# Constructing the relative Fukaya category

Timothy Perutz and Nick Sheridan

###### Abstract

We give a definition of Seidel’s ‘relative Fukaya category’, for a smooth
complex projective variety, under a semipositivity assumption. We use the
Cieliebak–Mohnke approach to transversality via stabilizing divisors. Two
features of our construction are noteworthy: that we work relative to a normal
crossings divisor that supports an effective ample divisor but need not have
ample components; and that our relative Fukaya category is linear over a
certain ring of multivariate power series with _integer_ coefficients.

##  1 Introduction

###  1.1 Definition

Let $X$ be a smooth complex projective variety of dimension $n$, and $D\subset
X$ a simple normal crossings ($\mathsf{snc}$) divisor, i.e., a normal
crossings divisor whose irreducible components $\\{D_{p}\\}_{p\in P}$ are
smooth. We consider $\mathrm{Div}(X,D)\simeq\Z^{P}$, the free abelian group
generated by divisors supported on $D$. Let
$\mathsf{N}\subset\mathrm{Div}(X,D)_{\Q}$ be a rational polyhedral cone—so
$\mathsf{N}=\\{\sum_{i=1}^{s}\lambda_{i}\delta_{i}:\lambda_{i}\geq 0\\}$ for a
set of generators $\delta_{i}\in\mathrm{Div}(X,D)$—satisfying the following
conditions:

  1. (Ample) $\mathsf{N}$ contains an ample class in its interior;

  2. (Semiample) $\mathsf{N}$ is contained in the cone of effective semiample divisors supported on $D$;

  3. (Semipositive) $\mathsf{N}$ contains a ‘homological anticanonical divisor’, i.e. one homologous to the anticanonical divisor modulo torsion.

We recall that a $\Q$-divisor $L$ is called semiample if there is a natural
number $m$ for which $|mL|$ is a basepoint-free linear system, and that
semiample implies nef. The (Semiample) assumption implies that $\mathsf{N}$ is
strictly convex: $\mathsf{N}\cap(-\mathsf{N})=\\{0\\}$.

Declare $(X,D)$ to satisfy the _non-negativity condition_ if such a cone
exists.

If $X$ is simply connected, the non-negativity condition implies that $-K_{X}$
is semiample. Hence $-K_{X}$ is nef; by [CH19], $X$ is therefore a product
$Y\times F$, where $Y$ has trivial canonical bundle and $F$ is rationally
connected.

Intersecting with components of $D$ defines a map

(1) $\displaystyle H_{2}(X,X\setminus D;\Z)$
$\displaystyle\to\Z^{P},\qquad\text{which we denote by}$ $\displaystyle u$
$\displaystyle\mapsto u\cdot D.$

We will arrange that all holomorphic curves we consider have non-negative
intersection number with the divisors in $\mathsf{N}$ (Lemma 22), so their
image under the map (1) lies in the cone
$\operatorname{\mathrm{NE}}\subset\Z^{P}$ dual to $\mathsf{N}$.

Any element $\alpha\in\operatorname{\mathrm{NE}}$ has only finitely many
decompositions $\alpha=\alpha_{1}+\alpha_{2}$ as a sum of two elements of
$\operatorname{\mathrm{NE}}$, since $\alpha(\delta_{i})\in\Z_{\geq 0}$ for all
$i$ where $(\delta_{1},\dots,\delta_{s})$ are generators of $\mathsf{N}$. One
can therefore form the completed monoid ring

(2) $R=\Z\,[\\![\operatorname{\mathrm{NE}}]\\!]$

of infinite series
$\sum_{\alpha\in\operatorname{\mathrm{NE}}}n_{\alpha}T^{\alpha}$ with
$n_{\alpha}\in\Z$. We think of $R$ as roughly analogous to the ring of non-
negative Novikov series used in [FOOO09].

Let $\mathfrak{m}\subset R$ be the ideal generated by $T^{\beta}$ for
$\beta\in\operatorname{\mathrm{NE}}\setminus\\{0\\}$, and equip $R$ with the
$\mathfrak{m}$-adic filtration.

The main construction of this paper is the _relative Fukaya category_
$\EuF(X,D,\mathsf{N})$. It is a $\Z/2$-graded curved filtered $R$-linear
$A_{\infty}$ category (this notion is defined precisely in Section 2.1). It is
independent of all auxiliary choices made in its construction, up to curved
filtered quasi-equivalence (this notion is defined precisely in Section 2.2).
Furthermore, if $\mathsf{N}_{0}\subset\mathsf{N}_{1}$, then there is a curved
filtered quasi-equivalence

(3)
$\EuF(X,D,\mathsf{N}_{1})\otimes_{\Z[\\![\operatorname{\mathrm{NE}}_{1}]\\!]}\Z[\\![\operatorname{\mathrm{NE}}_{0}]\\!]\simeq\EuF(X,D,\mathsf{N}_{0}).$

It was pointed out to us by Marco Castronovo that it should be possible (and
not particularly difficult) to relax the requirement that $D$ be simple normal
crossings.

###  1.2 Properties

The relative Fukaya category satisfies [She20, Assumption 5.1] (up to an
irrelevant change of grading conventions). This concerns the relationship
between the relative Fukaya category and the exact Fukaya category of the
affine variety $X\setminus D$, as defined in [Sei08].

We briefly outline our conventions concerning the exact Fukaya category. It is
a $\Z/2$-graded, $\Z$-linear $A_{\infty}$ category. Its objects are closed,
exact Lagrangian submanifolds $L\subset X\setminus D$, equipped with spin
structures. The morphism spaces $hom^{*}(L_{0},L_{1})$ are free, finite-rank,
$\Z/2$-graded $\Z$-modules generated by the intersection points betwen $L_{0}$
and a Hamiltonian perturbation of $L_{1}$. The $A_{\infty}$ structure maps
count certain pseudoholomorphic discs $u$ in $X\setminus D$.

######  Proposition 1 (Assumption 5.1 of [She20]).

There is a strict isomorphism

$\EuF(X,D,\mathsf{N})\otimes_{R}R/\mathfrak{m}\simeq\EuF(X\setminus D).$

In particular, the objects of $\EuF(X,D,\mathsf{N})$ are identical to those of
$\EuF(X\setminus D)$. The morphism spaces of $\EuF(X,D,\mathsf{N})$ are
obtained from those of $\EuF(X\setminus D)$ by tensoring with $R$: in
particular they are free, finite-rank, $\Z/2$-graded $R$-modules. The
$A_{\infty}$ structure maps count certain pseudoholomorphic discs $u$ in $X$,
weighted by $T^{[u]\cdot D}\in R$.

We remark that the relative Fukaya category also satisfies [She20, Assumptions
5.2 and 5.3].

###  1.3 Bounding cochains

For any curved filtered $A_{\infty}$ category $\EuA$, we can form the
associated uncurved $A_{\infty}$ category $\EuA^{\mathsf{bc}}$ of c-unital
bounding cochains in $\EuA$; and if two curved filtered $A_{\infty}$
categories are curved filtered quasi-equivalent, then their associated
categories of bounding cochains are quasi-equivalent (see Theorem 12).

Let $\BbK$ be a commutative ring,111All our rings, and homomorphisms thereof,
are taken to be unital. equipped with a complete, exhaustive, decreasing
filtration, and a filtered ring homomorphism
$\Z[\\![\operatorname{\mathrm{NE}}]\\!]\to\BbK$. Then
$\EuF(X,D,\mathsf{N})\otimes_{\Z[\\![\operatorname{\mathrm{NE}}]\\!]}\BbK$ is
a curved filtered $\BbK$-linear $A_{\infty}$ category, which is independent of
all choices made in its construction up to curved filtered quasi-equivalence.
In particular, the corresponding category of c-unital bounding cochains is
independent of choices up to quasi-equivalence.

We define an (uncurved) $A_{\infty}$ functor to be a quasi-equivalence if it
induces an equivalence on the level of cohomology, and say that two
$A_{\infty}$ categories are quasi-equivalent if they are connected by a zig-
zag of quasi-equivalences (Definition 10). This is a ‘good’ equivalence
relation if the morphism cochain complexes are cofibrant (see [GPS20, Section
3.1]); this is true for example if the coefficient ring $\BbK$ is a field, but
we have not checked it in general.

The most important example to consider is the following. Let $\Bbbk$ be a
commutative ring and $K\subset\R$ a submonoid. We define the corresponding
Novikov-type ring $\Lambda_{\Bbbk,K}$ to be the completion of the group ring
$\Bbbk[K]$ with respect to the filtration $F^{a}$, for any $a>0$, where
$F^{a}_{\geq\lambda}$ is spanned by the monomials $T^{\lambda^{\prime}}$ with
$\lambda^{\prime}\geq a\cdot\lambda$.

Now let $\kappa$ be a class in the interior of $\mathsf{N}$, so that
$\kappa(\operatorname{\mathrm{NE}})\subset K$; this determines a filtered ring
homomorphism

$\displaystyle\Z[\\![\operatorname{\mathrm{NE}}]\\!]$
$\displaystyle\to\Lambda_{\Bbbk,K},\qquad\text{sending}$ $\displaystyle
T^{\beta}$ $\displaystyle\mapsto T^{\kappa(\beta)}.$

(Recall that the filtration on $\Lambda_{\Bbbk,K}$ depends on a choice of
$a>0$; we choose $a\leq\min_{\beta\in\mathfrak{m}}\kappa(\beta)$, so that the
homomorphism is filtered.)

######  Definition 2.

We define

$\EuF(X,D,\kappa;\Lambda_{\Bbbk,K}):=\EuF(X,D,\mathsf{N})\otimes_{\Z[\\![\operatorname{\mathrm{NE}}]\\!]}\Lambda_{\Bbbk,K}.$

By our previous remark, $\EuF(X,D,\kappa;\Lambda_{\Bbbk,K})$ is independent of
all auxiliary choices made in the definition of $\EuF(X,D,\mathsf{N})$, up to
curved filtered quasi-equivalence. It is also independent of $\mathsf{N}$ up
to curved filtered quasi-equivalence, using equation (3); that is why we omit
$\mathsf{N}$ from the notation. It follows that
$\EuF(X,D,\kappa;\Lambda_{\Bbbk,K})^{\mathsf{bc}}$ is independent of these
choices up to quasi-equivalence.

The data on which $\EuF(X,D,\kappa;\Lambda_{\Bbbk,K})^{\mathsf{bc}}$ depends
(up to quasi-equivalence) are as follows. We need a pair $(X,D)$ such that
there exists an effective homological anticanonical divisor supported on $D$,
together with a class $\kappa\in H^{2}(X,X\setminus D;\R)\simeq\R^{P}$ such
that $\kappa_{p}>0$ for all $p$, and $\sum_{p}\kappa_{p}\cdot D_{p}$ is ample.
In order to make the construction, we need to choose a rational polyhedral
cone $\mathsf{N}$ satisfying the (Ample), (Semiample), and (Semipositive)
conditions, and containing $\kappa$ in its interior. For this it suffices to
observe that $\kappa$ lies in the interior of the convex cone of effective
semiample classes supported on $D$ (using the fact that ampleness is an open
condition, and ample implies semiample), and the homological anticanonical
divisor lies in the same cone.

###  1.4 Why we can work over $\Z$

In general, the Fukaya category can only be defined over a ring containing
$\Q$, because the moduli spaces of holomorphic discs may be orbifolds. The
orbifold points arise when the holomorphic disc admits a nontrivial finite
symmetry group. A smooth holomorphic disc won’t have such a symmetry group;
however in general the definition of the Fukaya category involves virtual
counts over moduli spaces of stable discs. A stable disc may admit such a
symmetry group, arising from symmetries of a spherical component. In order to
define the Fukaya category over $\Z$, one must rule out the appearance of such
spherical components in the moduli spaces. This can be done when the
symplectic manifold is exact [Sei08], or more generally, spherically positive
[FOOO13] (i.e., any $J$-holomorphic sphere $u$ with $c_{1}(u)\leq 0$ is
constant).

In this paper we show that the relative Fukaya category can be defined over
$\Z[\\![\operatorname{\mathrm{NE}}]\\!]$ when $(X,D)$ satisfies the non-
negativity condition. This encompasses the important Calabi–Yau case
($c_{1}=0$), which is not spherically positive. The non-negativity condition
ensures that all $J$-holomorphic spheres $u$ satisfy $c_{1}(u)\geq 0$. The
technique of ruling out such spheres in this case goes back to [HS95]: all
relevant moduli spaces of holomorphic discs have dimension $\leq 1$. Bubbling
off a sphere with $c_{1}>0$ subtracts at least two from the dimension of the
disc moduli space, leaving it with negative dimension and hence empty; on the
other hand spheres with $c_{1}=0$ sweep out a space of codimension $4$, and
hence generically avoid all one-dimensional moduli spaces of discs as they
sweep out a space of dimension $3$.

A number of authors have studied Lagrangian Floer theory and homological
mirror symmetry in finite characteristic in recent years, see e.g. [LP12,
EL15, LP17, LP20]. It has not possible to carry on this study for compact
Calabi–Yau varieties of dimension $\geq 3$, because the available
constructions of the Fukaya category relied on a characteristic-zero
assumption (c.f. [FOOO09, She15, Fuk17, CW19]). Our construction in this paper
removes this obstacle.

###  1.5 Applications

Homological mirror symmetry was proved for ‘generalized Greene–Plesser
mirrors’ in [SS21, Theorem D], contingent on a definition of the Fukaya
category satisfying certain properties (e.g., Abouzaid’s split-generation
criterion). This paper provides such a definition, namely
$\EuF(X,D,\kappa;\Lambda_{\C,\R})^{\mathsf{bc}}$, and the relevant properties
will be established in future work.222More precisely, because we only consider
embedded Lagrangians in this paper, our construction is only adequate to the
purposes of [SS21, Theorem D] under the ‘embeddedness condition’ [_op. cit._ ,
Definition 1.3].

The proof of homological mirror symmetry for generalized Greene–Plesser
mirrors goes through almost verbatim over a coefficient ring
$\Lambda_{\Bbbk,K}$ where $\Bbbk$ is a field of characteristic zero and
$K\subset\R$ is a subgroup containing the image of
$\kappa(\operatorname{\mathrm{NE}})$. The obstruction to extending the proof
to the case of finite characteristic is that the deformation theory techniques
used in [She20], on which [SS21] depends, use the characteristic-zero
assumption in a fundamental way; however this could possibly be circumvented
by using a different approach to the deformation theory (for example, [Sei15,
Lemma 3.9] works for a field of arbitrary characteristic). The characteristic-
zero assumption is also used in [SS21] in the verification of smoothness of
the mirror, but this can presumably be remedied by modifying the ‘MPCP
condition’ depending on the characteristic.

It was shown in [GPS15] that homological mirror symmetry for Calabi–Yau
varieties implied closed-string (Hodge-theoretic) mirror symmetry. The proof
required a definition of the relative Fukaya category satisfying certain
properties (e.g., the existence of a cyclic open–closed map respecting
connections). Again, this paper provides the necessary definition, namely
$\EuF(X,D,\kappa;\Lambda_{\C,\Z})^{\mathsf{bc}}$, but the relevant properties
are relegated to future work.

###  1.6 Outline

Section 2 defines the relevant notion of ‘curved filtered quasi-equivalence’
of curved filtered $A_{\infty}$ categories, gives the construction of the
uncurved $A_{\infty}$ category $\EuA^{\mathsf{bc}}$ of c-unital bounding
cochains on a curved filtered $A_{\infty}$ category $\EuA$, and proves that if
$\EuA$ and $\EuB$ are curved filtered quasi-equivalent then
$\EuA^{\mathsf{bc}}$ and $\EuB^{\mathsf{bc}}$ are quasi-equivalent. Section 3
introduces the notions of relative Kähler form, system of divisors $V$, and
the class of almost-complex structures we will work with (namely, those $J$
which preserve a chosen system of divisors, and are sufficiently close to the
integrable complex structure). Section 4 establishes basic results about the
moduli spaces of stable $J$-holomorphic spheres, which will allow us to rule
out their bubbling off a moduli space of pseudoholomorphic discs by the
argument sketched in Section 1.4 above. Section 5 introduces the relevant
moduli spaces of pseudoholomorphic discs. Section 6 gives the definition of
the relative Fukaya category, and proves Proposition 1. Section 7 proves
independence of the definition on the auxiliary choices made in its
construction, up to curved filtered quasi-equivalence, as well as the
‘independence of $\mathsf{N}$’ result described in equation (3).

Acknowledgments: N.S. is supported by a Royal Society University Research
Fellowship, ERC Starting Grant 850713 – HMS, and the Simons Collaboration on
Homological Mirror Symmetry. T.P. is partially supported by NSF CAREER grant
1455265. N.S. is grateful to Paul Seidel for very helpful conversations,
including suggesting the crucial notion of a system of divisors.

##  2 Curved filtered $A_{\infty}$ categories

###  2.1 Definition

All of our rings will be unital. In this section we will work over $\BbK$,
which will be a commutative ring equipped with an exhaustive, complete,
decreasing filtration $F_{\geq\bullet}\BbK$.

One key example of such a ring $\BbK$ will be
$\Z[\\![\operatorname{\mathrm{NE}}]\\!]$, equipped with the
$\mathfrak{m}$-adic filtration. The other key class of examples will be the
Novikov-type rings $\Lambda_{\Bbbk,K}$ defined in the Introduction.

We will regularly pass from $\BbK$ to its associated graded ring
$\mathrm{Gr}_{*}\BbK$; for instance, from
$\Z[\\![\operatorname{\mathrm{NE}}]\\!]$ to
$\mathrm{Gr}_{*}\Z[\\![\operatorname{\mathrm{NE}}]\\!]\cong\Z[\operatorname{\mathrm{NE}}]$.

######  Definition 3.

A _curved filtered $\BbK$-linear $A_{\infty}$ algebra_ is a $\Z/2$-graded
$\BbK$-module $A$, equipped with an exhaustive, complete, decreasing
filtration $F_{\geq\bullet}$, compatible with the module structure in the
sense that

$F_{\geq\lambda_{1}}\BbK\cdot F_{\geq\lambda_{2}}A\subset
F_{\geq\lambda_{1}+\lambda_{2}}A.$

This comes with $\BbK$-linear $A_{\infty}$ operations

$\mu^{k}:A^{\otimes_{\BbK}k}\to A$

for all $k\geq 0$, of degree $2-k$. These are required to respect the
filtration in the sense that

$\mu^{k}(F_{\geq\lambda_{1}}A,\ldots,F_{\geq\lambda_{k}}A)\in
F_{\geq\lambda_{1}+\ldots+\lambda_{k}}A,$

furthermore have $\mu^{0}\in F_{\geq 1}A$, and satisfy the curved $A_{\infty}$
equations.

Associated to a curved filtered $\BbK$-linear $A_{\infty}$ algebra $A$, there
is an uncurved $\Z/2\oplus\Z$-graded $\mathrm{Gr}_{*}\BbK$-linear $A_{\infty}$
algebra $\mathrm{Gr}_{*}A$ whose underlying $\mathrm{Gr}_{*}\BbK$-module is
the associated graded $\mathrm{Gr}_{*}A$, with the structure maps induced by
those for $A$. The structure map $\mu^{k}$ has degree $(2-k,0)$. The notion of
c(ohomological)-unitality [Sei08, Section 2a] makes sense for
$\mathrm{Gr}_{*}A$, as it is uncurved. If a c-unit exists, it necessarily has
degree $(0,0)$.

######  Definition 4.

We say that $A$ is c-unital if $\mathrm{Gr}_{*}A$ is.

We similarly define the notion of a curved filtered $A_{\infty}$ category, and
of c-unitality of such a category. Henceforth we will implicitly assume that
our curved filtered $A_{\infty}$ algebras and categories are c-unital.

######  Example 2.1.

The relative Fukaya category $\EuF(X,D,\mathsf{N})$, equipped with the
$\mathfrak{m}$-adic filtration, is a curved filtered
$\Z[\\![\operatorname{\mathrm{NE}}]\\!]$-linear $A_{\infty}$ category. The
fact that it is c-unital follows from the fact that
$\mathrm{Gr}_{*}\EuF(X,D,\mathsf{N})\simeq\EuF(X\setminus
D;\Z)\otimes_{\Z}\mathrm{Gr}_{*}\Z[\\![\operatorname{\mathrm{NE}}]\\!]$, and
the argument for c-unitality of $\EuF(X\setminus D)$ (see, e.g., [She16,
Section 2.4]) works with coefficients in an arbitrary commutative ring.

######  Example 2.2.

Let $\Z[\\![\operatorname{\mathrm{NE}}]\\!]\to\BbK$ be a filtered ring
homomorphism; then
$\EuF(X,D,\mathsf{N})\otimes_{\Z[\\![\operatorname{\mathrm{NE}}]\\!]}\BbK$ is
naturally a curved filtered $\BbK$-linear $A_{\infty}$ category. The argument
for c-unitality is the same as before.

###  2.2 Bounding cochains and c-unitality

######  Definition 5.

Let $A$ be a curved filtered $A_{\infty}$ algebra; we say that $a\in A$ is a
_Maurer–Cartan element_ if $a\in F_{\geq 1}A$ has degree $1$ and satisfies the
Maurer–Cartan equation:

$\sum_{k\geq 0}\mu^{k}(a,\ldots,a)=0.$

(Note that the infinite sum converges by completeness.)

Given a Maurer–Cartan element $a\in A$, we can use it to deform the
$A_{\infty}$ algebra $A$ to an uncurved, not-necessarily-c-unital $A_{\infty}$
algebra $A^{a}=(A,\mu^{*}_{a})$, in a standard way (see [She20, Section 2.7]).

######  Definition 6.

We say that a Maurer–Cartan element $a\in A$ is c-unital if $A^{a}$ is
c-unital and furthermore, there exists a cocycle $e^{a}\in F_{\geq 0}A^{a}$
representing the c-unit, whose class in $\mathrm{Gr}_{0}A$ represents the
c-unit of $\mathrm{Gr}_{*}A$.

######  Definition 7.

Let $\EuA$ be a curved filtered $A_{\infty}$ category. We define an uncurved
$A_{\infty}$ category $\EuA^{\mathsf{bc}}$ whose objects are pairs $(A,a)$,
where $a\in hom^{*}(A,A)$ is a c-unital Maurer–Cartan element. The morphism
spaces are

$hom_{\EuA^{\mathsf{bc}}}((A,a),(B,b)):=hom_{\EuA}(A,B).$

The formula for the $A_{\infty}$ structure maps, which get deformed by the
Maurer–Cartan elements, can be found in [Sei08, Equation (3.20)]; it converges
by completeness.

Since our convention is that all $A_{\infty}$ categories should be c-unital,
the following Lemma is required to show that Definition 7 is valid:

######  Lemma 8.

The $A_{\infty}$ category $\EuA^{\mathsf{bc}}$ is c-unital.

######  Proof 2.3.

By definition, each object $(A,a)$ comes equipped with $e^{a}\in F_{\geq
0}hom_{\EuA^{\mathsf{bc}}}((A,a),(A,a))$ which represents a c-unit for that
algebra, and whose class in $\mathrm{Gr}_{0}hom_{\EuA}(A,A)$ represents a
c-unit for $hom_{\mathrm{Gr}_{*}\EuA}(A,A)$. It suffices to show that the maps

$hom_{H^{*}(\EuA^{\mathsf{bc}})}((A,a),(B,b))\xrightarrow{[e^{a}]\bullet(-)}hom_{H^{*}(\EuA^{\mathsf{bc}})}((A,a),(B,b))\xleftarrow{(-)\bullet[e^{b}]}hom_{H^{*}(\EuA^{\mathsf{bc}})}((A,a),(B,b))$

are equal to the identity.

We will focus on the leftmost map, as the argument for the rightmost one is
identical. We start by showing that the map is an isomorphism. This follows
from a spectral sequence comparison argument, for the spectral sequence
induced by the filtration. Indeed, the map induced on the $E_{1}$ page is

$hom_{H^{*}(\mathrm{Gr}_{*}\EuA)}(A,B)\xrightarrow{[e^{a}]\bullet(-)}hom_{H^{*}(\mathrm{Gr}_{*}\EuA)}(A,B).$

This map is an isomorphism, in fact the identity, because $[e^{a}]$ represents
the c-unit in $\mathrm{Gr}_{*}\EuA$. Since the filtration is exhaustive and
complete, the Eilenberg–Moore comparison theorem [Wei94, Theorem 5.5.11]
implies that the original map $[e^{a}]\bullet(-)$ is an isomorphism.

Now observe that $[e^{a}]\bullet[e^{a}]=[e^{a}]$, because $e^{a}$ is a c-unit
for $H^{*}(hom_{\EuA^{\mathsf{bc}}}((A,a),(A,a)))$ by definition. For any
$f\in hom_{H^{*}(\EuA^{\mathsf{bc}})}((A,a),(B,b))$, we have

$[e^{a}]\bullet\left([e^{a}]\bullet f-f\right)=[e^{a}]\bullet f-[e^{a}]\bullet
f=0,$

and therefore $[e^{a}]\bullet f-f=0$, because $[e^{a}]\bullet(-)$ is an
isomorphism by our previous argument, and in particular injective. This
completes the proof.

###  2.3 Filtered curved quasi-equivalence

######  Definition 9.

A curved filtered $A_{\infty}$ functor between two curved filtered
$A_{\infty}$ categories is a curved $A_{\infty}$ functor $G$, respecting the
filtration in the obvious way, having curvature $G^{0}\in F_{\geq 1}$, such
that the uncurved $A_{\infty}$ functor $\mathrm{Gr}_{*}G$ is c-unital. (Note
that the curved $A_{\infty}$ functor equations converge by completeness.)

######  Definition 10.

We say that an uncurved $\Z/2\oplus\Z$-graded $A_{\infty}$ functor is a quasi-
equivalence if it induces an equivalence on the level of $\Z/2\oplus\Z$-graded
cohomological categories.333Note that the grading plays a role in the notion
of isomorphism of objects, and hence in the notion of essential surjectivity.
Specifically, a quasi-isomorphism is required to have degree $(0,0)$. We say
that two uncurved $\Z/2\oplus\Z$-graded $A_{\infty}$ categories are quasi-
equivalent if they are connected by a zig-zag of quasi-equivalences.

######  Definition 11.

A curved filtered $A_{\infty}$ functor $G$ is called a curved filtered quasi-
equivalence if $\mathrm{Gr}_{*}G$ is a $\Z/2\oplus\Z$-graded quasi-
equivalence. We say that two curved filtered $A_{\infty}$ categories are
curved filtered quasi-equivalent if they are connected by a zigzag of curved
filtered quasi-equivalences.

A curved filtered $A_{\infty}$ functor $G:\EuA\to\EuB$ induces a not-
necessarily-c-unital uncurved $A_{\infty}$ functor $G^{\mathsf{bc}}$, which
maps from $\EuA^{\mathsf{bc}}$ to the not-necessarily-c-unital category of
bounding cochains on $\EuB$, cf. [She20, Section 2.7].

######  Theorem 12.

If $G:\EuA\to\EuB$ is a curved filtered quasi-equivalence, then
$G^{\mathsf{bc}}$ defines an uncurved $A_{\infty}$ functor
$\EuA^{\mathsf{bc}}\to\EuB^{\mathsf{bc}}$ (in particular, the image of a
c-unital bounding cochain under $G^{\mathsf{bc}}$ is c-unital). Furthermore,
this functor is a quasi-equivalence.

The proof of Theorem 12 will be given in Section 2.5, but we record here an
immediate corollary:

######  Corollary 13.

If $\EuA$ and $\EuB$ are curved filtered quasi-equivalent, then
$\EuA^{\mathsf{bc}}$ and $\EuB^{\mathsf{bc}}$ are quasi-equivalent.

###  2.4 The triangle algebra

In this section we introduce a well-known construction (see [Kel03, Section
4.5]) which will be used in the proof of Theorem 12.

######  Definition 14.

Given $A_{\infty}$ algebras $A$, $B$, and an $A$-$B$ bimodule $M$, we can form
a new $A_{\infty}$ algebra

$\mathcal{T}(M):=A\oplus M[-1]\oplus B$

by defining

$\mu^{k}(c_{1},\ldots,c_{k}):=\left\\{\begin{array}[]{ll}\mu^{k}_{A}(c_{1},\ldots,c_{k})&\text{if
all $c_{i}\in A$;}\\\ \mu^{k}_{B}(c_{1},\ldots,c_{k})&\text{if all $c_{i}\in
B$;}\\\
\mu^{i|1|j}_{M}(c_{1},\ldots,c_{i-1};c_{i};c_{i+1},\ldots,c_{k})&\text{whenever
the expression makes sense;}\\\ 0&\text{otherwise}.\end{array}\right.$

We can think of elements of $\mathcal{T}(M)$ as arranged in a lower-triangular
two-by-two matrix:

$\mathcal{T}(M)=\left(\begin{array}[]{cc}A&0\\\ M&B\end{array}\right).$

######  Definition 15.

Let $M$ be an $A$-$B$ bimodule, and $f\in M[-1]\subset\mathcal{T}(M)$ have
degree $1$. Observe that the Maurer–Cartan equation for $f$ reduces to
$\mu^{0|1|0}(f)=0$, i.e., the equation for $f$ to be closed. Thus we can form
the $A_{\infty}$ algebra $\mathcal{T}(M)^{f}$, by deforming $\mathcal{T}(M)$
with the Maurer–Cartan element $f$. It is clear that there is no issue with
convergence, as $f$ can appear at most once in any $A_{\infty}$ product. It is
clear that the projections to $A$ and $B$ define strict $A_{\infty}$
homomorphisms

(4) $A\xleftarrow{\pi_{A}}\mathcal{T}(M)^{f}\xrightarrow{\pi_{B}}B.$

######  Definition 16.

If $M$ is an $A$-$B$ bimodule, we say that $f\in M$ is a _quasi-isomorphism_
if $\mu^{0|1|0}(f)=0$, and the chain maps

$A\xrightarrow{\mu^{1|1|0}(-;f;)}M\xleftarrow{\mu^{0|1|1}(;f;-)}B$

are quasi-isomorphisms of cochain complexes.

######  Lemma 17.

If $f\in M$ is a quasi-isomorphism, then the projections (4) both induce
isomorphisms on cohomology; in particular, $A$ is quasi-isomorphic to $B$.

######  Proof 2.4.

The kernel of $\pi_{B}$ is the cochain complex

$Cone\left(A\xrightarrow{\mu^{1|1|0}(-;f;)}M\right)$

which is acyclic, because $f$ is a quasi-isomorphism. Similarly for $\pi_{A}$.

######  Corollary 18.

Suppose that $X$ and $Y$ are quasi-isomorphic objects in an $A_{\infty}$
category (i.e., they become isomorphic objects in the cohomological category).
Then the $A_{\infty}$ algebras $hom(X,X)$ and $hom(Y,Y)$ are quasi-isomorphic.

######  Proof 2.5.

This follows from Lemma 17, where we take $A=hom(X,X)$, $B=hom(Y,Y)$,
$M=hom(X,Y)$, and $f\in M$ a cochain-level representative of the isomorphism
between $X$ and $Y$ in the cohomological category.

###  2.5 Proof of Theorem 12

Let $A$ and $B$ be curved filtered $A_{\infty}$ algebras and $M$ a filtered
$A$-$B$ bimodule. If $a\in A$ and $b\in B$ are solutions of the respective
Maurer–Cartan equations, we denote by $A^{a}$ and $B^{b}$ the corresponding
uncurved $A_{\infty}$ algebras; we can also define an $A^{a}$-$B^{b}$ bimodule
${}^{a}M^{b}$ in a similar way.

Note that $\mathrm{Gr}_{*}M$ is a $\Z/2\oplus\Z$-graded
$\mathrm{Gr}_{*}A$-$\mathrm{Gr}_{*}B$ bimodule.

######  Lemma 19.

Suppose that $m_{0}\in\mathrm{Gr}_{0}M$ is a quasi-isomorphism, and $b\in B$ a
solution of the Maurer–Cartan equation. Then there exists $a\in A$ a solution
of the Maurer–Cartan equation, and a closed element $m\in F_{\geq
0}{}^{a}M^{b}$ representing $m_{0}$. If $b$ is c-unital, then so is $a$.

######  Proof 2.6.

Observe that the conditions required of $a$ and $m$ are equivalent to

$t:=\left(\begin{array}[]{cc}a&0\\\
m&b\end{array}\right)\in\left(\begin{array}[]{cc}F_{\geq 1}A&0\\\ F_{\geq
0}M&F_{\geq 1}B\end{array}\right)\subset\mathcal{T}(M)$

satisfying the Maurer–Cartan equation. This is not quite the same as being a
Maurer–Cartan element for $\mathcal{T}(M)$, as $m\notin F_{\geq 1}M$.
Nevertheless, the Maurer–Cartan equation still converges, because $m$ is
lower-triangular.

We solve for $a$ and $m$, inductively in the filtration. Suppose we have
solved to order $k-1$; i.e., the Maurer–Cartan equation for $t$ holds modulo
$F_{\geq k}$. Let

$\mathbb{T}_{k}=\left(\begin{array}[]{cc}\mathbb{A}_{k}&0\\\
\mathbb{M}_{k}&\mathbb{B}_{k}\end{array}\right):=\sum_{s}\mu^{s}(t,\ldots,t)\in\mathrm{Gr}_{k}\mathcal{T}(M).$

The fact that $b$ is a bounding cochain implies that $\mathbb{B}_{k}=0$. The
$A_{\infty}$ equations

$\sum\mu^{*}(t,\ldots,t,\mu^{*}(t,\ldots,t),t,\ldots,t)=0$

imply that

$\left(\begin{array}[]{cc}\mu^{1}\left(\mathbb{A}_{k}\right)&0\\\
\mu^{0|1|0}\left(\mathbb{M}_{k}\right)+\mu^{2}_{0}(m_{0},\mathbb{A}_{k})&0\end{array}\right)=0,$

which is equivalent to

$\mu^{1}_{\mathrm{Gr}_{*}\mathcal{T}(M)^{m_{0}}}(\mathbb{T}_{k})=0.$

Similarly, adding $a_{k}\in F_{\geq k}A$ to $a$ and $m_{k}\in F_{\geq k}M$ to
$m$ changes $\mathbb{T}_{k}$ by

$\mathbb{T}_{k}\mapsto\mathbb{T}_{k}+\mu^{1}_{\mathrm{Gr}_{k}\mathcal{T}(M)^{m_{0}}}\left(\begin{array}[]{cc}a_{k}&0\\\
m_{k}&0\end{array}\right).$

We now observe that because $\mathbb{B}_{k}=0$, $\mathbb{T}_{k}$ lies in
$\ker(\pi_{\mathrm{Gr}_{k}B})$. This complex is acyclic, because $m_{0}$ is a
quasi-isomorphism; we have shown that $\mathbb{T}_{k}$ is closed, and that we
are free to modify it by adding exact terms. Therefore we can choose $a_{k}$
and $m_{k}$ so that $\mathbb{T}_{k}$ vanishes. The inductive construction of
$a$ and $m$ converges, by completeness.

It remains to check that if $b$ is c-unital, then so is $a$. For this we
consider the strict $A_{\infty}$ homomorphisms

$A^{a}\xleftarrow{\pi_{A^{a}}}\mathcal{T}(M)^{m}\xrightarrow{\pi_{B^{b}}}B^{b}.$

These are filtered quasi-isomorphisms, by a spectral sequence comparison
argument. Therefore the unit $e^{b}\in F_{\geq 0}B^{b}$ admits a lift to a
closed element in $F_{\geq 0}\mathcal{T}(M)^{m}$, whose projection to $F_{\geq
0}A^{a}$ we denote by $e^{a}$. This represents a c-unit for $A^{a}$, using the
fact that $\pi_{A^{a}}$ and $\pi_{B^{b}}$ induce algebra isomorphisms on the
level of cohomology.

######  Proof 2.7 (Proof of Theorem 12).

Because $\mathrm{Gr}_{*}G$ is fully faithful, so is $G^{\mathsf{bc}}$, by a
comparison argument for the spectral sequence induced by the filtration. It
follows immediately that $G^{\mathsf{bc}}$ is c-unital. It remains to check
that $G^{\mathsf{bc}}$ is essentially surjective.

In other words, given $(Y,y)\in\ob\EuB^{\mathsf{bc}}$, we must find
$(X,x)\in\ob\EuA^{\mathsf{bc}}$ such that $G^{\mathsf{bc}}(X,x)\simeq(Y,y)$.
Because $\mathrm{Gr}_{*}G$ is a $\Z/2\oplus\Z$-graded quasi-equivalence, and
in particular essentially surjective, there exists $X\in\ob\EuA$ such that
$GX$ is quasi-isomorphic to $Y$ in $\mathrm{Gr}_{*}\EuB$. Let
$f_{0}\in\mathrm{Gr}_{0}hom^{*}_{\EuB}(X,Y)$ be a quasi-isomorphism.

We now apply Lemma 19, with $A=hom_{\EuA}(X,X)$, $B=hom_{\EuB}(Y,Y)$,
$M=(G\otimes\id)^{*}(hom_{\EuB}(X,Y))$, $b=y$, $m_{0}=f_{0}$. The outcome is a
c-unital Maurer–Cartan element $a\in A$, together with a closed element $m\in
F_{\geq 0}{}^{a}M^{b}$ representing $m_{0}$. Setting $x=a$, and $f=m\in
F_{\geq 0}hom_{\EuB^{\mathsf{bc}}}(G^{\mathsf{bc}}(X,x),(Y,y))$, we find that
$f$ is closed and represents $f_{0}$.

We now consider the maps

$\displaystyle hom((Y,y),G^{\mathsf{bc}}(X,x))$
$\displaystyle\xrightarrow{(-)\bullet[f]}hom((Y,y),(Y,y))\qquad\text{and}$
$\displaystyle hom((Y,y),G^{\mathsf{bc}}(X,x))$
$\displaystyle\xrightarrow{[f]\bullet(-)}hom(G^{\mathsf{bc}}(X,x),G^{\mathsf{bc}}(X,x)),$

where all morphism spaces are taken in the cohomological category
$H^{*}(\EuB^{\mathsf{bc}})$. Using the fact that $f_{0}$ is a quasi-
isomorphism, a comparison argument for the spectral sequence induced by the
filtration shows that these maps are isomorphisms. In particular we can solve
$[g]\bullet[f]=[e^{y}]$ and $[f]\bullet[h]=[G^{\mathsf{bc}}(e^{x})]$, from
which one easily deduces that $[h]=[g]$, so that $f$ defines a quasi-
isomorphism between $G^{\mathsf{bc}}(X,x)$ and $(Y,y)$.

##  3 Target geometry

Our construction of $\EuF(X,D,\mathsf{N})$ will depend on various auxiliary
choices, which we now start to describe. We start by choosing a finite set of
divisors $\\{\EuV_{w}\\}_{w\in W}\in\mathrm{Div}(X,D)$ generating the convex
cone $\mathsf{N}$, such that the linear systems $|\EuV_{w}|$ are basepoint
free. That such a set exists follows from the (Semiample) assumption on
$\mathsf{N}$; that it may be taken to be finite follows from the assumption
that $\mathsf{N}$ is rational polyhedral.

###  3.1 Relative Kähler form

We recall the notion of a _relative Kähler form_ from [She20, Definition 3.2].
It is a Kähler form $\omega$ on $X$, equipped with a Kähler potential
$h:X\setminus D\to\R$ having a prescribed form near $D$. The Kähler potential
$h$ determines a Liouville one-form $\alpha:=-d^{c}h$, so that any regular
sublevel set $\\{h\leq c\\}$ is a Liouville domain.

There is a corresponding cohomology class $[\omega;\alpha]\in
H^{2}(X,X\setminus D;\R)\simeq\R^{P}$. Such a cohomology class is represented
by a relative Kähler form if and only if the corresponding $\R$-divisor
$\sum_{p}a_{p}\cdot D_{p}$ is ample, and has all $a_{p}>0$ [She20, Lemma 3.3].

By the (Ample) condition on $\mathsf{N}$, there exists a relative Kähler form
with cohomology class

$[\omega;\alpha]=\sum_{w}\lambda_{w}\cdot\EuV_{w}\qquad\text{with
$\lambda_{w}\geq 0$}.$

We choose such a relative Kähler form, whose cohomology class lies in the
interior of $\mathsf{N}$. We also choose a regular sublevel set $\\{h\leq
c\\}$ of the Kähler potential, which contains all critical points of $h$.

###  3.2 System of divisors

We will denote $\\{h>c\\}:=X\setminus\\{h\leq c\\}$. Note that
$D\subset\\{h>c\\}$ is a deformation retract, so any divisor $V$ contained in
$\\{h>c\\}$ has a class

$[V]\in H_{2n-2}(D;\Z)\simeq\Z^{P}.$

######  Definition 20 (Definition 3.5 of [She20]).

A _system of divisors_ is a $\mathsf{snc}$ divisor $V\subset\\{h>c\\}$ with
irreducible components $\\{V_{q}\\}_{q\in Q}$, such that
$[V_{q}]=[\EuV_{w(q)}]$ where the function $w:Q\to W$ has fibres of size $\geq
n+1$.

There exists a system of divisors by Bertini’s theorem [She20, Lemma 3.8]. We
choose such a system of divisors $V$.

###  3.3 Almost complex structures

######  Definition 21 (cf. Definition 4.1 of [She20]).

We denote the space of $\omega$-tame almost complex structures on $X$ by
$\EuJ(X)$. We denote the space of $\omega$-tame almost complex structures
which make each $V_{q}$ into an almost complex submanifold by
$\EuJ(X,V)\subset\EuJ(X)$.

######  Lemma 22 (Lemma 4.2 of [She20]).

If $J\in\EuJ(X,V)$, then any $J$-holomorphic curve
$u:(\Sigma,\partial\Sigma)\to(X,X\setminus D)$ satisfies $[u]\cdot\EuV_{w}\geq
0$. In particular, $[u]\cdot D\in\operatorname{\mathrm{NE}}$.

######  Proof 3.1.

If $u$ is not contained inside $V_{q}$, then $u\cdot V_{q}\geq 0$ by
positivity of intersection (cf. [MS04, Exercise 2.6.1] or [CM07, Prop. 7.1]).
There are $\geq n+1$ divisors $V_{q}$ with $[V_{q}]=[\EuV_{w}]$, and $u$ can’t
be contained in all of them because their common intersection is empty by
transversality.

We recall that $J\in\EuJ(X)$ is called _semipositive_ if every $J$-holomorphic
sphere has nonnegative first Chern number [MS04, Definition 6.4.5].

######  Corollary 23.

Any $J\in\EuJ(X,V)$ is semipositive.

######  Proof 3.2.

Let $u:\mathbb{CP}^{1}\to X$ be a $J$-holomorphic sphere. By the
(Semipositivity) assumption on $\EuV$, we have

$c_{1}(TX)(u)=\sum_{w}\kappa_{w}\cdot([u]\cdot\EuV_{w})$

for some $\kappa_{w}\geq 0$. The result now follows by Lemma 22.

We introduce notation, for any subset $K\subset Q$:

(5) $V_{K}:=\bigcap_{q\in K}V_{q},\qquad V^{K}:=\bigcup_{q\notin K}V_{q}.$

(We set $V_{\emptyset}:=X$.)

######  Lemma 24 (Adjunction formula).

If $u\in H_{2}(V_{K})$, then

(6) $c_{1}(TV_{K})(u)=c_{1}(TX)(u)-\sum_{q\in K}u\cdot V_{q}.$

######  Proof 3.3.

Follows from the adjunction formula, together with the isomorphism

$NV_{K}\simeq\bigoplus_{q\in K}NV_{q}|_{V_{K}}$

which holds by transversality.

The following lemma will be used to show that holomorphic curves contained
inside $V$ do not have ‘excess dimension’:

######  Lemma 25 (Lemma 4.5 of [She20]).

If $J\in\EuJ(X,V)$, then any $J$-holomorphic curve $u$ contained inside
$V_{K}$ satisfies $c_{1}(TV_{K})(u)\leq c_{1}(TX)(u)$.

######  Proof 3.4.

Follows from Lemmas 24 and 22.

######  Definition 26.

We define $\EuJ^{\mathsf{max}}(X,V)\subset\EuJ(X,V)$ to be the subset of
almost-complex structures $J$ satisfying $\alpha=-dh\circ J$ on the set
$\\{c\leq h\leq\delta\\}$, for some $\delta>0$.

The superscript in $\EuJ^{\mathsf{max}}$ refers to the applicability of the
maximum principle, as in the following lemma:

######  Lemma 27 (Lemma 4.3 of [She20]).

Let $J\in\EuJ^{\mathsf{max}}(X,V)$, and

$u:(\Sigma,\partial\Sigma)\to(X,\\{h\leq c\\})$

be a $J$-holomorphic map, such that $[u]=0$ in $H_{2}(X,X\setminus D)$. Then
$u$ is contained in $\\{h\leq c\\}$.

######  Proof 3.5.

Follows from the integrated maximum principle of [AS10, Lemma 7.2]. Namely,
find a sequence $(\epsilon_{k})$ of numbers in $(0,\delta)$, decreasing to
zero, such that for each $k$, $\Sigma$ intersects $\\{h=c+\epsilon_{k}\\}$
transversely. For any $k$, let
$u_{k}:(\Sigma_{k},\partial\Sigma_{k})\to(\\{h\geq
c+\epsilon_{k}\\},\\{h=c+\epsilon_{k}\\})$ be the part of $u$ mapping to
$\\{h\geq c+\epsilon_{k}\\}$. We have $[u_{k}]=[u]=0$ in $H_{2}(X,X\setminus
D)$, by assumption; hence, $[\omega;\alpha](u_{k})=0$. It follows by
definition that

$\omega(u_{k})=\alpha(\partial u_{k}).$

But now, $\omega(u_{k})\geq 0$ because $J$ is $\omega$-tame, and
$\alpha(\partial u_{k})\leq 0$ because $\alpha=-dh\circ J$ along
$\\{h=c+\epsilon_{k}\\}$. So $\omega(u_{k})=0$, hence $u_{k}$ is constant, so
$u$ is contained in $\\{h\leq c+\epsilon_{k}\\}$. Hence $\im u\subset\\{h\leq
c\\}$.

######  Lemma 28 (Lemma 4.4 of [She20]).

Let $J\in\EuJ(X,V)$, and $u:(\Sigma,\partial\Sigma)\to(X,L)$ be a non-constant
$J$-holomorphic curve with boundary on a closed, exact Lagrangian submanifold
$L\subset\\{h\leq c\\}$. Then $u$ intersects $V$, and in particular $u\cdot
V_{q}>0$ for some $q$ by positivity of intersection.

######  Proof 3.6.

We prove the contrapositive: suppose that $u$ does not intersect $V$, so
$u\cdot V_{q}=0$ for all $q$. Because the classes $[V_{q}]=[\EuV_{w(q)}]$ span
a convex cone with nonempty interior by the (Ample) assumption, they span
$H_{2n-2}(D;\Q)$. It follows that $[u]=0$, and hence $u\subset\\{h\leq c\\}$
by Lemma 27. Because $L$ is exact, this implies that $u$ has zero energy and
hence is constant.

###  3.4 Space of almost complex structures

We introduce a class of perturbations of the integrable complex structure
$J_{0}\in\EuJ(X,V)$, which will be sufficiently large to achieve
transversality for all holomorphic curves we consider. Set

(7) $\displaystyle\EuY=\EuY(X,V)$ $\displaystyle:=\left\\{Y\in
C^{\infty}\left(\End TX\right):YJ_{0}+J_{0}Y=0;\,Y(TV_{q})\subset TV_{q}\text{
for all $q\in Q$}\right\\},$ (8) $\displaystyle\EuY_{*}=\EuY_{*}(X,V)$
$\displaystyle:=\left\\{Y\in\EuY:\|Y\|_{C^{0}}<\log\frac{3}{2}\right\\}.$

Here $\|Y\|_{C^{0}}$ means $\sup_{x\in X}\|Y_{x}\|$, where $\|Y_{x}\|$ is the
operator norm, taken with respect to the Kähler metric on $TX$. We will
usually fix $V$ and drop it from the notation. Note that $\EuY_{*}$ is a
convex (hence contractible) subset of the vector space $\EuY$, and is open
with respect to the $C^{\infty}$ topology.

Any $Y\in\EuY_{*}$ determines an almost complex structure
$J_{Y}:=J_{0}\exp(Y)$. The bound on $\|Y\|_{C^{0}}$ implies that $\exp(Y)$ is
defined. The two conditions in the definition of $\EuY$ imply respectively
that $J_{Y}$ is an almost complex structure on $X$; and that it makes each
component $V_{q}$ into an almost complex submanifold. The bound on
$\|Y\|_{C^{0}}$ also implies that $J_{Y}$ tames $\omega$ (to see this, write
$e^{Y}=I+(e^{Y}-I)$ and observe that $\|e^{Y}-I\|\leq e^{\|Y\|}-1\leq 1/2$).
Therefore $J_{Y}\in\EuJ(X,V)$.

For positive integers $\ell$, we write $\EuY^{\ell}$ and $\EuY_{*}^{\ell}$ for
the counterparts of these spaces in which $Y$ is $C^{\ell}$ rather than
$C^{\infty}$.

###  3.5 Extending almost complex structures

Observe that $V^{K}\cap V_{K}$ (notation from (5)), the points of $V_{K}$
which lie in $V_{q}$ for some $q\notin K$, is a $\mathsf{snc}$ divisor in
$V_{K}$. We define $\EuY_{K}=\EuY(V_{K},V^{K}\cap V_{K})$, and observe that
there are restriction maps $r_{K}\colon\EuY\to\EuY_{K}$; similarly define
$\EuY_{K}^{\ell}$ and the restriction map
$r_{K}\colon\EuY^{\ell}\to\EuY_{K}^{\ell}$.

######  Lemma 29.

Take $Y\in\EuY_{*,K}^{\ell}$ (where $1\leq\ell\leq\infty$), and $x\in V_{K}$.
Then there exists $\widetilde{Y}\in\EuY_{*}^{\ell}$ such that
$r_{K}(\widetilde{Y})=Y$ on a neighborhood of $x$ in $V_{K}$.

######  Proof 3.7.

We may choose local complex coordinates $(z_{1},\ldots,z_{n})$ in a
neighborhood $U$ of $x$, such that $U\cap V$ is identified with
$\\{z_{1}\cdots z_{i+j}=0\\}$, $V_{K}$ with $\\{z_{1}=\ldots=z_{i}=0\\}$, and
$x$ with $0$. The complex coordinates give a trivialization
$(TX|_{U},J_{0})\simeq(\C^{n}\times U,J_{\C^{n}})$. The complex coordinates
also define a splitting $TU\simeq TV_{K}\oplus
NV_{K}\simeq\underline{\C^{n-i}}\oplus\underline{\C^{i}}$. On $U$, we define
the infinitesimal almost complex structure $\widehat{Y}$ to equal $Y$ on the
$TV_{K}$ summand and zero on the $NV_{K}$ summand. Notice that
$\widehat{Y}(T_{y}V_{q})\subset T_{y}V_{q}$ for $y\in U\cap V_{q}$. The
$C^{0}$ norm of $\widehat{Y}$ (over $U$) is bounded by $\|Y\|_{C^{0}}$. Define
$\widetilde{Y}=\chi\cdot\widehat{Y}$ on $U$, where $\chi\colon X\to[0,1]$ is a
smooth function that is compactly supported in $U$ and equal to $1$ near $x$;
extend $\widetilde{Y}$ by zero to $X$. Since the constraints on $\tilde{Y}$
are fibrewise linear, this extension has the desired properties.

######  Lemma 30.

The restriction maps $r_{K}\colon\EuY_{*}^{\ell}\to\EuY_{*,K}^{\ell}$ are
surjective for $1\leq\ell\leq\infty$.

######  Proof 3.8.

Given $Y_{K}\in\EuY_{*,K}^{\ell}$, and given $x\in V_{K}$, there exists, by
Lemma 29, $\widetilde{Y}_{x}\in\EuY_{*}^{\ell}$ such that
$r_{K}(\widetilde{Y}_{x})$ agrees with $Y_{K}$ over $U_{x}$, for some
neighborhood $U_{x}$ of $x$ in $V_{K}$. Choose a finite set of points
$x_{\alpha}$ such that $\bigcup_{\alpha}U_{x_{\alpha}}=V_{K}$. Let
$\\{\rho_{\alpha}:V_{K}\to[0,1]\\}$ be a smooth partition of unity subordinate
to the open cover $\\{U_{\alpha}\\}$, and let
$\widetilde{\rho}_{\alpha}:X\to[0,1]$ be a smooth extension of $\rho_{\alpha}$
to $X$, for each $\alpha$. If we then set
$Y=\sum_{\alpha}\widetilde{\rho}_{\alpha}\widetilde{Y}_{x_{\alpha}}$, then
$Y\in\EuY_{*}^{\ell}$ and $r_{K}(Y)=Y_{K}$ as required.

##  4 Holomorphic spheres

###  4.1 Bubble trees

For $J\in\EuJ(X,V)$, let us consider the topological space

(9) $\overline{\mathcal{M}}_{k}(A;J)$

of stable $J$-holomorphic curves with $k$ marked points, with homology class
$A\in H_{2}(X;\Z)$, in Gromov’s topology. We follow the notation of [MS04,
Sections 5 and 6] with minor changes.

The topological type of the domain of a stable curve in
$\overline{\mathcal{M}}_{k}(A;J)$ is prescribed by a $k$-labelled tree
$(T,E,\Lambda)$: $T$ is the set of vertices $\alpha$ corresponding to
components $\Sigma_{\alpha}$ of the domain, $E\subset T\times T$ the set of
edges $\alpha E\beta$ corresponding to points
$z_{\alpha\beta}\in\Sigma_{\alpha}$ at which $\Sigma_{\alpha}$ is attached to
$z_{\beta\alpha}\in\Sigma_{\beta}$, and $\Lambda:\\{1,\ldots,k\\}\to T$ is a
function which prescribes the distribution of marked points among the
components. We associate a function

$T\to\mathsf{subsets}(Q):\quad\alpha\mapsto K_{\alpha}$

to each stable curve, where $K_{\alpha}\subset Q$ is defined to be the set of
divisors $V_{q}$ in which the image of the component
$u_{\alpha}:\Sigma_{\alpha}\to X$ of the stable curve is contained. (Because
$J\in\EuJ(X,V)$, every smooth $J$-holomorphic curve is either contained in
$V_{q}$ or intersects it positively in finitely many points, for all $q\in
Q$.) The homology class of the components of the stable curve is the
collection of spherical homology classes

$A_{\alpha}\in\im\left(\pi_{2}(V_{K_{\alpha}})\to
H_{2}(V_{K_{\alpha}};\Z)\right).$

The images of these classes in $H_{2}(X;\Z)$ sum to $A$, and they satisfy a
stability condition: if $A_{\alpha}=0$ then $\Sigma_{\alpha}$ has at least
three marked points.

We define the _combinatorial type_ of an element of
$\overline{\mathcal{M}}_{k}(A;J)$ to be the five-tuple
$\Gamma=(T,E,\Lambda,\\{K_{\alpha}\\},\\{A_{\alpha}\\})$. We decompose the
moduli space into a disjoint union according to combinatorial type:

(10)
$\overline{\mathcal{M}}_{k}(A;J)=\coprod_{\Gamma}\mathcal{M}_{\Gamma}(J).$

We can describe $\mathcal{M}_{\Gamma}(J)$ as the quotient of a space of stable
_maps_ by a reparametrization group. Explicitly, let

$\mathcal{M}(\Gamma,J)=\\{\mathbf{u}=\\{u_{\alpha}\\}_{\alpha\in T}\colon
u_{\alpha}:\mathbb{CP}^{1}_{\alpha}\to V_{K_{\alpha}}\;J\text{-holomorphic,
}[u_{\alpha}]=A_{\alpha}\\}.$

Let

$Z(\Gamma)\subset(\mathbb{CP}^{1})^{E}\times(\mathbb{CP}^{1})^{k}$

denote the set of tuples
$\mathbf{z}:=\left(\\{z_{\alpha\beta}\in\mathbb{CP}^{1}_{\alpha}\\}_{\alpha
E\beta},\\{z_{i}\in\mathbb{CP}^{1}_{\alpha_{i}}\\}_{1\leq i\leq k}\right)$ of
marked points, where the marked points on each sphere
$\mathbb{CP}^{1}_{\alpha}$ are distinct. We denote

(11) $(X,V)^{\Gamma}:=\prod_{\alpha E\beta}V_{K_{\alpha}},$

so that there is an evaluation map

(12) $\displaystyle\mathrm{ev}^{E}:\mathcal{M}\left(\Gamma,J\right)\times
Z(\Gamma)$ $\displaystyle\to$ $\displaystyle(X,V)^{\Gamma},$
$\displaystyle(\mathbf{u},\mathbf{z})$ $\displaystyle\mapsto$
$\displaystyle\\{u_{\alpha}(z_{\alpha\beta})\\}.$

We denote

(13)
$\Delta^{\Gamma}:=\left\\{(x_{\alpha\beta})\in(X,V)^{\Gamma}:x_{\alpha\beta}=x_{\beta\alpha}\right\\}\subset(X,V)^{\Gamma}.$

We then denote

$\widetilde{\mathcal{M}}_{\Gamma}\left(J\right):=\left(\mathrm{ev}^{E}\right)^{-1}\left(\Delta^{\Gamma}\right).$

We have
$\mathcal{M}_{\Gamma}(J):=\widetilde{\mathcal{M}}_{\Gamma}(J)/G_{\Gamma}$,
where $G_{\Gamma}:=\prod_{\alpha}\mathrm{PSL}(2,\C)$ is the reparametrization
group.

We denote by

(14) $\mathcal{M}^{*}_{k}(A;J)\subset\overline{\mathcal{M}}_{k}(A;J)$

the subset of _simple_ stable curves (see [MS04, Definition 6.1.1]), and
similarly $\mathcal{M}^{*}_{\Gamma}(J)\subset\mathcal{M}_{\Gamma}(J)$,
$\widetilde{\mathcal{M}}^{*}_{\Gamma}(J)\subset\widetilde{\mathcal{M}}_{\Gamma}(J)$,
and $\mathcal{M}^{*}(\Gamma,J)\subset\mathcal{M}(\Gamma,J)$. Explicitly, the
latter is the subset of tuples $\mathbf{u}$ such that
$u_{\alpha}(\mathbb{CP}^{1})\neq u_{\beta}(\mathbb{CP}^{1})$ for
$\alpha\neq\beta$ such that $u_{\alpha}$ and $u_{\beta}$ are non-constant, and
non-constant $u_{\alpha}$ are simple.

######  Definition 31.

An element $u\in\widetilde{\mathcal{M}}^{*}_{\Gamma}(J)$ is called _regular_
if

  * •

$u_{\alpha}:\mathbb{CP}^{1}\to V_{K_{\alpha}}$ is regular, as a
$J$-holomorphic curve inside $V_{K_{\alpha}}$.

  * •

The evaluation map $\mathrm{ev}^{E}$ is transverse to $\Delta^{\Gamma}$.

The moduli space $\widetilde{\mathcal{M}}^{*}_{\Gamma}(J)$ is called regular
if every element in it is regular. An element $Y\in\EuY_{*}$ is called regular
if $\widetilde{\mathcal{M}}^{*}_{\Gamma}(J_{Y})$ is regular for every
$\Gamma$. The set of regular $Y$ is denoted
$\EuY^{\mathrm{reg}}_{*}\subset\EuY_{*}$.

######  Lemma 32.

If $Y\in\EuY^{\mathrm{reg}}_{*}$, then $\mathcal{M}^{*}_{\Gamma}(J_{Y})$ is a
smooth manifold of dimension $\mathrm{dim}(\Gamma)$, where

(15) $\displaystyle\half\mathrm{dim}(\Gamma)$ $\displaystyle=k+\sum_{\alpha\in
T}\left(n-|K_{\alpha}|-3\right)+c_{1}(TV_{K_{\alpha}})(A_{\alpha})+\sum_{\alpha
E\beta}\left(|K_{\alpha}\cap K_{\beta}|-n+2\right)$
$\displaystyle=k+n-2+\sum_{\alpha\in
T}\left(c_{1}(TV_{K_{\alpha}})(A_{\alpha})-|K_{\alpha}|-1\right)+\sum_{\alpha
E\beta}|K_{\alpha}\cap K_{\beta}|.$

In the sums $\sum_{\alpha E\beta}$, we take $E$ to be the set of _undirected_
edges.

######  Proof 4.1.

Follows from standard dimension counting, cf. [MS04, Theorem 6.2.6 (i)]. The
second equality uses the Euler characteristic of the tree.

######  Lemma 33.

The dimension (15) is bounded above by

$\displaystyle\mathrm{dim}(\Gamma)$ $\displaystyle\leq$ $\displaystyle
2\left(k+n+c_{1}(TX)(A)-3-|E|-\max_{\alpha}|K_{\alpha}|\right)$
$\displaystyle=$
$\displaystyle\mathrm{dim}(\mathcal{M}_{k}(A))-2|E|-2\max_{\alpha}|K_{\alpha}|$

(again, we recall that $|E|$ is the number of undirected edges).

######  Proof 4.2.

The inequality follows by combining two inequalities. First,

$c_{1}(TV_{K_{\alpha}})(A_{\alpha})\leq c_{1}(TX)(A_{\alpha})$

by Lemma 25. Second,

$\sum_{\alpha}|K_{\alpha}|-\sum_{\alpha E\beta}|K_{\alpha}\cap
K_{\beta}|\geq\max_{\alpha}|K_{\alpha}|.$

To see this, we first choose a vertex $\alpha^{\prime}$ such that
$|K_{\alpha^{\prime}}|$ is maximal; then we pair each vertex $\alpha$ with the
edge $\alpha E\beta$ that points towards the vertex $\alpha^{\prime}$. The
term $|K_{\alpha^{\prime}}|$ in the sum does not have a partner, and the other
terms pair off as $|K_{\alpha}|-|K_{\alpha}\cap K_{\beta}|\geq 0$, giving the
result.

###  4.2 Parametrized moduli spaces

We will need to consider a parametric generalization of the framework of the
previous section; cf. [Sei08, Section 9h], [MS04, Section 6.7]. Suppose that
$\EuS$ is a smooth manifold and $\mathbf{Y}:\EuS\to\EuY_{*}$ a smooth
function. We denote

$\mathcal{M}_{\Gamma}(J_{\mathbf{Y}}):=\coprod_{s\in\EuS}\mathcal{M}_{\Gamma}(J_{\mathbf{Y}(s)}).$

It comes with a map

$\pi_{\EuS}:\mathcal{M}_{\Gamma}(J_{\mathbf{Y}})\to\EuS$

recording the parameter. In slightly more detail, we let $\mathcal{B}(V_{K})$
be the Fréchet manifold of smooth maps $\mathbb{CP}^{1}\to V_{K}$, and define
$\widetilde{\mathcal{M}}(\Gamma,J_{\mathbf{Y}})\subset\EuS\times\prod_{\alpha\in
T}\mathcal{B}(V_{K_{\alpha}})$ to be the subspace of tuples
$\left(s,\\{u_{\alpha}\\}\right)$ defined by the Cauchy–Riemann equation
$\bar{\partial}_{J_{\mathbf{Y}}}=0$ and the discrete condition
$(u_{\alpha})_{*}[\mathbb{CP}^{1}]=A_{\alpha}$. Proceed to construct
$\widetilde{\mathcal{M}}_{\Gamma}(J_{\mathbf{Y}})$, its quotient
$\mathcal{M}_{\Gamma}(J_{\mathbf{Y}})$, and
$\mathcal{M}^{*}_{\Gamma}(J_{\mathbf{Y}})\subset\mathcal{M}_{\Gamma}(J_{\mathbf{Y}})$,
by analogy with their non-parametric counterparts.

A first use of parametric moduli spaces (albeit in an infinite-dimensional
context) is in proving that there are plenty of regular almost complex
structures:

######  Lemma 34.

The subset $\EuY^{\mathrm{reg}}_{*}\subset\EuY_{*}$ is comeagre.

######  Proof 4.3.

The proof is that of [MS04, Theorem 6.2.6 (ii)], with adaptations. The main
adaptation is that Lemma 30 shows we have enough perturbations along $V_{K}$
to achieve transversality for curves contained entirely inside $V_{K}$.

By the argument of [MS04, proof of Theorem 6.2.6 (ii)], it suffices to prove
that the subset $\EuY_{*}^{\ell,\mathrm{reg}}\subset\EuY^{\ell}_{*}$ of
regular elements is comeagre for each $\ell\geq 2$.

Fix $\ell$ and a set of indices $K$. The space $\EuY_{K}^{\ell}$ has a
comeagre subset $\EuY_{K,*}^{\ell,\mathrm{reg}}\subset\EuY_{*,K}^{\ell}$ whose
elements $Y$ are defined by the condition that all simple $J_{Y}$-spheres
$u\colon\C P^{1}\to V_{K}$, not contained in $V^{K}\cap V_{K}$, are regular
(standard adaptation of the argument of [MS04, Theorem 3.1.5]).

Define

$\EuY_{*}^{\ell,\mathrm{reg\,for\,}K}=\\{Y\in\EuY_{*}^{\ell}:\,r_{K}(Y)\in\EuY_{*,K}^{\ell,\mathsf{reg}}\\}.$

The restriction map $r_{K}\colon\EuY_{*}^{\ell}\to\EuY_{*,K}^{\ell}$ is
surjective by Lemma 30. The preimage of a comeagre subset under a continuous
linear surjection is again comeagre; hence
$\EuY_{*}^{\ell,\mathrm{reg\,for\,}K}$ is comeager in $\EuY_{*}^{\ell}$. Thus
$\bigcap_{K}\EuY_{*}^{\ell,\mathrm{reg\,for\,}K}$ is also comeager. This deals
with the first bullet point in the definition of regularity (Definition 31).

One next needs to prove that the evaluation map

$\ev^{E}\colon\mathcal{M}^{*}(\Gamma,\EuJ_{\ell})\times
Z(\Gamma)\to(X,V)^{\Gamma}$

(cf. (12)) is transverse to the diagonal $\Delta^{\Gamma}$; here
$\EuJ^{\ell}=\\{J_{Y}:Y\in\EuY^{\ell}_{*}\\}$ is the family of almost complex
structures parametrized by $\EuY^{\ell}_{*}$, and
$\mathcal{M}^{*}(\Gamma,\EuJ_{\ell})=\bigcup_{J\in\EuJ^{\ell}}\mathcal{M}^{*}(\Gamma,J)$.
This can be accomplished by an adaptation of [MS04, Prop. 6.2.8]. The proof in
_op. cit._ hinges on [MS04, Prop. 3.4.3], and it suffices to use the tweak to
the latter proposition indicated in [MS04, Remark 3.4.8]. With that point in
place, the proof of [MS04, Theorem 6.2.6 (ii)] goes through.

We return to the discussion of parametric moduli spaces. An element
$u\in\widetilde{\mathcal{M}}^{*}_{\Gamma}(J_{\mathbf{Y}})$ is called regular
if its cut out transversely by its defining equations. That is, each
$u_{\alpha}$ is parametrized-regular, i.e., a transverse zero of the
Cauchy–Riemann operator on $\mathcal{B}(V_{K_{\alpha}})\times\EuS$, and the
evaluation map

$\displaystyle(\pi_{\EuS},\mathrm{ev}^{E}):\mathcal{M}\left(\Gamma,J_{\mathbf{Y}}\right)\times
Z(\Gamma)$ $\displaystyle\to$ $\displaystyle\EuS\times(X,V)^{\Gamma}$

is transverse to $\EuS\times\Delta^{\Gamma}$.

The moduli space $\widetilde{\mathcal{M}}^{*}_{\Gamma}(J_{\mathbf{Y}})$ is
called regular if every element in it is regular; and $\mathbf{Y}$ is called
regular if $\widetilde{\mathcal{M}}^{*}_{\Gamma}(J_{\mathbf{Y}})$ is regular
for every $\Gamma$. The proof of Lemma 34 generalizes easily to show that the
set of regular $\mathbf{Y}$ is comeagre in $C^{\infty}(\EuS,\EuY_{*})$. In
this situation, $\mathcal{M}^{*}_{\Gamma}(J_{\mathbf{Y}})$ is a smooth
manifold of dimension $\dim(\Gamma)+\dim(\EuS)$.

##  5 Holomorphic discs

###  5.1 Deligne–Mumford moduli space

If $\ell\geq 3$, we consider the Deligne–Mumford moduli space
$\overline{\EuR}_{\ell}$ of stable curves of genus zero with interior marked
points labelled $z_{1},\ldots,z_{\ell}$. It is a complex manifold, and carries
a universal family $\overline{\EuS}_{\ell}\to\overline{\EuR}_{\ell}$ with
sections $\\{z_{i}\\}_{i=1,\ldots,\ell}$ given by the interior marked points.
We denote the restriction of the universal family to the smooth locus by
$\EuS_{\ell}\to\EuR_{\ell}$.

If $k+2\ell\geq 2$, we consider the Deligne–Mumford moduli space
$\overline{\EuR}_{k,\ell}$ of stable discs equipped with boundary marked
points labelled $\zeta_{0},\ldots,\zeta_{k}$ (in order) and interior marked
points labelled $z_{1},\ldots,z_{\ell}$. It is a smooth manifold with corners
and carries a universal family
$\overline{\EuS}_{k,\ell}\to\overline{\EuR}_{k,\ell}$ with sections
$\\{z_{i}\\}_{i=1,\ldots,\ell}$ and $\\{\zeta_{i}\\}_{i=0,\ldots,k}$ given by
the marked points. We denote the restriction of the universal family to the
smooth locus by $\EuS_{k,\ell}\to\EuR_{k,\ell}$.

The strata of the Deligne–Mumford compactification

$\overline{\EuR}_{k,\ell}=\coprod_{\Gamma}\EuR^{\Gamma}_{k,\ell}$

are indexed by certain decorated trees $\Gamma$. Specifically, each $\Gamma$
consists of a tree with vertex set $V=V_{\mathsf{sph}}\sqcup
V_{\mathsf{disc}}$ decomposed into ‘sphere’ and ‘disc’ vertices; a set of
‘finite’ edges $E=E_{\mathsf{sph}}\sqcup E_{\mathsf{disc}}\subset V\times V$
decomposed into ‘disc’ edges (those connecting disc vertices) and ‘sphere’
edges (the rest); a set of ‘semi-infinite sphere edges’ indexed by
$\\{1,\ldots,\ell\\}$, which is equivalent to an $\ell$-marking of $(V,E)$; a
set of ‘semi-infinite disc edges’ indexed by $\\{0,\ldots,k\\}$ and attached
to disc vertices, which is equivalent to a $k+1$-marking of
$(V_{\mathsf{disc}},E_{\mathsf{disc}})$; the graph
$(V_{\mathsf{disc}},E_{\mathsf{disc}})$ together with the semi-infinite disc
edges is required to be a tree, and must come equipped with an embedding in
the plane so that the semi-infinite edges are labelled in order from $0$ to
$k$ around the boundary. For each vertex $\alpha$, we denote by
$\ell_{\alpha}$ the number of sphere edges incident to $\alpha$; and for each
disc vertex $\alpha$, we denote by $k_{\alpha}$ the number of disc edges
incident to $\alpha$, minus one. $\Gamma$ is required to be _stable_ , which
means that $k_{\alpha}+2\ell_{\alpha}\geq 2$ for all $\alpha$.

The stratum indexed by such a $\Gamma$ has the form

$\EuR^{\Gamma}_{k,\ell}\simeq\prod_{\alpha\in
V_{\mathsf{sph}}}\EuR_{\ell_{\alpha}}\times\prod_{\alpha\in
V_{\mathsf{disc}}}\EuR_{k_{\alpha},\ell_{\alpha}},$

where the diffeomorphism is determined by a choice of ordering of the sphere
edges incident to each vertex. Specifically, given such an ordering, each
finite sphere edge $\alpha E\beta$ determines a section $z_{\alpha\beta}$ of
the universal family $\EuS_{\alpha}\to\EuR_{\alpha}$ of spheres or discs; each
finite disc edge determines a section $\zeta_{\alpha\beta}$ of the universal
family $\EuS_{\alpha}\to\EuR_{\alpha}$ of discs. Letting
$\mathrm{pr}_{\alpha}:\EuR^{\Gamma}\to\EuR_{\alpha}$ denote the projection,
the restriction of the universal family to such a stratum has the form

$\EuS^{\Gamma}_{k,\ell}\simeq\left(\coprod_{\alpha\in
V_{\mathsf{sph}}}\mathrm{pr}_{\alpha}^{*}\EuS_{\ell_{\alpha}}\sqcup\coprod_{\alpha\in
V_{\mathsf{disc}}}\mathrm{pr}_{\alpha}^{*}\EuS_{k_{\alpha},\ell_{\alpha}}\right)/\sim$

where ‘$\sim$’ identifies
$\mathrm{pr}_{\alpha}^{*}z_{\alpha\beta}\sim\mathrm{pr}_{\beta}^{*}z_{\beta\alpha}$
for all finite spherical edges $\alpha E\beta$ and
$\zeta_{\alpha\beta}\sim\zeta_{\beta\alpha}$ for all finite disc edges $\alpha
E\beta$.

We now observe that the group $\mathrm{Sym}(\ell)$ acts on
$\overline{\EuS}_{k,\ell}\to\overline{\EuR}_{k,\ell}$ by relabelling the
interior marked points, and the action preserves the smooth locus
$\EuS_{k,\ell}\to\EuR_{k,\ell}$.

######  Lemma 35.

The action of $\mathrm{Sym}(\ell)$ on $\EuR_{k,\ell}$ is free, and therefore
properly discontinuous.

######  Proof 5.1.

Since $\mathrm{Sym}(\ell)$ is finite and $\EuR_{k,\ell}$ is Hausdorff, a free
action is properly discontinuous. To say that the action is free is to say
that if $f:\Sigma\to\Sigma$ is an isomorphism which fixes the boundary
markings and permutes the interior markings, then $f$ is the identity. Note
that $f^{\ell!}$ must act as the identity on all marked points, hence be equal
to the identity, since all domains we consider have trivial automorphism
group. But the group of automorphisms of the disc fixing a single boundary
point has no nontrivial roots of the identity;444To prove this claim, observe
that if the boundary fixed-point set
$P=\\{t\in\partial\mathbb{D}:\alpha(t)=t)\\}$ of an automorphism $\alpha$ of
the disc $\mathbb{D}$ has cardinality $|P|\geq 3$, then $\alpha$ is the
identity; if $|P|\in\\{1,2\\}$ then $\alpha$ preserves each connected
component $I$ of $\partial\mathbb{D}\setminus P$, because $\alpha$ preserves
the boundary orientation. For each component $I$, either $\alpha(t)<t$ for all
$t\in I$, or $\alpha(t)>t$ for all $t\in I$ ($<$ is the order on an oriented
interval); either property persists in iterates. therefore $f$ itself is the
identity as required.

The argument extends to show that, for $\Sigma$ a stable disc, if
$f:\Sigma\to\Sigma$ is an isomorphism which fixes the boundary markings and
permutes the interior markings, then $f$ acts as the identity on each disc
component of $\Sigma$. Thus the action of $\mathrm{Sym}(\ell)$ on strata of
$\EuR_{k,\ell}$ without spherical components is again free and properly
discontinuous.

The action of $\mathrm{Sym}(\ell)$ on $\overline{\EuR}_{k,\ell}$ is not free,
in general. For example, if there is a spherical component with exactly one
node and exactly two marked points, there is an automorphism interchanging
those marked points. There is a quotient orbifold-with-corners
$\overline{\EuR}_{k,\ell}/\mathrm{Sym}(\ell)$—locally the quotient of a
manifold with corners by the action of a finite group—parametrizing stable
discs whose interior marked points are unordered. The interior stratum
$\EuR_{k,\ell}/\mathrm{Sym}(\ell)$ is both a manifold and a fine moduli space;
and the same goes for the codimension-1 strata, since they parametrize stable
discs with two disc components but no spherical components.

###  5.2 Strip-like ends

For any $r\in\overline{\EuR}_{k,\ell}$, we denote the corresponding fibre of
the universal family by $\Sigma_{r}$. We denote by

(16) $\Sigma^{\circ}_{r}\subset\Sigma_{r}$

the complement of the boundary marked points and boundary nodes. Recall the
notion of a set of _strip-like ends_ for $\Sigma^{\circ}_{r}$: it consists of
holomorphic embeddings

$\displaystyle\epsilon_{i}:Z^{\pm}$ $\displaystyle\to\Sigma^{\circ}_{r}\qquad
i=0,1,\ldots,k$

where $Z^{\pm}:=\R_{\pm}\times[0,1]$ and the sign is $-$ for $i=0$ and $+$
otherwise. These maps are required to have disjoint images, be proper, send
the boundary intervals $\R_{\pm}\times\\{0,1\\}$ to $\partial\Sigma_{r}$, and
limit to $\zeta_{i}$ as $s\to\pm\infty$ (see [Sei08, Section 8d]). We will
furthermore require that our strip-like ends be disjoint from the interior
marked points.

We denote by $\overline{\EuS}^{\circ}_{k,\ell}\subset\overline{\EuS}_{k,\ell}$
the union of subsets $\Sigma^{\circ}_{r}$ over all $r$. We recall the notion
of a _universal choice of strip-like ends_ , which consists of smooth maps

$\epsilon_{i}:\overline{\EuR}_{k,\ell}\times
Z^{\pm}\to\overline{\EuS}^{\circ}_{k,\ell}$

for all $k,\ell$ and $i=0,\ldots,k$, which restrict to a choice of strip-like
ends on each fibre of the universal family; and what it means for such a
choice to be _consistent_ [Sei08, Section 9g]. We will require our universal
choice of strip-like ends to be consistent, and furthermore
$\mathrm{Sym}(\ell)$-equivariant; a straightforward modification of [Sei08,
Lemma 9.3] shows that such a choice exists.

Given an equivariant consistent universal choice of strip-like ends, we may
choose a _universal thick-thin decomposition_ (cf. [Sei08, Remark 9.1] – note
that this is an additional choice). This consists of an open set

(17) $\overline{\EuS}^{thin}_{k,\ell}\subset\overline{\EuS}^{\circ}_{k,\ell},$

whose intersection with each fibre is parametrized as the union of strip-like
ends for each disc component of the fibre, together with ‘gluing regions’
$\epsilon_{e}([0,l_{e}]\times[0,1])$, which are also required to be
$\mathrm{Sym}(\ell)$-equivariant and disjoint from the marked points. (In
Seidel’s terminology, we may take $\overline{\EuS}^{thin}_{k,\ell}$ to be the
union of all strip-like ends together with the union of gluing regions lying
over subsets where the gluing parameters are sufficiently small.)

###  5.3 Floer data

We set

$\EuY^{\mathsf{max}}:=\\{Y\in\EuY_{*}:Y|_{h\geq c}=0\\}.$

Note that if $Y\in\EuY^{\mathsf{max}}_{*}$, then
$J_{Y}\in\EuJ^{\mathsf{max}}(X,V)$.

######  Definition 36.

A _Lagrangian brane_ is a closed, exact Lagrangian submanifold
$L\subset\\{h<c\\}$, equipped with a spin structure.

Let $\EuH\subset C^{\infty}(X;\R)$ denote the subset of functions supported in
$\\{h<c\\}$. For each pair of Lagrangian branes $(L_{0},L_{1})$ we choose a
smooth function

$H_{01}:[0,1]\to\EuH.$

We assume that the time-1 flow of the corresponding Hamiltonian vector field
$X_{H_{01}(t)}$, when applied to $L_{0}$, makes it transverse to $L_{1}$.
Thus, the set of Hamiltonian chords $y$ from $L_{0}$ to $L_{1}$ is finite.

Following [She20, Appendix B] (which is based on [Sei08, Section 12f]), to
each chord $y$ we can associate the orientation line $o_{y}$, which is a
$\Z/2$-graded free $R$-module of rank $1$.

######  Definition 37.

The $hom$-spaces in the relative Fukaya category are the direct sums of all
orientation lines associated to Hamiltonian chords:

$hom^{*}_{\EuF(X,D,\mathsf{N})}(L_{0},L_{1}):=\bigoplus_{y}o_{y}.$

They are $\Z/2$-graded free $R$-modules.

Next, for each pair of Lagrangian branes $(L_{0},L_{1})$ we consider a smooth
function

$\displaystyle Y_{01}:[0,1]$
$\displaystyle\to\EuY_{*}^{\max{}},\qquad\text{giving rise to}$ $\displaystyle
J_{01}:[0,1]$ $\displaystyle\to\EuJ^{\max{}}(X,V),\quad\text{via
$J_{01}:=J_{Y_{01}}$.}$

The corresponding equation for Floer trajectories between chords $y_{0}$ and
$y_{1}$ in the Liouville domain $\\{h\leq c\\}$ is

$\displaystyle u:\R\times[0,1]$ $\displaystyle\to\\{h\leq c\\}$
$\displaystyle\partial_{s}u+J_{01}(t)(\partial_{t}u-X_{H_{01}(t)}\circ u)$
$\displaystyle=0$ $\displaystyle u(s,i)$ $\displaystyle\in
L_{i}\qquad\text{for $i=0,1$}$ $\displaystyle\lim_{s\to+\infty}u(s,\cdot)$
$\displaystyle=y_{1}$ $\displaystyle\lim_{s\to-\infty}u(s,\cdot)$
$\displaystyle=y_{0}.$

For a comeagre subset of the space of choices of $Y_{01}$, the moduli space of
such Floer trajectories is regular [FHS95]. We choose such a regular $Y_{01}$,
for each pair of Lagrangian branes $(L_{0},L_{1})$ equipped with $H_{01}$.

Taken together, the choice of $(H_{01},Y_{01})$ for each pair $(L_{0},L_{1})$
is called a choice of _Floer data_.

###  5.4 Perturbation data

A _Lagrangian labelling_ $\mathbf{L}$ is a tuple of Lagrangian branes
$L_{0},\ldots,L_{k}$ (with $k\geq 0$). Given a family of stable discs
$\overline{\EuR}_{k,\ell}$, a Lagrangian labelling $\mathbf{L}$ will be
thought of as a labelling of the boundary components of discs in the family by
the branes $L_{i}$. Let $r\in\overline{\EuR}_{k,\ell}$, $\Sigma_{r}$ the fibre
of the universal family over $r$, and $\Sigma^{\circ}_{r}$ the complement of
all boundary marked points and nodes. Following [Sei08, Section 8e], we make
the following:

######  Definition 38.

A _perturbation datum_ for $\Sigma_{r}^{\circ}$ equipped with boundary
labelling $\mathbf{L}$ consists of a pair $P=(Y,K)$ where

$Y\in C^{\infty}(\Sigma^{\circ}_{r},\EuY_{*}),\quad
K\in\Omega^{1}(\Sigma^{\circ}_{r},\EuH),$

satisfying

$Y(\epsilon(s,t))=Y_{\ell,r}(t)\qquad\epsilon^{*}K=H_{\ell,r}(t)dt$

over every thin region (strip-like end or gluing region) parametrized by
$\epsilon$, having boundary labels $L_{\ell},L_{r}$, and furthermore

$K(\xi)|_{L_{C}}=0\quad\text{for all $\xi\in TC\subset
T(\partial\Sigma^{\circ}_{r})$}$

over each component $C$ of the boundary labelled by $L_{C}$.

######  Definition 39.

A _universal choice of perturbation data_ consists of pairs
$\mathbf{P}_{k,\ell,\mathbf{L}}=(\mathbf{Y}_{k,\ell,\mathbf{L}},\mathbf{K}_{k,\ell,\mathbf{L}})$
for all $k+2\ell\geq 2$ and all Lagrangian labellings $\mathbf{L}$, where

$\mathbf{Y}_{k,\ell,\mathbf{L}}\in
C^{\infty}\left(\overline{\EuS}^{\circ}_{k,\ell},\EuY_{*}\right),\qquad\mathbf{K}_{k,\ell,\mathbf{L}}\in\Omega^{1}_{\overline{\EuS}^{\circ}_{k,\ell}/\overline{\EuR}_{k,\ell}}\left(\EuH\right)$

so that $\mathbf{K}_{k,\ell,\mathbf{L}}$ is an $\EuH$-valued, fiberwise linear
map defined on the vertical tangent bundle of the universal curve, satisfying:

  1. (Thin regions) The restriction to the thin regions is given by the Floer data (i.e., the restriction to each fibre is a perturbation datum for that fibre, in the above sense);

  2. (Equivariant) $\mathbf{P}_{k,\ell,\mathbf{L}}$ is $\mathrm{Sym}(\ell)$-equivariant;

  3. (Consistent on discs) The restriction of $\mathbf{P}_{k,\ell,\mathbf{L}}$ to $\mathrm{pr}_{\alpha}^{*}\overline{\EuS}^{\circ}_{k_{\alpha},\ell_{\alpha}}$ is equal to $\mathrm{pr}_{\alpha}^{*}\mathbf{P}_{k_{\alpha},\ell_{\alpha},\mathbf{L}_{\alpha}}$ for any disc vertex $\alpha\in V_{\mathsf{disc}}$ of a tree $\Gamma$;

  4. (Constant on spheres) For any spherical component $\Sigma_{\alpha}$ of a stable disc $\Sigma_{r}$, the restriction of $\mathbf{Y}_{k,\ell,\mathbf{L}}$ to $\Sigma_{\alpha}$ is constant and the restriction of $\mathbf{K}_{k,\ell,\mathbf{L}}$ to $\Sigma_{\alpha}$ vanishes;

  5. (Maximum principle) When $\ell=0$, we have

$\mathbf{Y}_{k,0,\mathbf{L}}\in
C^{\infty}\left(\overline{\EuS}^{\circ}_{k,0},\EuY^{\max{}}_{*}\right)\subset
C^{\infty}\left(\overline{\EuS}^{\circ}_{k,0},\EuY_{*}\right).$

Recall that the embedding
$\overline{\EuS}^{\circ}_{k_{\alpha},\ell_{\alpha}}\hookrightarrow\overline{\EuS}^{\circ}_{k,\ell}$
depends on a choice of ordering of the finite spherical edges incident to
$\alpha$. By the (Equivariant) condition satisfied by
$\mathbf{P}_{k_{\alpha},\ell_{\alpha},\mathbf{L}_{\alpha}}$, the (Consistent
on discs) condition on $\mathbf{P}_{k,\ell,\mathbf{L}}$ does not depend on
this ordering.

######  Lemma 40.

Suppose $k+2\ell=N$, and that
$\mathbf{P}_{k^{\prime},\ell^{\prime},\mathbf{L}^{\prime}}$ satisfying the
conditions of Definition 39 have been chosen for all
$k^{\prime}+2\ell^{\prime}<N$. Then there exist
$\mathbf{P}_{k,\ell,\mathbf{L}}$ satisfying the conditions of Definition 39.

######  Proof 5.2.

The (Consistent on discs) condition uniquely specifies the restriction of
$(\mathbf{Y}_{k,\ell,\mathbf{L}},\mathbf{K}_{k,\ell,\mathbf{L}})$ to all
boundary strata of the form $\EuS_{k_{\alpha},\ell_{\alpha}}$ for $\alpha\in
V_{\mathsf{disc}}$; the (Constant on spheres) condition then uniquely
specifies the restriction of
$(\mathbf{Y}_{k,\ell,\mathbf{L}},\mathbf{K}_{k,\ell,\mathbf{L}})$ to the
remaining boundary strata of the form $\EuS_{\ell_{\alpha}}$ for $\alpha\in
V_{\mathsf{sph}}$, because each maximal tree of sphere components intersects a
disc component in a unique point. Thus the restriction of
$(\mathbf{Y}_{k,\ell,\mathbf{L}},\mathbf{K}_{k,\ell,\mathbf{L}})$ to all
boundary strata is uniquely specified by the preceding choices. It is clear
that these uniquely-specified restrictions satisfy the (Equivariant) condition
(using the fact that a constant map is equivariant) and the (Thin regions)
condition.

Having specified
$(\mathbf{Y}_{k,\ell,\mathbf{L}},\mathbf{K}_{k,\ell,\mathbf{L}})$ over the
boundary strata, we need to show that it extends over all of
$\overline{\EuS}^{\circ}_{k,\ell}$ (cf. [Sei08, Lemma 9.5]). Such an extension
can be constructed using appropriate cutoff functions, because $\EuY_{*}$ and
$\EuH$ are convex, and it can easily be arranged to satisfy the (Thin regions)
condition. The (Equivariant) condition can then be achieved by averaging,
again using convexity. Note that it is important that the parametrization of
the thin regions is chosen to be equivariant, so that the averaged
perturbation data continues to respect the (Thin regions) condition.

###  5.5 Pseudoholomorphic curves

Given a Lagrangian labelling $\mathbf{L}=(L_{0},\ldots,L_{k})$, a _choice of
Hamiltonian chords_ $\mathbf{y}=(y_{0},\ldots,y_{k})$ consists of chords
$y_{0}$ from $L_{0}$ to $L_{k}$ and $y_{i}$ from $L_{i-1}$ to $L_{i}$ for
$i=1,\ldots,k$. We will be considering maps $u:\Sigma^{\circ}\to X$, where
$\Sigma^{\circ}$ is a disc with $k+1$ boundary punctures, each boundary
component is constrained to lie on the Lagrangian by which it is labelled, and
$u$ limits to the chords $y_{i}$ at the punctures. We denote the set of
homotopy classes of such maps by $\pi_{2}(\mathbf{y})$. To each
$A\in\pi_{2}(\mathbf{y})$ we can associate certain numerical invariants: a
Maslov-type index $i(A)$, equal to the Fredholm index of the linearization of
the pseudo-holomorphic curve equation in the form to be discussed
shortly,555In the case that our Floer groups are $\Z$-graded, we have
$i(A)=i(y_{0})-\sum_{j=1}^{k}i(y_{j})$, which $i(y)$ denoting the degree of
the generator $y$. and topological intersection numbers $A\cdot V_{q}$ for
$q\in Q$. We denote
$\pi_{2}^{\mathrm{num}}(\mathbf{y}):=\pi_{2}(\mathbf{y})/\sim$, where the
equivalence relation identifies two classes if they have the same numerical
invariants. Note that given $A\in\pi_{2}^{\mathrm{num}}(\mathbf{y})$ and a
spherical homology class $A^{\prime}\in\im\pi_{2}(X)\to H_{2}(X;\Z)$, we can
form the connect sum $A+A^{\prime}\in\pi_{2}^{\mathrm{num}}(\mathbf{y})$: the
connect sum has Maslov index $i(A)+2c_{1}(A^{\prime})$ and intersection
numbers $A\cdot V_{q}+A^{\prime}\cdot V_{q}$, so the resulting equivalence
class is independent of the choices involved in forming the connect sum.

Because we choose our almost complex structures to respect the components
$V_{q}$ of the system of divisors $V$, any holomorphic curve which is not
contained in $V$ (such as a disc whose boundary is contained in Lagrangians
disjoint from $V$) will intersect each $V_{q}$ positively in a finite set of
points. We will record the local intersection number at $z$ of a holomorphic
curve $u$ with the components of $V$ as a ‘tangency vector’
$\iota(u,z)\in(\Z_{\geq 0})^{Q}$.

Now, we will only need to consider classes
$A\in\pi_{2}^{\mathrm{num}}(\mathbf{y})$ such that $A\cdot V_{q}\geq 0$ for
all $q$ (because of Lemma 22). Given such an $A$, a choice of _tangency data_
is a number $\ell\geq 0$ and a function
$\mathsf{t}:\\{1,\ldots,\ell\\}\to\Z_{\geq 0}^{Q}$ such that
$\sum_{i=1}^{\ell}\mathsf{t}(i)_{q}=A\cdot V_{q}$.

Given a Lagrangian labelling $\mathbf{L}$, a universal choice of perturbation
data associates an almost complex structure
$J_{z}:=J_{\mathbf{Y}_{k,\ell,\mathbf{L}}(z)}\in\EuJ(X,V)$ to every point $z$
on a fibre $\Sigma_{r}^{\circ}$ of the universal family
$\overline{\EuS}^{\circ}_{k,\ell}\to\overline{\EuR}_{k,\ell}$. It also
determines a one-form $\nu$ on $\Sigma_{r}^{\circ}$ with values in
(Hamiltonian) vector fields on $X$: $\nu(\xi)$ is the Hamiltonian vector field
corresponding to the Hamiltonian function
$\mathbf{K}_{k,\ell,\mathbf{L}}(\xi)$.

Now, suppose we are given a Lagrangian labelling $\mathbf{L}$, a choice of
Hamiltonian chords $\mathbf{y}$, an element
$A\in\pi_{2}^{\mathrm{num}}(\mathbf{y})$, a choice of tangency data
$\mathsf{t}$, and a universal choice of perturbation data $\mathbf{P}$. We
define a moduli space $\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ if
$k+2\ell\geq 1$; the moduli space is undefined if $k=\ell=0$. If $k+2\ell\geq
2$, we define the moduli space of pseudo-holomorphic discs with tangency
conditions

(18) $\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P})$

to consist of pairs $(r,u)$ where $r\in\EuR_{k,\ell}$ and
$u:\Sigma_{r}^{\circ}\to X$ satisfies the resulting pseudoholomorphic curve
equation:

$\displaystyle(Du-\nu)\circ j_{z}$ $\displaystyle=J_{z}\circ(Du-\nu)$
$\displaystyle u(C_{i})$ $\displaystyle\subset L_{i}$
$\displaystyle\lim_{s\to\pm\infty}u(\epsilon_{i}(s,\cdot))$
$\displaystyle=y_{i}(\cdot)$ $\displaystyle\iota(u,z_{i})$
$\displaystyle\geq\mathsf{t}(i)\text{ for $i=1,\ldots,\ell$}$
$\displaystyle[u]$ $\displaystyle=A.$

Note that $\mathrm{Sym}(\ell)$ acts on
$\bigcup_{\mathsf{t}}\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P})$, by the
(Equivariant) condition on the perturbation data. The action is free by Lemma
35.

In the remaining case $k+2\ell=1$, we must have $(k,\ell)=(1,0)$ (so
$\EuR_{k,\ell}$ is undefined). In this case we have
$\mathbf{y}=(y_{0},y_{1})$, and we define
$\mathcal{M}(\mathbf{y},A,\emptyset,\mathbf{P})$ to be the moduli space of
Floer trajectories from $y_{0}$ to $y_{1}$, modulo translation.

######  Lemma 41.

If $u\in\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ then $u$ intersects
$V_{q}$ at the marked points $z_{i}$ with $\mathsf{t}(i)_{q}\geq 1$, with
multiplicity $\mathsf{t}(i)_{q}$, and nowhere else.

######  Proof 5.3.

By [CM07, Proposition 7.1], each intersection point $z$ of $u$ with $V_{q}$
contributes $\iota(u,z)_{q}\geq 1$ to $[u]\cdot V_{q}=A\cdot V_{q}$, so we
have

$A\cdot
V_{q}\geq\sum_{i}\iota(u,z_{i})_{q}\geq\sum_{i}\mathsf{t}(i)_{q}=A\cdot
V_{q}.$

Thus we have equality, which implies the result.

If we set $\ell=A\cdot V$ and let $\mathsf{q}:\\{1,\ldots,\ell\\}\to Q$ be a
function with $|\mathsf{q}^{-1}(q)|=A\cdot V_{q}$, we can define a function
$\mathsf{t}^{\mathsf{can}}:\\{1,\ldots,\ell\\}\to\Z_{\geq 0}^{Q}$ by setting
$\mathsf{t}^{\mathsf{can}}(i)_{q}=1$ if $\mathsf{q}(i)=q$ and $0$ otherwise.
We denote the resulting moduli space by

(19)
$\mathcal{M}(\mathbf{y},A,\mathbf{P}):=\mathcal{M}(\mathbf{y},A,\mathsf{t}^{\mathsf{can}},\mathbf{P}).$

It is independent of the choice of $\mathsf{q}$ up to the action of
$\mathrm{Sym}(\ell)$. We denote by
$\mathrm{Sym}(\mathsf{q})\subset\mathrm{Sym}(\ell)$ the subgroup preserving
$\mathsf{q}$, and observe that it acts freely on
$\mathcal{M}(\mathbf{y},A,\mathbf{P})$.

###  5.6 Transversality

In this section we will define what it means for our perturbation data to be
‘regular’; in the next we will show how this leads to the desired Gromov-type
compactification of our moduli spaces.

Let $(\mathbf{y},A_{0},\mathsf{t},\mathbf{P})$ be data as in the previous
section. Let $I\subset\\{1,\ldots,\ell\\}$ be a subset containing all $i$ such
that $\mathsf{t}(i)=0$, and let $\\{\Gamma_{i}\\}_{i\in I}$ be combinatorial
types of bubble trees with $1$ marked point, in the sense of Section 4.1. We
will denote the image of the total homology class of $\Gamma_{i}$ in
$\pi_{2}^{\mathrm{num}}(\mathbf{y})$ by $A_{i}$, and set

$A:=A_{0}+\sum_{i\in I}A_{i}.$

Associated to this data we have a moduli space of discs with
‘$J_{z_{i}}$-holomorphic bubble trees attached at the points $z_{i}$, for
$i\in I$’. To be precise, we introduce notation: $K_{i}\subset Q$ is the set
of non-zero coordinates of $\mathsf{t}(i)$; and $K^{\prime}_{i}\subset Q$ is
the subset attached to the component of $\Gamma_{i}$ containing the marked
point. Starting with the moduli spaces (18), we consider the evaluation maps

$\displaystyle\ev_{i}:\mathcal{M}(\mathbf{y},A_{0},\mathsf{t},\mathbf{P})$
$\displaystyle\to\EuS^{\circ}_{k,\ell}\times V_{K_{i}}$ $\displaystyle(r,u)$
$\displaystyle\mapsto(z_{i}(r),u(z_{i}))$

and

$\displaystyle(\pi_{\EuS^{\circ}_{k,\ell}},\mathrm{ev}_{1}):\mathcal{M}_{\Gamma_{i}}(\mathbf{Y}_{k,\ell})$
$\displaystyle\to\EuS^{\circ}_{k,\ell}\times V_{K^{\prime}_{i}}$

(see Section 4.2). Here ‘$\mathrm{ev}_{1}$’ denotes evaluation at the point
$z_{1}$. Recall (cf. Definition 39) that $\mathbf{Y}_{k,\ell}$ is a family of
infinitesimal almost complex structures parametrized by
$\EuS^{\circ}_{k,\ell}$; the map $\pi_{\EuS^{\circ}_{k,\ell}}$ records the
parameter. We put these together to define

$\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P},\\{\Gamma_{i}\\}):=\mathcal{M}(\mathbf{y},A_{0},\mathsf{t},\mathbf{P})\times\prod_{i\in
I}\mathcal{M}_{\Gamma_{i}}(\mathbf{Y}_{k,\ell}),$

with an evaluation map

$\ev:\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P},\\{\Gamma_{i}\\})\to\prod_{i\in
I}V_{K_{i}}\times
V_{K^{\prime}_{i}}\times\EuS^{\circ}_{k,\ell}\times\EuS^{\circ}_{k,\ell}.$

We define the following subset of the RHS:

$\Delta:=\prod_{i\in I}\left\\{(x_{i},x^{\prime}_{i},s,s^{\prime})\in
V_{K_{i}}\times
V_{K^{\prime}_{i}}\times\EuS^{\circ}_{k,\ell}\times\EuS^{\circ}_{k,\ell}:x_{i}=x^{\prime}_{i},\;s=s^{\prime}\right\\}.$

Now define the moduli space of pseudo-holomorphic discs with tangencies and
unmarked bubble-trees

(20)
$\mathcal{M}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P}):=\ev^{-1}(\Delta).$

We denote by

(21) $\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}\subset\mathcal{M}_{\\{\Gamma_{i}\\}}$

the subset consisting of discs with unmarked bubble trees attached, each of
which is simple.

Let us clear up a potential confusion. A disc with unmarked bubble trees
attached will typically _not_ be the Gromov limit of a family of discs.
Rather, a Gromov limit of discs will have spherical components contain
interior marked points at which tangency conditions to $V$ are imposed; a disc
with bubble trees attached is obtained by forgetting these marked points,
which is possible by our (Constant on spheres) assumption from Definition 39
(cf. Lemma 50).

The tangency constraint on an element
$u\in\mathcal{M}(\mathbf{y},A_{0},\mathsf{t},\mathbf{P})$ can be formulated as
follows: Let $x_{q}$ be a holomorphic function defining $V_{q}$ in a
neighborhood of $u(z_{i})$. Then tangency amounts to the vanishing of jets of
the functions $x_{q}\circ u$:

(22) $\mathsf{Jet}^{\mathsf{t}(i)_{q}}_{z_{i}}(x_{q}\circ u)=0.$

The choice of the coordinates $x_{q}$ does not affect these conditions.

######  Definition 42.

A disc $u\in\mathcal{M}(\mathbf{y},A_{0},\mathsf{t},\mathbf{P})$ is called
parametrized-regular if it is cut out transversely by the parametric
Cauchy–Riemann equation together with the jet-vanishing condition (22). An
element of
$\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ is
called regular if the disc is parametrized-regular, the bubble trees are
parametrized-regular (cf. Section 4.2), and the evaluation map is transverse
to $\Delta$. The moduli space is called regular if all of its points are
regular. A universal choice of perturbation data is called regular if all
moduli spaces having negative virtual dimension are regular (i.e., empty); and
furthermore, all moduli spaces $\mathcal{M}(\mathbf{y},A,\mathbf{P})$ of
virtual dimension $\leq 1$ are regular.

######  Lemma 43.

Let $\mathbf{y}=(y_{0},\ldots,y_{k})$ be a choice of Hamiltonian chords
corresponding to a Lagrangian labelling $\mathbf{L}$,
$A_{0}\in\pi_{2}^{\mathrm{num}}(\mathbf{y})$,
$\mathsf{t}:\\{1,\ldots,\ell\\}\to\Z_{\geq 0}^{Q}$ a choice of tangency data,
and $\\{\Gamma_{i}\\}_{i\in I}$ a set of combinatorial types of bubble trees
as above. Let $k+2\ell=N$. Suppose we have fixed
$\mathbf{P}_{k^{\prime},\ell^{\prime},\mathbf{L}^{\prime}}$ for all $2\leq
k^{\prime}+2\ell^{\prime}\leq N$ and all Lagrangian labellings
$\mathbf{L}^{\prime}$; and suppose that these satisfy the conditions required
of a universal choice of perturbation data. Then the set of
$\mathbf{P}_{k,\ell,\mathbf{L}}$ satisfying those conditions, and such that
$\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ is
regular, is comeagre.

######  Proof 5.4.

The argument is essentially standard. In the absence of higher tangency
conditions (i.e., if $\mathsf{t}(i)\leq 1$ for all $i$), it is a combination
of [Sei08, Section 9k] with [MS04, Section 6.7] (modified as in Section 4 to
account for the divisor $V$). The necessary techniques to deal with higher
tangency conditions were developed in [CM07, Section 6], and implemented in a
setting very similar to ours in [CW19, Theorem 4.19] and [CW17, Theorem 4.1].
The only slightly non-standard aspect of our setup is our requirement that the
perturbation data be chosen to be $\mathrm{Sym}(\ell)$-equivariant.

We start by choosing a countable open cover of $\EuR_{k,\ell}$ by open sets
$C_{i}$, each of which comes equipped with a smooth trivialization
$\EuS^{\circ}_{k,\ell}|_{C_{i}}\simeq C_{i}\times\Sigma_{i}^{\circ}$ with
respect to which the strip-like ends and marked points are constant.

We fix one of the open sets $C_{i}$ in our cover, and drop the $i$ from the
notation. We will prove regularity of
$\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})|_{C}$
(i.e., the set of all
$(r,u,\\{u_{i}\\})\in\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$
such that $r\in C$), for a comeagre subset of the set of perturbation data
$\mathbf{P}$; taking the intersection over the countable set of sets in our
cover yields the result.

We will consider maps of Sobolev class $W^{q,p}$, and perturbation data of
class $C^{s}$, where $q-\frac{2}{p}\geq\max_{i,j}\\{\mathsf{t}(i)_{j}\\}$ and
$s\geq q+1$. We define $\mathrm{Map}^{q,p}(\Sigma^{\circ},X,\mathbf{L},V)$ to
be the Banach manifold of maps $u:\Sigma^{\circ}\to X$ of Sobolev class
$W^{q,p}$, satisfying the boundary constraints $\mathbf{L}$, converging to the
Hamiltonian chords $\mathbf{y}$ in $W^{q,p}$-sense along the strip-like ends,
and having the prescribed orders of tangency at the marked points. We set

$\EuB^{q,p}_{C}:=\mathrm{Map}^{q,p}(\Sigma^{\circ},X,\mathbf{L},V)\times\prod_{1\leq
i\leq\ell,\alpha\in
T(\Gamma_{i})}\mathrm{Map}^{q,p}(\mathbb{CP}^{1}_{\alpha},V_{K_{\alpha}}).$

Let $\mathbb{P}^{s}=\mathbb{Y}^{s}\times\mathbb{K}^{s}$ denote the space of
all
$\mathbf{P}_{k,\ell,\mathbf{L}}=(\mathbf{Y}_{k,\ell,\mathbf{L}},\mathbf{K}_{k,\ell,\mathbf{L}})$
satisfying the conditions required of a universal choice of perturbation data,
but having class $C^{s}$ instead of $C^{\infty}$. We have a $C^{s-q}$ Banach
vector bundle $\EuE_{C}\to C\times\EuB_{C}^{q,p}\times\mathbb{P}^{s}$,
together with a $C^{s-q}$ Fredholm section $\bar{\partial}$. There is also an
evaluation map

$\mathrm{ev}^{E}:\EuB_{C}^{q,p}\times\prod_{1\leq
i\leq\ell}Z(\Gamma_{i})\to\prod_{1\leq i\leq\ell}V_{K_{i}}\times\prod_{1\leq
i\leq\ell}(X,V)^{\Gamma_{i}},$

together with a natural diagonal subset $\Delta$ of the target. We define

$\widetilde{\mathcal{M}}_{\\{\Gamma_{i}\\}}^{q,p}(\mathbf{y},A,\mathsf{t},\mathbb{P}^{s})|_{C}\subset
C\times\EuB^{q,p}_{C}\times\mathbb{P}^{s}\times\prod_{1\leq
i\leq\ell}Z(\Gamma_{i})$

to be the subset cut out by the equations
$\bar{\partial}(r,\mathbf{u},\mathbf{P})=0$ and
$\mathrm{ev}^{E}(\mathbf{u},\mathbf{z})\in\Delta$, and
$\mathcal{M}^{q,p}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbb{P}^{s})|_{C}$
to be its quotient by the reparametrization group $\prod_{1\leq
i\leq\ell}G_{\Gamma_{i}}$. The fibre of
$\mathcal{M}_{\\{\Gamma_{i}\\}}^{q,p}(\mathbf{y},A,\mathsf{t},\mathbb{P}^{s})|_{C}$
over $\mathbf{P}\in\mathbb{P}^{s}$ is precisely
$\mathcal{M}_{\\{\Gamma_{i}\\}}^{q,p}(\mathbf{y},A,\mathsf{t},\mathbf{P})|_{C}$.
By definition, it is regular if and only if the corresponding fibre
$\widetilde{\mathcal{M}}_{\\{\Gamma_{i}\\}}^{q,p}(\mathbf{y},A,\mathsf{t},\mathbf{P})|_{C}$
is cut out transversely.

We now argue that the universal moduli space
$\widetilde{\mathcal{M}}_{\\{\Gamma_{i}\\}}^{q,p,*}(\mathbf{y},A,\mathsf{t},\mathbb{P}^{s})|_{C}$
(of pseudoholomorphic discs with _simple_ bubble trees attached) is cut out
transversely, for each $C$. We focus on the only slightly non-routine step,
which is regularity of the disc component.

Transversality is a local condition at each point
$(r,u,\mathbf{Y},\mathbf{K})$. Given such a point, we consider the space
$\mathbb{Y}^{s}_{r}$ of $C^{s}$ domain-dependent almost-complex structures on
the fibre $\Sigma^{\circ}_{r}$, satisfying the conditions required to
constitute part of a perturbation datum for that fibre. By [CM07, Lemma 6.5],
in order to prove regularity, it suffices to show that there exists a non-
empty open set $W\times U\subset\Sigma^{\circ}\times X\setminus V$, such there
exists a point $z\in W$ with $u(z)\in U$, and the image of the linearization
of the restriction map $\mathbb{P}^{s}\to\mathbb{Y}^{s}_{r}$ contains all
perturbations $\delta Y$ which are supported in $W\times U$.

To arrange this, we first choose a neighbourhood $N$ of $r\in\EuR_{k,\ell}$,
such that $\sigma(N)\cap N=\emptyset$ for all $\sigma\in\mathrm{Sym}(\ell)$.
This is possible by Lemma 35. We choose $W\subset\Sigma^{\circ}$ to be the
complement of the thin regions and marked points, and $U$ to be equal to
$\\{h<c\\}$. Note that $W\cap\partial\Sigma^{\circ}$ is non-empty, and gets
mapped into $U$; thus there exists $z\in W$ with $u(z)\in U$, as required.

Now, given a perturbation $\delta Y$ supported in $W\times U$, we explain how
to construct a perturbation
$\delta\mathbf{P}=(\delta\mathbf{K},\delta\mathbf{Y})$ such that the
restriction of $\delta\mathbf{Y}$ to $\Sigma^{\circ}_{r}$ is $\delta Y$.
First, we choose a bump function $\eta$ with support in $N$, so that
$\eta(r)=1$; then we consider the perturbation $\eta(n)\cdot\delta Y$ with
support in $N\times W\times U$; then we set

$\delta\mathbf{Y}:=\sum_{\sigma\in\mathrm{Sym}(\ell)}\sigma^{*}\left(\eta(n)\cdot\delta
Y\right).$

We claim that $\mathbf{Y}+\epsilon\cdot\delta\mathbf{Y}\in\mathbb{Y}^{s}$ for
$\epsilon$ sufficiently small: it clearly lies in $\EuY_{*}$ for $\epsilon$
sufficiently small, because $\delta\mathbf{Y}$ is compactly supported so its
$C^{0}$ norm is bounded; it extends to $\overline{\EuS}^{\circ}_{k,\ell}$ and
satisfies the (Consistent on discs) and (Constant on spheres) hypotheses,
because it has compact support in $\EuS^{\circ}_{k,\ell}$ and in particular
vanishes on the boundary; it satisfies the (Thin regions) hypothesis because
$W$ is disjoint from the thin regions, hence so is its orbit under
$\mathrm{Sym}(\ell)$ (by equivariance of the thin regions); it satisfies the
(Equivariant) hypothesis by construction. The image of
$\partial/\partial\epsilon(\mathbf{Y}+\epsilon\cdot\delta\mathbf{Y})$ under
the linearization of the restriction map $\mathbb{Y}^{s}\to\mathbb{Y}^{s}_{r}$
is equal to $\delta Y$, because $\left(\eta(n)\cdot\delta
Y\right)|_{\Sigma^{\circ}_{r}}=\delta Y$ by construction, but
$\sigma^{*}\left(\eta(n)\cdot\delta Y\right)|_{\Sigma^{\circ}_{r}}=0$ for
$\sigma$ not equal to the identity, using the fact that $\sigma(N)\cap
N=\emptyset$.

Having established regularity of the universal moduli space, we can apply
Sard–Smale to the projection
$\widetilde{\mathcal{M}}^{q,p,*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbb{P}^{s})|_{C}\to\mathbb{P}^{s}$.
We conclude that
$\widetilde{\mathcal{M}}^{q,p,*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})|_{C}$
(and hence, by definition,
$\mathcal{M}^{q,p,*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})|_{C}$)
is regular for a comeagre subset of the set of choices of $C^{s}$ perturbation
data $\mathbb{P}^{s}$, for each $C$. It follows by an argument due to Taubes
(as in [MS04, Proof of Theorem 3.5.1(ii)]) that
$\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})|_{C}$
is regular for a comeagre subset of the set of choices of smooth perturbation
data. Taking the intersection over all $C$, we conclude that
$\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ is
regular for a comeagre subset of the set of smooth perturbation data, as
required.

######  Corollary 44.

There exists a regular universal choice of perturbation data.

######  Proof 5.5.

In fact, we can construct a universal choice of perturbation data such that
_all_ moduli spaces
$\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ are
regular; such a choice is _a fortiori_ regular. The choice can be constructed
inductively in $k+2\ell$. At each step, there is a nonempty space of (possibly
non-regular) choices of $\mathbf{P}_{k,\ell,\mathbf{L}}$, by Lemma 40. The
subset of regular choices is the intersection of the subsets making
$\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$
regular for all compatible $\mathbf{y},A,\mathsf{t},\\{\Gamma_{i}\\}$. This is
a countable intersection of subsets, each of which is comeagre by Lemma 43;
thus it is itself a comeagre subset of a nonempty Baire space, in particular
it is nonempty.

######  Lemma 45.

If $\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ is
regular, it has dimension

(23)
$\mathrm{dim}=i(A_{0})+k-2+2\ell-2\sum_{i=1}^{\ell}|\mathsf{t}(i)|+\sum_{i\in
I}\mathrm{dim}(\Gamma_{i})-2n+2|K_{i}\cap K^{\prime}_{i}|.$

Here $|\mathsf{t}(i)|=\sum_{q\in Q}\mathsf{t}(i)_{q}$.

######  Proof 5.6.

The dimension count goes as follows:

$\underbrace{i(A_{0})+k-2+2\ell-2\sum_{i=1}^{\ell}|\mathsf{t}(i)|}_{\dim\mathcal{M}(\mathbf{y},A_{0},\mathsf{t},\mathbf{P})}+\underbrace{\sum_{i\in
I}\left(\dim(\Gamma_{i})+\dim\EuS_{k,l}^{\circ}\right)}_{\sum_{i}\dim\mathcal{M}_{\Gamma_{i}}(\mathbf{Y}_{k,\ell})}-\underbrace{\sum_{i\in
I}\left(2n-2|K_{i}\cap
K^{\prime}_{i}|+\dim\EuS_{k,l}^{\circ}\right)}_{\mathrm{codim}\,\Delta}.$

The formula follows from the assumption that $\ev$ is transverse to $\Delta$.

######  Lemma 46.

If $\mathsf{t}=\mathsf{t}^{\mathsf{can}}$ and $I=\emptyset$, then the
dimension (23) is equal to

$\mathrm{dim}(\mathcal{M}(\mathbf{y},A,\mathbf{P}))=i(A)+k-2.$

In any other case, it is bounded above by
$\mathrm{dim}(\mathcal{M}(\mathbf{y},A,\mathbf{P}))-2$.

######  Proof 5.7.

Applying Lemmas 45 and 33 (we set $k=1$ in the latter, because by definition
all bubble trees carry only a single marked point, namely the one at which
they are attached to the disc), we find

$\displaystyle\dim$ $\displaystyle\leq
i(A)+k-2+\sum_{i=1}^{\ell}2(1-|\mathsf{t}(i)|)+\sum_{i\in I}2(|K_{i}\cap
K^{\prime}_{i}|-\max_{\alpha\in T_{i}}|K_{\alpha}|-2)$
$\displaystyle\leq\dim(\mathcal{M}(\mathbf{y},A,\mathbf{P}))+\sum_{i=1}^{\ell}2(1-|\mathsf{t}(i)|-2I_{i}),$

where $I_{i}:=1$ if $i\in I$, $0$ if $i\notin I$. Because $I$ contains all $i$
such that $\mathsf{t}(i)=0$, we have $1-|\mathsf{t}(i)|-2I_{i}\leq 0$, with
equality if and only if $|\mathsf{t}(i)|=1$ and $I_{i}=0$. The result follows.

######  Corollary 47.

If $\mathbf{P}$ is regular and
$\mathrm{dim}(\mathcal{M}(\mathbf{y},A,\mathbf{P}))\leq 1$, then
$\mathcal{M}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})=\emptyset$
unless $\mathsf{t}=\mathsf{t}^{\mathsf{can}}$ and $I=\emptyset$.

######  Proof 5.8.

Observe that the subset
$\mathcal{M}^{*}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ of
discs with simple bubble trees attached is empty by Lemma 46, since it is
regular and has negative virtual dimension. On the other hand, one can always
replace a non-simple bubble tree with a simple one (see [MS04, Proposition
6.1.2]). The procedure involves replacing multiply-covered spheres by their
image, deleting some spheres, and rearranging the marked points; in
particular, it cannot increase the Chern number by Corollary 23. It follows
that the moduli spaces of discs with non-simple bubble trees attached are also
empty.

One would expect a similar result to hold for
$\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P})$, if one changed the
definition of the bubble trees to incorporate higher tangency conditions
(otherwise the analogue of Lemma 46 would not hold).

###  5.7 Compactness

Because we chose our perturbation data $\mathbf{P}_{k,\ell}$ to extend
smoothly to $\overline{\EuS}^{\circ}_{k,\ell}$, the moduli space
$\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P})$ of discs with tangency
conditions admits a Gromov compactification
$\overline{\mathcal{M}}^{\mathrm{tot}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$.
Let

(24)
$\overline{\mathcal{M}}(\mathbf{y},A,\mathsf{t},\mathbf{P})\subset\overline{\mathcal{M}}^{\mathrm{tot}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$

denote the closure of the ‘smooth locus’
$\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P})$. Write

(25) $\overline{\mathcal{M}}(\mathbf{y},A,\mathbf{P})$

to mean
$\overline{\mathcal{M}}(\mathbf{y},A,\mathsf{t}^{\mathsf{can}},\mathbf{P})$.

######  Lemma 48.

If $u$ is a disc component of an element of
$\overline{\mathcal{M}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$, then any point
at which $u$ intersects $V$ is special, i.e., an internal marked point or
node.

######  Proof 5.9.

Follows from Lemma 41, as isolated intersection points persist under small
perturbations.

Let us call a disc with $k+1$ boundary marked points and $\ell$ interior
marked points ‘stable’ if $k+2\ell\geq 2$, ‘semistable’ if $k+2\ell\geq 1$,
and ‘unstable’ otherwise.

######  Corollary 49.

Every disc component of an element of
$\overline{\mathcal{M}}(\mathbf{y},A,\mathbf{P})$ is semistable.

######  Proof 5.10.

Follows from Lemmas 48 and 28.

######  Lemma 50.

Suppose $\mathbf{P}$ is regular, and
$\dim(\mathcal{M}(\mathbf{y},A,\mathbf{P}))\leq 1$. Then elements of
$\overline{\mathcal{M}}(\mathbf{y},A,\mathbf{P})$ have no (constant or non-
constant) sphere bubbles.

######  Proof 5.11.

Given an element $u\in\overline{\mathcal{M}}(\mathbf{y},A,\mathbf{P})$, let us
cut it along its boundary nodes. By Corollary 49, each disc component is
semistable; therefore, by the (Consistent on discs) condition on $\mathbf{P}$,
each component defines an element $u_{\alpha}$ of some
$\overline{\mathcal{M}}(\mathbf{y}_{\alpha},A_{\alpha},\mathbf{P})$, without
boundary nodes. We have

$\sum_{\alpha}\dim(\mathcal{M}(\mathbf{y}_{\alpha},A_{\alpha},\mathbf{P})+\\#\,(\text{boundary
nodes})=\dim(\mathcal{M}(\mathbf{y},A,\mathbf{P}))$

by Lemma 46. It follows from the assumption
$\dim(\mathcal{M}(\mathbf{y},A,\mathbf{P}))\leq 1$, then, that
$\dim(\mathcal{M}(\mathbf{y}_{\alpha},A_{\alpha},\mathbf{P})\leq 1$ for each
$\alpha$. Thus we may assume, without loss of generality, that $u$ has no
boundary nodes.

Let $u_{\mathsf{disc}}$ be the disc component of $u$. By Lemma 48, every
intersection point of $u_{\mathsf{disc}}$ with $V$ is a marked point or node.
We attach the tangency data $\mathsf{t}(i):=\iota(u_{\mathsf{disc}},z_{i})$ to
each marked point or node $z_{i}$, so that $u_{\mathsf{disc}}$ defines an
element of some $\mathcal{M}(\mathbf{y},A_{0},\mathsf{t},\mathbf{P})$ by the
(Consistent on discs) condition on $\mathbf{P}$. By the (Constant on spheres)
condition on $\mathbf{P}$, forgetting the internal marked points on the sphere
bubbles of $u$ and collapsing unstable components gives a set of bubble trees
attached to the marked points of $u_{\mathsf{disc}}$. In order to check that
it defines an element of some
$\mathcal{M}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$, we need
to check that the subset $I$ of all points with non-trivial bubble trees
attached, contains all $i$ such that $\mathsf{t}(i)=0$; or equivalently, if
the bubble tree attached to the $i$th point is trivial, then
$\mathsf{t}(i)\neq 0$. Indeed, if the bubble tree attached to point $i$
becomes trivial after forgetting all marked points, it must originally have
been constant; therefore $\mathsf{t}(i)$ is greater than or equal to the sum
of $\mathsf{t}^{\mathsf{can}}(j)$ over all points $j$ on the bubble tree, by
[CM07, Lemma 7.2]. Since $\mathsf{t}^{\mathsf{can}}(j)\neq 0$ for all $j$,
this implies that $\mathsf{t}(i)\neq 0$ as required.

Thus we obtain an element of some
$\mathcal{M}_{\\{\Gamma_{i}\\}}(\mathbf{y},A,\mathsf{t},\mathbf{P})$, so we
must have $\mathsf{t}=\mathsf{t}^{\mathsf{can}}$ and $I=\emptyset$ by
Corollary 47. Because $I=\emptyset$, we conclude that all sphere bubbles on
$u$ become trivial after forgetting the marked points they contain, so they
must be constant. In order for a constant tree of sphere bubbles to be stable,
it must contain at least two marked points in addition to the point at which
it is attached to the disc; however then the tangency vector of the point $i$
at which the tree of sphere bubbles is attached to the disc would have to be
greater than or equal to a sum of $\mathsf{t}^{\mathsf{can}}(j)$ over at least
two points $j$, so could not possibly be equal to
$\mathsf{t}^{\mathsf{can}}(i)$. Therefore $u$ contains no sphere bubbles.

##  6 The relative Fukaya category

###  6.1 Definition

We already defined the objects and morphism spaces in Definitions 36 and 37.
To define the $A_{\infty}$ structure maps, we choose a regular universal
choice of perturbation data $\mathbf{P}$. Each zero-dimensional moduli space
$\mathcal{M}(\mathbf{y},A,\mathbf{P})$ is then finite (because compact). Each
element $u$ of such a moduli space determines an isomorphism

(26) $\sigma_{u}:o_{y_{1}}\otimes\ldots\otimes
o_{y_{k}}\xrightarrow{\sim}o_{y_{0}},$

in accordance with the conventions of [She20, Appendix B]. Observe that
$\mathrm{Sym}(\mathsf{q})$ acts on $\mathcal{M}(\mathbf{y},A,\mathbf{P})$
because of the (Equivariant) condition on our perturbation data; and the
action is free by Lemma 35. It is easy to see from the definition that
$\sigma_{g\cdot u}=\sigma_{u}$ for any $g\in\mathrm{Sym}(\mathsf{q})$. This
allows us to make the following

######  Definition 51.

We define the $A_{\infty}$ structure map

$\displaystyle\mu^{k}$
$\displaystyle:hom^{*}(L_{0},L_{1})\otimes_{R}\ldots\otimes_{R}hom^{*}(L_{k-1},L_{k})\to
hom^{*}(L_{0},L_{k}),$ (27) $\displaystyle\mu^{k}$
$\displaystyle:=\sum_{u\in\mathcal{M}(\mathbf{y},A,\mathbf{P})/\mathrm{Sym}(\mathsf{q})}T^{[u]\cdot
D}\cdot\sigma_{u},$

where the sum is over all $\mathrm{Sym}(\mathsf{q})$-orbits of zero-
dimensional moduli spaces $\mathcal{M}(\mathbf{y},A,\mathbf{P})$.

Observe that $[u]\cdot D\in\operatorname{\mathrm{NE}}$ by Lemma 22. We also
observe that $\mu^{0}$ is of order $\mathfrak{m}$ by definition: we at no
point consider discs with a single boundary marked point and no interior
marked point (we are able to do this by Corollary 49).

Finally, $\mu^{*}\circ\mu^{*}=0$ by the standard argument [Sei08, Section
12g]. The only boundary strata appearing in a one-dimensional moduli space
$\overline{\mathcal{M}}(\mathbf{y},A,\mathbf{P})$ consist of stable discs with
two disc components and no spheres, by Lemma 50. The moduli space has the
structure of a one-dimensional manifold with boundary in a neighbourhood of
these points, by a standard gluing argument. The action of $\mathrm{Sym}(q)$
on such boundary strata remains free (cf. the two remarks following Lemma 35),
and so the moduli space of $\mathrm{Sym}(q)$-orbits is again a one-dimensional
manifold with boundary. The signed count of boundary points of a compact one-
dimensional manifold with boundary is zero; and these boundary points
correspond precisely to the terms in the expression $\mu^{*}\circ\mu^{*}$.

###  6.2 Assumptions

Proposition 1 states that
$\EuF(X,D,\mathsf{N})\otimes_{R}R/\mathfrak{m}\simeq\EuF(X\setminus D)$: this
is immediate from Lemma 27.

We also remark that [She20, Assumptions 5.2 and 5.3] are satisfied. The
argument is a straightforward variation on the argument, given in Section 7,
that the relative Fukaya category is independent of the choice of system of
divisors, so we omit it. (For the purposes of the argument, one can treat a
single component of $D_{p}$ as if it were a system of divisors, by [She20,
Lemma 4.9].)

##  7 Independence of choices

We made the following choices in the definition of the relative Fukaya
category, in order:

  1. 1.

A relative Kähler form;

  2. 2.

A regular sublevel set $\\{h\leq c\\}$ of the Kähler potential, containing all
critical points;

  3. 3.

A system of divisors $V$ contained in $\\{h>c\\}$;

  4. 4.

A choice of strip-like ends and thick-thin decomposition;

  5. 5.

A choice of Floer data;

  6. 6.

A choice of perturbation data.

We will show that the relative Fukaya category is independent of these choices
up to non-unique quasi-equivalence (we have a ‘weak system’ of equivalent
categories, in the terminology of [Sei08, Section 10a]).

###  7.1 Independence of perturbation data and system of divisors

Let us fix a relative Kähler form and regular sublevel set $\\{h\leq c\\}$ of
the Kähler potential, containing all critical points. We denote the relative
Fukaya category, defined using system of divisors $V$ and regular universal
choice of perturbation data $\mathbf{P}$, by $\EuF(X,D,V,\mathbf{P})$.
Implicit in the choice of $\mathbf{P}$ is a universal choice of strip-like
ends and a choice of Floer data, on which the definition of perturbation data
depends.

Now let $\mathsf{N}_{0}\subset\mathsf{N}_{1}$ be cones satisfying the
conditions from Section 1.1. Let $V_{1}=\cup_{q\in Q_{1}}V_{q}$ be a system of
divisors for $\mathsf{N}_{1}$, $Q_{0}\subset Q_{1}$ a subset, such that
$V_{0}:=\cup_{q\in Q_{0}}V_{q}$ is a system of divisors for $\mathsf{N}_{0}$.

######  Proposition 52.

For any regular $\mathbf{P}^{1}$, and a non-empty subset of the set of regular
$\mathbf{P}^{0}$, there exists a curved filtered $R(\mathsf{N}_{1})$-linear
$A_{\infty}$ category $\EuC$ which admits strict curved filtered quasi-
equivalences

(28)
$\EuF(X,D,V_{1},\mathbf{P}^{1})\lhook\joinrel\longrightarrow\EuC\longleftarrow\joinrel\rhook\EuF(X,D,V_{0},\mathbf{P}^{0})\otimes_{R(\mathsf{N}_{0})}R(\mathsf{N}_{1}).$

If $V_{0}=V_{1}$, then the the subset of the regular $\mathbf{P}^{0}$ consists
of all regular $\mathbf{P}^{0}$.

Note that the case $V_{0}=V_{1}$ of the Proposition shows that
$\EuF(X,D,V,\mathbf{P})$ is independent of the choice of strip-like ends,
Floer data, and perturbation data, up to non-unique quasi-equivalence; we
denote this quasi-equivalence class by $\EuF(X,D,V)$. The full statement of
the proposition then yields a quasi-equivalence

$\EuF(X,D,V_{1})\simeq\EuF(X,D,V_{0})\otimes_{R(\mathsf{N}_{0})}R(\mathsf{N}_{1}).$

The case $\mathsf{N}_{0}=\mathsf{N}_{1}$ shows that $\EuF(X,D,V)$ is
independent of $V$ up to non-unique quasi-equivalence; we denote this quasi-
equivalence class by $\EuF(X,D,\mathsf{N})$, where
$\mathsf{N}=\mathsf{N}_{0}=\mathsf{N}_{1}$.

The basic idea behind the construction of $\EuC$ is to show that
$\EuF(X,D,V_{1})$ can be defined using the pullback of the perturbation data
$\mathbf{P}^{0}$ via the map forgetting all marked points labelled by $q\in
Q_{1}\setminus Q_{0}$. The difficulty lies in the fact that the pulled-back
perturbation data will never satisfy the conditions (Thin regions),
(Equivariant), (Consistent on discs) required of a universal choice of
perturbation data. Therefore, we enlarge the allowed class of perturbation
data so that it includes the pullback of $\mathbf{P}^{0}$, as well as the
perturbation data $\mathbf{P}^{1}$; then a doubled category construction
allows us to interpolate between the two.

###  7.2 Construction of the doubled category: domain data

We give the definition of the doubled category $\EuC$ appearing in Proposition
52. The objects are Lagrangian branes $L^{i}$, equipped with a label $i=0$ or
$1$. On the level of objects, the left embedding in (28) send $L\mapsto
L^{0}$, while the right embedding sends $L\mapsto L^{1}$. A _Lagrangian
labelling_ $\mathbf{L}$ will be a tuple of objects
$(L_{0}^{i_{0}},\ldots,L_{k}^{i_{k}})$ of $\EuC$. Forgetting the Lagrangians
$L_{j}$, we obtain a _0/1 labelling_
$\mathbf{i}(\mathbf{L})=(i_{0},\ldots,i_{k})$.

######  Definition 53.

Suppose that $2\leq k+2\ell$, $\ell_{\mathsf{for}}\leq\ell$,
$K^{0}_{\mathsf{pun}}\subset\\{1,\ldots,k\\}$ is a subset, and
$\mathbf{i}=(i_{0},\ldots,i_{k})$ is a 0/1 labelling, having the property that
whenever $j\notin K_{\mathsf{pun}}$, we have $i_{j-1}=i_{j}=0$ (where $j$ is
taken modulo $k+1$). We define
$\EuS_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\to\EuR_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$
to coincide with $\EuS_{k,\ell}\to\EuR_{k,\ell}$. We define
$K_{\mathsf{pun}}:=K^{0}_{\mathsf{pun}}\cap\\{1,\ldots,k\\}$, and require that
$0\notin K^{0}_{\mathsf{pun}}$ if and only if $K_{\mathsf{pun}}$ is empty and
$\ell=\ell_{\mathsf{for}}$. We define
$\EuS^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$ to be
the complement of all boundary marked points labelled by $j\in
K_{\mathsf{pun}}^{0}$.

We now explain how to ‘think about’ Definition 53, and preview the main
construction. The moduli spaces
$\EuR_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$ parametrize
discs with $k+1$ boundary marked points and $\ell$ interior marked points. We
call the last $\ell_{\mathsf{for}}$ of the internal marked points
‘forgettable’, and the remaining ones ‘unforgettable’. One should think of
$\mathbf{i}$ as a 0/1 labelling of the boundary intervals between the boundary
marked points.

We call the boundary marked points corresponding to $j\in
K_{\mathsf{pun}}^{0}$ ‘puncturable’, and the remaining ones ‘unpuncturable’.
If a boundary point is unpuncturable, then the two boundary intervals on
either side of it must be labeled $0$. The output boundary marked point is
unpuncturable if and only if all input boundary marked points are
unpuncturable and all interior marked points are forgettable.

We will consider pseudoholomorphic discs
$u\in\mathcal{M}(\mathbf{y},A,\mathsf{t},\mathbf{P}$), cf. (18), whose domain
is a fibre $\Sigma^{\circ}_{r}$ of the family
$\EuS^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$, with
the following properties:

  * •

The disc $u$ maps unforgettable internal marked points to $V_{0}$, and
forgettable ones to $V_{q}$ for $q\in Q_{1}\setminus Q_{0}$.

  * •

The disc $u$ limits to Hamiltonian chords at the puncturable boundary marked
points, which are removed from $\Sigma_{r}$ to obtain $\Sigma^{\circ}_{r}$.

  * •

We have required that unpuncturable boundary marked points $z_{j}$ always
separate two boundary components carrying the label $0$. These boundary
components will furthermore _be labeled by the same Lagrangian_ :
$L_{j-1}=L_{j}$. Unpuncturable boundary points $z_{j}$ are not removed, and
$u$ will extend smoothly across $z_{j}$.

  * •

Our perturbation data will be constructed to be compatible with the map
forgetting the forgettable marked points, over certain subsets of the space of
domains. It will similarly be compatible with the map forgetting the
unpuncturable marked points.

#### The role of unpuncturable marked points.

The $A_{\infty}$ structure maps in the doubled category $\EuC$ will be defined
via pseudoholomorphic discs with only puncturable boundary marked points.
Nevertheless, unpuncturable marked points play an unavoidable role in the
construction, as we now explain.

The following are two fundamental aspects of the construction of our moduli
spaces of pseudoholomorphic curves:

  * •

the perturbation data $\mathbf{P}$ on $\EuS^{\circ}$ should extend smoothly
over the boundary $\overline{\EuS}^{\circ}$, so that a Gromov compactification
exists;

  * •

over moduli spaces all of whose boundary components are labelled ‘$0$’, the
perturbation data $\mathbf{P}$ must be pulled back from the perturbation data
$\mathbf{P}^{0}$ used to define $\EuF(X,D,V_{0},\mathbf{P}^{0})$, via the map
$\mathfrak{f}$ which ‘forgets’ all of the forgettable marked points.

Combining these two, we see that the perturbation data over the boundary
$\partial\overline{\EuS}^{\circ}$ must also be pulled back via the forgetful
map. Now let us consider a stable disc in $\overline{\EuR}$ which has a disc
component $\mathbb{D}$ with a single boundary marked point which is a node,
and a single interior marked point which is forgettable. The forgetful map
$\mathfrak{f}$ collapses $\mathbb{D}$ to a smooth boundary point $p$. In
particular, the pulled back perturbation data $\mathfrak{f}^{*}\mathbf{P}^{0}$
take the form $(\mathbf{Y},\mathbf{K})$ where $\mathbf{K}=0$ over
$\mathbb{D}$, and $\mathbf{Y}$ is constant equal to $\mathbf{Y}^{0}(p)$ over
$\mathbb{D}$. This raises the following issues:

  * •

The perturbation data over $\mathbb{D}$ cannot satisfy the (Thin regions)
condition, as $\mathbf{K}=0$.

  * •

The perturbation data over $\mathbb{D}$ can only satisfy the (Consistent on
discs) condition if $\mathbf{Y}^{0}$ is constant along the corresponding
boundary component.

In particular, when considering pseudoholomorphic maps from this source, we
are forced to consider the disc $\mathbb{D}$ as bubbling off at a node, rather
than breaking along a Hamiltonian chord.

In particular, we are forced to treat certain boundary marked points and
components differently from others: the marked point described above is an
example of an unpuncturable marked point (whereas in the previous sections we
only ever considered puncturable marked points), and the disc $\mathbb{D}$ is
an example of what we will call an ‘$\mathfrak{f}$-disclike’ component in the
sequel.

Although we are forced to introduce unpuncturable marked points in order to
construct moduli spaces admitting Gromov compactifications, _a posteriori_ ,
the corresponding components of the Gromov compactification will actually be
empty (Lemma 78). For example, the disc $u(\mathbb{D})$ considered above must
by non-constant, since it intersects $V_{1}$ at its interior marked point; but
that implies that it must also intersect $V_{0}$ by Lemma 28, as $V_{0}$ is a
system of divisors, and in particular must contain an additional, stabilizing
marked point mapping to $V_{0}$.

#### Deligne–Mumford compactification.

Now we describe the structure of the Deligne–Mumford compactification of
$\EuR_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$. Of course we
already have a description with strata isomorphic to products of
$\EuR_{\ell_{\alpha}}$ and $\EuR_{k_{\alpha},\ell_{\alpha}}$; the aim is to
extend the $0/1$ labelling of boundary components, designation of interior
marked points as forgettable/unforgettable, and designation of boundary marked
points as puncturable/unpuncturable, to the components of a stable curve. This
is necessary in order to specify the consistency conditions we require on our
choices of perturbation data.

We define
$\overline{\EuS}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\to\overline{\EuR}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$
to be equal to $\overline{\EuS}_{k,\ell}\to\overline{\EuR}_{k,\ell}$. For the
following definitions, let $\Sigma_{r}$ be a fibre of
$\overline{\EuS}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\to\overline{\EuR}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$.

######  Definition 54.

Let $z_{j}\in\Sigma_{r}$ be an interior marked point or an interior node on a
disc component. We define $z_{j}$ to be forgettable if it is a forgettable
marked point, or if it is a node and all marked points lying on the tree of
spheres attached at $z_{j}$ are forgettable; otherwise define it to be
unforgettable.

Let us call the following procedure the ‘$\mathfrak{f}$-process’. Start with
the stable curve $\Sigma_{r}$, and modify it as follows.

  1. 1.

Delete all forgettable interior marked points.

  2. 2.

Delete all unpuncturable boundary marked points.

  3. 3.

Collapse any spheres with $\leq 2$ special points (i.e., marked points or
nodes).

  4. 4.

Collapse any discs which have no internal special points, which have only one
boundary marked point, and whose unique boundary component is labelled ‘$0$’.

  5. 5.

Return to (3) and iterate.

The iteration stops when there are no more spheres or discs to collapse. The
output is a curve $\Sigma_{r}^{\prime}$ without unforgettable or unpuncturable
marked points, which comes with a collapsing map
$c\colon\Sigma_{r}\to\Sigma_{r}^{\prime}$.

######  Definition 55.

The disc-components of $\Sigma_{r}$ are partitioned into three types, as
follows: The discs which are collapsed in the $\mathfrak{f}$-process are
called ‘$\mathfrak{f}$-disclike’. Discs which, in the output curve
$\Sigma_{r}^{\prime}$, have no internal marked points but exactly two boundary
special points, separated by two boundary components both labelled ‘$0$’, are
defined to be ‘$\mathfrak{f}$-striplike’. Designate all other disc-components
of $\Sigma_{r}$ as ‘$\mathfrak{f}$-stable’.

######  Definition 56.

We define a boundary marked point or node of $\Sigma_{r}$ to be unpuncturable
if it is an unpuncturable marked point, or if it is a boundary node lying on
an $\mathfrak{f}$-disclike component; otherwise define it to be puncturable.

As before, strata of
$\overline{\EuR}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$ are
indexed by certain decorated trees $\Gamma$. The conventions established in
Definitions 54 and 56 (together with the obvious convention for determining
boundary labellings $\mathbf{i}_{\alpha}$ from $\mathbf{i}$) determine
isomorphisms

$\displaystyle\EuR^{\Gamma}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$
$\displaystyle\simeq\prod_{\alpha\in
V_{\mathsf{sph}}}\EuR_{\ell_{\alpha}}\times\prod_{\alpha\in
V_{\mathsf{disc}}}\EuR_{k_{\alpha},K_{\alpha,\mathsf{pun}},\ell_{\alpha},\ell_{\alpha,\mathsf{for}},\mathbf{i}_{\alpha}},$
(29)
$\displaystyle\EuS^{\Gamma}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$
$\displaystyle\simeq\left(\coprod_{\alpha\in
V_{\mathsf{sph}}}\mathrm{pr}_{\alpha}^{*}\EuS_{\ell_{\alpha}}\sqcup\coprod_{\alpha\in
V_{\mathsf{disc}}}\mathrm{pr}_{\alpha}^{*}\EuS_{k_{\alpha},K_{\alpha,\mathsf{pun}},\ell_{\alpha},\ell_{\alpha,\mathsf{for}},\mathbf{i}_{\alpha}}\right)/\sim$

(depending on an appropriate choice of ordering of the interior marked points
on each component). It is straightforward to check that
$(k_{\alpha},K_{\alpha,\mathsf{pun}},\ell_{\alpha},\ell_{\alpha,\mathsf{for}},\mathbf{i}_{\alpha})$
satisfies the conditions required by Definition 53 in order for the
corresponding moduli spaces to be defined.

We define
$\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\subset\overline{\EuS}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$
to be the complement of all puncturable points. One easily checks that

$\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\cap\mathrm{pr}_{\alpha}^{*}\EuS_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}=\mathrm{pr}_{\alpha}^{*}\EuS^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$

in (29).

Now we need to make an appropriate universal choice of strip-like ends and
thick/thin decomposition for our moduli spaces, satisfying slightly different
conditions from those in Section 5.2. A choice of strip-like ends for a fibre
$\Sigma_{r}^{\circ}$ of
$\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\to\overline{\EuR}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$
consists of a choice of strip-like end for each puncturable point
(unpuncturable points do not get a strip-like end), disjoint from each other
and all internal and boundary marked points. The notions of consistency is
defined as before.

Observe that
$\mathrm{Sym}(\ell-\ell_{\mathsf{for}})\times\mathrm{Sym}(\ell_{\mathsf{for}})\subset\mathrm{Sym}(\ell)$
acts on
$\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\to\overline{\EuR}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$,
preserving the forgettability status of the marked points. We will require our
universal choice of strip-like ends to be equivariant for this subgroup.

######  Definition 57.

Suppose that $2\leq|K_{\mathsf{pun}}|+2(\ell-\ell_{\mathsf{for}})$ and
$\mathbf{i}=(0,\ldots,0)$. We define the map

$\mathfrak{f}:\overline{\EuS}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\to\overline{\EuS}_{|K_{\mathsf{pun}}|,\ell-\ell_{\mathsf{for}}}$

which forgets all forgettable and unpuncturable marked points and stabilizes.
That is, one applies the $\mathfrak{f}$-process to obtain a collapsing map
$c\colon\Sigma_{r}\to\Sigma_{r}^{\prime}$, and then also collapses the images
in $\Sigma_{r}^{\prime}$ of the $\mathfrak{f}$-striplike components to obtain
the stabilized curve $\Sigma_{r}^{\prime\prime}$ with a collapsing map
$c^{\prime}\colon\Sigma_{r}^{\prime}\to\Sigma_{r}^{\prime\prime}$; and
$\mathfrak{f}|_{\Sigma_{r}}=c^{\prime}\circ c$.

######  Lemma 58.

We have

$\mathfrak{f}\left(\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\right)\subset\overline{\EuS}^{\circ}_{|K_{\mathsf{pun}}|,\ell-\ell_{\mathsf{for}}}.$

On the other hand, let

$\overline{\EuS}^{\mathfrak{f}\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\subset\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$

denote the complement of all $\mathfrak{f}$-striplike components and their
attached trees of spheres, together with all trees of $\mathfrak{f}$-disclike
components attached to $\mathfrak{f}$-striplike components and their attached
trees of spheres. Then

$\mathfrak{f}^{-1}\left(\overline{\EuS}^{\circ}_{|K_{\mathsf{pun}}|,\ell-\ell_{\mathsf{for}}}\right)=\overline{\EuS}^{\mathfrak{f}\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}.$

######  Proof 7.1.

This is straightforward from the definition of $\mathfrak{f}$.

######  Lemma 59.

There exists a universal choice of strip-like ends for the moduli spaces
$\overline{\EuS}_{k,L_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$,
having the following properties:

  * •

It is consistent;

  * •

It is
$\mathrm{Sym}(\ell-\ell_{\mathsf{for}})\times\mathrm{Sym}(\ell_{\mathsf{for}})$-equivariant;

  * •

If $K_{\mathsf{pun}}=\\{1,\ldots,k\\}$ and $\mathbf{i}=(1,\ldots,1)$, then the
strip-like ends coincide with those used to define
$\EuF(X,D,V_{1},\mathbf{P}^{1})$, under the isomorphism
$\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\simeq\overline{\EuS}^{\circ}_{k,\ell}$;

  * •

If $2\leq|K_{\mathsf{pun}}|+2(\ell-\ell_{\mathsf{for}})$ and
$\mathbf{i}=(0,\ldots,0)$, then

(30)
$\mathfrak{f}(\tilde{\epsilon}_{j}(r,s,t))=\epsilon_{j}(\mathfrak{f}(r),s+S_{j}(r),t),$

where $\tilde{\epsilon}_{j}$ is the strip-like end for
$\overline{\EuS}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}$,
$\epsilon_{j}$ is the corresponding strip-like end used to define
$\EuF(X,D,V_{0},\mathbf{P}^{0})$, and
$S_{j}:\overline{\EuR}_{k,\ell_{0},\ell_{\mathsf{for}},\mathbf{i}}\to\R$ are
smooth functions which are strictly positive for $j=0$ and strictly negative
for $j=1,\ldots,k$. (It follows from the first clause of Lemma 59 that the
image of $\mathfrak{f}\circ\tilde{\epsilon}_{j}$ lies in
$\overline{\EuS}^{\circ}_{|K_{\mathsf{pun}}|,\ell-\ell_{\mathsf{for}}}$.)

  * •

If $|K_{\mathsf{pun}}|=1$, $\ell=\ell_{\mathsf{for}}$, and
$\mathbf{i}=(0,\ldots,0)$, then $\tilde{\epsilon}_{j}(r,\cdot,\cdot)$ extends
to a parametrization of the whole surface
$\Sigma_{r}^{\circ}\simeq\R\times[0,1]$.

######  Proof 7.2.

It is simple to show that one can choose such strip-like ends on the moduli
spaces labelled by $(0,\ldots,0)$ or $(1,\ldots,1)$: in the former case, the
$j$th strip-like ends is uniquely specified by the choice of $S_{j}(r)$, so
one simply needs to choose $S_{j}(r)$ larger (for $j=0$) or smaller
(otherwise) than the $s$-coordinate of any forgettable marked point on the
$i$th strip-like end of $\Sigma_{r}$, and equivariant. (The case
$|K_{\mathsf{pun}}|=1,\ell=\ell_{\mathsf{for}}$ is similar.) It is then
straightforward to extend these choices to all moduli spaces with mixed
labels, following the procedure outlined in [Sei08, Section 9g].

We now introduce the appropriate notion of a thick-thin decomposition: it is
an open set
$\overline{\EuS}_{k,\ell_{0},\ell_{\mathsf{for}},\mathbf{i}}^{thin}\subset\overline{\EuS}^{\circ}_{k,\ell_{0},\ell_{\mathsf{for}},\mathbf{i}}$
whose intersection with each fibre consists of the strip-like ends, together
with a _subset_ of the gluing regions: namely, those formed by gluing two
puncturable boundary points together (gluing regions formed by gluing two
unpuncturable boundary points together do not get included in the thin
region). As with the strip-like ends, the parametrizations of the gluing
regions are required to be
$\mathrm{Sym}(\ell_{\mathsf{for}})\times\mathrm{Sym}(\ell-\ell_{\mathsf{for}})$-equivariant.
We also require that our thick-thin decomposition coincides with that chosen
in the definition of $\EuF(X,D,V_{1},\mathbf{P}^{1})$, when
$\mathbf{i}=(1,\ldots,1)$, and that the image of the thin region under
$\mathfrak{f}$ lies inside the thin region chosen in the definition of
$\EuF(X,D,V_{0},\mathbf{P}^{0})$. By a straightforward extension of Lemma 59,
such a choice of thick-thin decomposition exists. (We remark that the
parametrizations of the gluing regions will automatically satisfy the analogue
of (30), because they are constructed from the strip-like ends which have this
property.)

###  7.3 Construction of the doubled category: perturbation data

We now choose Floer data for each pair of objects in $\EuC$, coinciding with
those used to define $\EuF(X,D,V_{0},\mathbf{P}^{0})$ on pairs of objects
$(K^{0},L^{0})$, coinciding with those used to define
$\EuF(X,D,V_{1},\mathbf{P}^{1})$ on pairs of objects $(K^{1},L^{1})$, and
chosen arbitrarily on pairs of objects with different labels. This allows us
to define the morphism spaces in $\EuC$, precisely as in Definition 37. We
have already defined the embeddings (28) on the level of objects; it is clear
from this convention how to define it on the level of morphisms.

######  Definition 60.

A _universal choice of perturbation data_ consists of pairs

$\mathbf{P}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}=(\mathbf{Y}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}},\mathbf{K}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}})$

for all $(k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L})$ such that

  * •

$(k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i})$ satisfies the
conditions of Definition 53, where $\mathbf{i}=\mathbf{i}(\mathbf{L})$;

  * •

$K_{\mathsf{pun}}\neq\emptyset$;

  * •

$L_{j-1}=L_{j}$ whenever $j\notin K_{\mathsf{pun}}$;

where

$\displaystyle\mathbf{Y}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}$
$\displaystyle\in
C^{\infty}\left(\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}},\EuY_{*}(X,V_{1})\right),$
$\displaystyle\mathbf{K}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}$
$\displaystyle\in\Omega^{1}_{\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}/\overline{\EuR}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}}\left(\EuH\right).$

Notice that the perturbation data are defined at the unpuncturable marked
points.

These perturbation data are required to satisfy the following conditions:

  1. (Floer data) The restriction to the thin regions, and to $\mathfrak{f}$-striplike components (which are isomorphic to $\R\times[0,1]$), is given by the Floer data;666Note that the final condition imposed in Lemma 59 ensures that the requirements imposed over thin regions and $\mathfrak{f}$-striplike components are compatible.

  2. (Equivariant) $\mathbf{P}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}$ is $\mathrm{Sym}(\ell-\ell_{\mathsf{for}})\times\mathrm{Sym}(\ell_{\mathsf{for}})$-equivariant;

  3. (Consistent on $\mathfrak{f}$-stable discs) The restriction of $\mathbf{P}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}$ to $\mathrm{pr}_{\alpha}^{*}\overline{\EuS}^{\circ}_{k_{\alpha},K_{\alpha,\mathsf{pun}},\ell_{\alpha},\ell_{\alpha,\mathsf{for}},\mathbf{L}_{\alpha}}$ is equal to $\mathrm{pr}_{\alpha}^{*}\mathbf{P}_{k_{\alpha},K_{\alpha,\mathsf{pun}},\ell_{\alpha},\ell_{\alpha,\mathsf{for}},\mathbf{L}_{\alpha}}$, for any $\mathfrak{f}$-stable disc vertex $\alpha\in V_{\mathsf{disc}}$ of a tree $\Gamma$;

  4. (Constant on spherical and $\mathfrak{f}$-disclike components) For any spherical or $\mathfrak{f}$-disclike component $\Sigma_{\alpha}$ of a fibre $\Sigma^{\circ}_{r}$, the restriction of $\mathbf{Y}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}$ to $\Sigma_{\alpha}$ is constant and the restriction of $\mathbf{K}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}$ to $\Sigma_{\alpha}$ vanishes;

  5. (Maximum principle) In the case $\ell=0$, we have

$\mathbf{Y}_{k,K_{\mathsf{pun}},0,0,\mathbf{L}}\in
C^{\infty}(\overline{\EuS}^{\circ}_{k,K_{\mathsf{pun}},0,0,\mathbf{i}},\EuY^{\max{}}_{*})\subset
C^{\infty}(\Sigma^{\circ}_{r},\EuY_{*}).$

  6. (Compatibility with $\mathbf{P}^{1}$) In the case $\mathbf{i}=(1,\ldots,1)$ (which determines $K_{\mathsf{pun}}=\\{1,\ldots,k\\}$ uniquely), we have

$\mathbf{P}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}=\mathbf{P}^{1}_{k,\ell,\mathbf{L}^{\prime}},$

where $\mathbf{L}^{\prime}$ is obtained from $\mathbf{L}$ by forgetting the
$0/1$ labels;

  7. (Compatibility with $\mathbf{P}^{0}$) Let $\mathbf{i}=(0,\ldots,0)$; then, under the map $\mathfrak{f}\colon\overline{\EuS}^{\mathfrak{f}\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}\to\overline{\EuS}^{\circ}_{|K_{\mathsf{pun}}|,\ell-\ell_{\mathsf{for}}}$ (cf. Lemma 59) one has

$\left.\mathbf{P}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{L}}\right|_{\overline{\EuS}^{\mathfrak{f}\circ}_{k,K_{\mathsf{pun}},\ell,\ell_{\mathsf{for}},\mathbf{i}}}=\mathfrak{f}^{*}\mathbf{P}^{0}_{|K_{\mathsf{pun}}|,\ell-\ell_{\mathsf{for}},\mathbf{L}^{\prime}},$

where $\mathbf{L}^{\prime}$ is obtained from $\mathbf{L}$ by merging all
$0$-labelled Lagrangians separated by an unpuncturable marked point, then
forgetting the $0/1$ labels.

######  Definition 61.

We say that $\mathbf{P}^{0}$ is _$V_{1}$ -compatible_ if the almost-complex