mathcal{M}}={\mathcal{M}}{}_{0,k+2}^{\rm reg}(J_{B},A)$ and
$\tilde{\mathcal{M}}=\overline{\mathcal{M}}{}_{0,k+2}^{\rm
reg}(J_{P},\tilde{A})$. Identify the domain of the pseudocycles
$f_{i}:W_{i}\to B$ resp. $\tilde{f}_{i}:\tilde{W}_{i}\to P$ with their images.
Suppose

$\tilde{q}\in{\rm
ev}_{P}^{-1}\left(\tilde{W}_{0}\times\tilde{W}_{\infty}\times\prod_{i=1}^{k}\tilde{W}_{i}\right)\subset\tilde{\mathcal{M}}$

which projects down to the points

$q\in{\rm
ev}_{B}^{-1}\left(\\{p_{0}\\}\times\\{p_{\infty}\\}\times\prod_{i=1}^{k}W_{i}\right)\subset{\mathcal{M}}.$

If $q$ is represented by a smooth $J_{B}$-holomorphic map $u:S^{2}\to B$ with
$k$ markings ${\bf z}$, then $\tilde{q}$ corresponds to an element in the
graph moduli

$\tilde{\mathcal{M}}^{\rm vert}:=\overline{\mathcal{M}}{}_{0,2;{\bf z}}^{\rm
graph}(u^{*}P,J_{P}|_{u^{*}P},\tilde{A}).$

To identify the signs, it suffices to consider the case that all moduli spaces
are transverse. In this case, the tangent spaces are kernels of the
corresponding linearized operators. Then by Lemma 5.10 and Lemma 5.11, there
is a natural exact sequence

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