# Fukaya Categories of Hyperplane Arrangements

Sukjoo Lee, Yin Li, Si-Yang Liu and Cheuk Yu Mak

###### Abstract

To a simple polarized hyperplane arrangement (not necessarily cyclic)
$\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a
Liouville sector) $\left(M(\mathbb{V}),\xi\right)$, where $M(\mathbb{V})$ is
the complement of finitely many hyperplanes in $\mathbb{C}^{d}$, obtained as
the complexifications of the real hyperplanes in $\mathbb{V}$. The Liouville
structure on $M(\mathbb{V})$ comes from a very affine embedding, and the stop
$\xi$ is determined by the polarization. In this article, we study the
symplectic topology of $\left(M(\mathbb{V}),\xi\right)$. In particular, we
prove that their partially wrapped Fukaya categories are generated by
Lagrangian submanifolds associated to the bounded and feasible chambers of
$\mathbb{V}$. A computation of the Fukaya $A_{\infty}$-algebra of these
Lagrangians then enables us to identify the partially wrapped Fukaya
categories $\mathcal{W}(M(\mathbb{V}),\xi)$ with the
$\mathbb{G}_{m}^{d}$-equivariant hypertoric convolution algebras
$\widetilde{B}(\mathbb{V})$ associated to $\mathbb{V}$. This confirms a
conjecture of Lauda-Licata-Manion [26] and provides evidence for the general
conjecture of Lekili-Segal [30] on the equivariant Fukaya categories of
symplectic manifolds with Hamiltonian torus actions.

## 1\. Introduction

### (1a) Main Results.

Very affine varieties are local models of divisor complements in proper
Calabi-Yau varieties, so they provide important examples for the study of
mirror symmetry and symplectic topology, and are studied extensively in the
literature [2, 3, 7, 20, 19, 29]. In this paper, we consider a particular
class of very affine varieties, defined via the combinatorial data of a
polarized hyperplane arrangement.

A polarized hyperplane arrangement in $\mathbb{R}^{d}$ consists of a triple
$\mathbb{V}=(V,\eta,\xi)$, where $V\subset\mathbb{R}^{n}$ is a $d$-dimensional
subspace, $\eta\in\mathbb{R}^{n}/V$, and the polarization
$\xi\in(\mathbb{R}^{n})^{\ast}/V^{\perp}$ (see Definition 3.1). Assume that
$\mathbb{V}$ is simple, meaning that the non-empty intersection of any $k$
hyperplanes is codimension $k$, we can define a Weinstein sector
$(M(\mathbb{V}),\xi)$ as follows. Let $M(\mathbb{V})$ be the complement

$M(\mathbb{V}):=\mathbb{C}^{d}\setminus\bigcup_{i=1}^{n}H_{i},$

where $H_{i}\subset\mathbb{C}^{d}$ is the complexification of the hyperplane
$H_{\mathbb{R},i}\subset\mathbb{R}^{d}$ in the arrangement $\mathbb{V}$. It
follows from the definition that we have an affine algebraic embedding
$M(\mathbb{V})\hookrightarrow(\mathbb{C}^{\ast})^{n}$, equipping
$M(\mathbb{V})$ with the structure of a very affine variety. The Weinstein
structure on $M(\mathbb{V})$ is inherited from the standard Stein structure on
$(\mathbb{C}^{\ast})^{n}$. Moreover, since the polarization $\xi$ can be
interpreted as a superpotential $\xi:M(\mathbb{V})\rightarrow\mathbb{C}$, it
defines a stop on the ideal boundary $\partial_{\infty}M(\mathbb{V})$, which
we will still denote by $\xi$ by abuse of notations. Our goal in this paper is
to identify the partially wrapped Fukaya category
$\mathcal{W}(M(\mathbb{V}),\xi)$ with a convolution algebra associated to the
combinatorial data $\mathbb{V}$. To do this, we will find a generating set of
objects in $\mathcal{W}(M(\mathbb{V}),\xi)$ and show that their endomorphism
$A_{\infty}$-algebra in $\mathcal{W}(M(\mathbb{V}),\xi)$ is formal.

To describe the generating Lagrangians in $\mathcal{W}(M(\mathbb{V}),\xi)$,
note that the hyperplanes in the arrangement $\mathbb{V}$ divide
$\mathbb{R}^{d}$ into a finite number of chambers. Since these hyperplanes
come with their co-orientations, each of these chamber in $\mathbb{R}^{d}$
corresponds to a sign sequence $\alpha\in\\{+,-\\}^{n}$, which we call
feasible. Among the set of feasible sign sequences $\mathscr{F}(\mathbb{V})$,
those for which the linear functional
$\xi:\Delta_{\alpha}\rightarrow\mathbb{R}$ is bounded above are called
bounded, where $\Delta_{\alpha}\subset\mathbb{R}^{d}$ is the (closed) chamber
corresponding to the sign sequence $\alpha\in\mathscr{F}(\mathbb{V})$.. Denote
by $\mathscr{P}(\mathbb{V})\subset\mathscr{F}(\mathbb{V})$ the set bounded-
feasible sign sequences (oe equivalently, chambers). See Figure 1 for a
polarized hyperplane arrangement consisting of three lines in
$\mathbb{R}^{2}$. To each $\alpha\in\mathscr{P}(\mathbb{V})$, we can associate
a Lagrangian submanifold $L_{\alpha}\subset M(\mathbb{V})$, which is the
connected component (open chamber)
$\Delta_{\alpha}^{\circ}\subset\Delta_{\alpha}$ in the real locus

$M(\mathbb{V})\cap\mathbb{R}^{d}\subset\mathbb{C}^{d}.$

This is a non-compact, exact Lagrangian submanifold which gives an object of
the Fukaya category $\mathcal{W}(M(\mathbb{V}),\xi)$.

##### Convention

In this paper, we will work over the field $\mathbb{R}$ of real numbers. This
matches the convention on the algebraic side [12, 13], where the hypertoric
convolution algebras are defined.

We can now state the main results of this paper.

######  1.1 Theorem (Generation, Theorem 4.12).

The partially wrapped Fukaya category $\mathcal{W}(M(\mathbb{V}),\xi)$ is
generated by the Lagrangian submanifolds
$\\{L_{\alpha}\\}_{\alpha\in\mathscr{P}(\mathbb{V})}$.

The key ingredient of the proof is the Sectoral Descent established in [22]
(cf. Theorem 2.18). In the next subsection, we will present the main idea
behind the proof explicitly with the example given in Figure 1.

Theorem 1.1 enables us to identify the partially wrapped Fukaya category
$\mathcal{W}(M(\mathbb{V}),\xi)$ with the endomorphism $A_{\infty}$-algebra of
its objects $\\{L_{\alpha}\\}_{\alpha\in\mathscr{P}(\mathbb{V})}$. Consider
the Fukaya $A_{\infty}$-algebra

$\widetilde{\mathcal{B}}_{\mathbb{V}}:=\bigoplus_{\alpha,\beta\in\mathscr{P}(\mathbb{V})}\mathit{CW}^{\ast}(L_{\alpha},L_{\beta})$

where $\mathit{CW}^{\ast}(-,-)$ denotes the partially wrapped Floer cochain
complex with respect to the stop $\xi$. We will show that
$\widetilde{\mathcal{B}}_{\mathbb{V}}$ is quasi-isomorphic to a convolution
algebra $\widetilde{B}(\mathbb{V})$ introduced in [12], as the universal flat
graded deformation of the hypertoric convolution algebra $B(\mathbb{V})$ [13].
We will provide more background and motivation for this prediction in Section
(1b).

######  1.2 Theorem (Formality, Theorem 5.5).

There is a quasi-isomorphism between $A_{\infty}$-algebras over $\mathbb{R}$:

$\widetilde{\mathcal{B}}_{\mathbb{V}}\cong\widetilde{B}(\mathbb{V}).$

In particular, the Fukaya $A_{\infty}$-algebra
$\widetilde{\mathcal{B}}_{\mathbb{V}}$ is formal.

This proves [26, Conjecture 1.1] in the general case when the hyperplane
arrangement $\mathbb{V}$ is not necessarily cyclic (refer to [26] for the
precise definition of a cyclic hyperplane arrangement, in particular, such a
hyperplane arrangement cannot contain two parallel hyperplanes). When
$\mathbb{V}$ is cyclic, $M(\mathbb{V})$ has a convenient interpretation as a
symmetric product of punctured spheres
$\mathbb{C}\setminus\\{z_{1},\cdots,z_{n}\\}$, and Theorem 1.2 follows
essentially from the computation of the (fully) wrapped Fukaya category
$\mathcal{W}(M(\mathbb{V}))$ by Lekili-Polischuk [29].

In [13, 26], some functorial properties of the category of perfect
$\widetilde{B}(\mathbb{V})$-modules with respect to the deletion and
restriction of hyperplane arrangements are discussed. In Section 6, we discuss
these functors and describe some other natural geometric functors between
Fukaya categories.

$\xi$$---$$-+-$$-++$$++-$$-+-$$+--$$+-+$ Figure 1: A polarized hyperplane
arrangement in $\mathbb{R}^{2}$,
with the bounded-feasible chambers shaded

### (1b) Motivations.

To a polarized hyperplane arrangement $\mathbb{V}$, one can associate a non-
compact hyperkähler variety $\mathfrak{M}_{\mathbb{V}}$, called a hypertoric
variety [8]. Under certain assumptions on $\mathbb{V}$, the variety
$\mathfrak{M}_{\mathbb{V}}$ is smooth. Under a suitable hyperkähler twist,
$\mathfrak{M}_{\mathbb{V}}$ admits the structure of an affine variety. The
induced Liouville structure on $\mathfrak{M}_{\mathbb{V}}$, together with a
(singular) Legendrian $\mathfrak{f}_{\mathbb{V}}$ in the ideal boundary
$\partial_{\infty}\mathfrak{M}_{\mathbb{V}}$ determined by the polarization
$\xi$, provides a stopped Liouville manifold
$(\mathfrak{M}(\mathbb{V}),\mathfrak{f}_{\mathbb{V}})$. Moreover, there is the
complex moment map

$\bar{\mu}_{\mathbb{C}}:\mathfrak{M}_{\mathbb{V}}\rightarrow\mathbb{C}^{d}$

which defines an algebraic $(\mathbb{C}^{\ast})^{d}$-fibration whose
descriminant loci are precisely the union of the complex hyperplanes
$\bigcup_{i=1}^{n}H_{i}$. In this situation, each feasible chamber
$\Delta_{\alpha}$ turns out to define a Lagrangian $V_{\alpha}$ that is
diffeomorphic to the toric variety associated to the polytope
$\Delta_{\alpha}$.

In [13], the authors introduced the hypertoric convolution algebra
$B(\mathbb{V})$, which can be interpreted as the topological convolution
algebra formed by the cohomologies of the complex Lagrangians in
$\mathfrak{M}_{\mathbb{V}}$ associated to the bounded feasible chambers in
$\mathbb{V}$. They also predicted that $B(\mathbb{V})$ can be realized as an
appropriate version of the Fukaya category of $\mathfrak{M}_{\mathbb{V}}$. In
the ongoing project [27], we investigate this question by examining the
infinitesimal (forward-stopped) Fukaya category
$\mathcal{F}(\mathfrak{M}_{\mathbb{V}},\mathfrak{f}_{\mathbb{V}})$ and show
that $\mathcal{F}(\mathfrak{M}_{\mathbb{V}},\mathfrak{f}_{\mathbb{V}})$ is
quasi-equivalent to the category of perfect modules over $B(\mathbb{V})$.

It is a general philosophy due to Aganagic [5] and Teleman [38] that the
symplectic topology of $\mathfrak{M}_{\mathbb{V}}$ is closely related to a
half-dimensional space, which in our case is the Weinstein manifold
$M(\mathbb{V})$ that we study in this paper. The precise connections between
the Fukaya categories of $\mathfrak{M}_{\mathbb{V}}$ and $M(\mathbb{V})$ are
explained in Lekili-Segal [30, Conjecture A]. They conjectured that there
should be a quasi-equivalence

$\mathcal{W}_{\mathbb{G}_{m}^{d}}(\mathfrak{M}_{\mathbb{V}})_{-1}\cong\mathcal{W}(M(\mathbb{V})),$
(1)

where
$\mathcal{W}_{\mathbb{G}_{m}^{d}}(\mathfrak{M}_{\mathbb{V}})_{-1}\subset\mathcal{W}_{\mathbb{G}_{m}^{d}}(\mathfrak{M}_{\mathbb{V}})$
is the spectral component of the $\mathbb{G}_{m}^{d}$-equivariant wrapped
Fukaya category at $-1$. We refer the reader to [30, Section 1.2] for more
details (see also [33] for a related recent work on multiplicative hypertoric
varieties).

Here, what is relevant for us is a partially wrapped version of the
conjectural equivalence (1). Then the corresponding conjecture relating
partially wrapped Fukaya categories takes the form

$\mathcal{F}_{\mathbb{G}_{m}^{d}}(\mathfrak{M}_{\mathbb{V}},\mathfrak{f}_{\mathbb{V}})\cong\mathcal{W}(M(\mathbb{V}),\xi),$
(2)

where
$\mathcal{F}_{\mathbb{G}_{m}^{d}}(\mathfrak{M}_{\mathbb{V}},\mathfrak{f}_{\mathbb{V}})$
consists of objects in
$\mathcal{F}(\mathfrak{M}_{\mathbb{V}},\mathfrak{f}_{\mathbb{V}})$ which are
invariant under the fiberwise $\mathbb{G}_{m}^{d}$-action with respect to the
complex moment map $\bar{\mu}_{\mathbb{C}}$. Note that performing
localizations (stop removal) on both sides of (2) would imply the equivalence
(1) conjectured by Lekili-Segal.

######  1.3 Remark.

There are two versions of equivariant Fukaya categories that can be associated
to the hypertoric variety $\mathfrak{M}_{\mathbb{V}}$, defined respectively by
using the geometric Hamiltonian $T^{d}$-action of the real torus
$T^{d}\subset\mathbb{G}_{m}^{d}$ on the Weinstein manifold
$\mathfrak{M}_{\mathbb{V}}$, and the algebraic $\mathbb{G}_{m}^{d}$-action on
the $A_{\infty}$-category
$\mathcal{F}(\mathfrak{M}_{\mathbb{V}},\mathfrak{f}_{\mathbb{V}})$. For the
former version, see [14, 17, 23, 40, 24, 25], and for the later one, see [28,
35, 36]. For hypertoric varieties, these two versions are expected to be the
same. An oversimplified explanation for this coincidence is that mirror
symmetry exchanges the geometric group action on a space $X$ and the algebraic
group action on the category associated to the mirror of $X$, and hyperkähler
varieties are in some sense “self-mirror”.

Consider the Fukaya $A_{\infty}$-algebra

$\mathcal{B}_{\mathbb{V}}:=\bigoplus_{\alpha,\beta\in\mathscr{P}(\mathbb{V})}\mathit{CF}^{\ast}(V_{\alpha},V_{\beta})$

where $\mathit{CF}^{\ast}$ denotes the infinitesimally wrapped Floer cochain
complexes with respect to the stop
$\mathfrak{f}_{\mathbb{V}}\subset\partial_{\infty}\mathfrak{M}_{\mathbb{V}}$
mentioned above. In [27], we will show that there is a quasi-isomorphism
between $\mathcal{B}_{\mathbb{V}}$ and the hypertoric convolution algebra
$B(\mathbb{V})$, and that the Lagrangians
$\\{V_{\alpha}\\}_{\alpha\in\mathscr{P}(\mathbb{V})}$ split-generate the
Fukaya category
$\mathcal{F}(\mathfrak{M}_{\mathbb{V}},\mathfrak{f}_{\mathbb{V}})$. Together
with conjectural quasi-equivalences (2), it follows that there should be a
quasi-isomorphism between the partially wrapped Fukaya $A_{\infty}$-algebra
$\widetilde{\mathcal{B}}_{\mathbb{V}}$ of $(M(\mathbb{V}),\xi)$ and the
$\mathbb{G}_{m}^{d}$-equivariant convolution algebra
$\widetilde{B}(\mathbb{V})$. Theorem 1.2 confirms that this is indeed the
case.

### (1c) Idea of the Proof.

In this subsection, we give an outline for the proof of our generation result
(Theorem 1.1), by taking $M(\mathbb{V})$ to be the two-dimensional pair-of-
pants (cf. Figure 1). Using the polarization $\xi$, we can cut the Weinstein
sector $(M(\mathbb{V}),\xi)$ into four subsectors as shown in Figure 2 (cf.
Propositions 3.14, 3.22). For $i=1,\dots,4$, we denote by $\mathfrak{s}_{i}$
the $i$-th sectorial hypersurface and by $X_{i,i+1}$ the subsectors after
cutting. Each of these sectors looks pretty similar in the figure, consisting
of a single crossing point and several non-intersecting lines. After curving
the picture, we get four sectors on the right-hand side of Figure 2, and the
generation problem in this case reduces to that of the wrapped Fukaya category
for each subsector.

$\xi$$\Rightarrow$$\mathfrak{s}_{1}$$\mathfrak{s}_{2}$$\mathfrak{s}_{3}$$\mathfrak{s}_{4}$$X_{12}$$X_{23}$$X_{34}$$X_{45}$
Figure 2: Decomposing $2$-dimensional pair-of-pants into sectors

For each of these subsectors, we perform a further cut by hyperplanes which
are transverse to the hyperplane defined by the polarization $\xi$, cutting
each sector into “cubes” as shown in Figure 3. Each cube is then easily seen
to be either isomorphic to $(\mathbb{C}^{\ast})^{2}$ with four stops, or the
stabilization of the $2$-dimensional pair-of-pants
$\mathbb{C}\setminus\\{z_{1},z_{2},z_{3}\\}$. It is well-known that the
wrapped Fukaya category of $(\mathbb{C}^{\ast})^{2}$ is generated by one of
the four real quadrants (a cotangent fiber), and adding stops adds linking
disks associated to these stops as new generators. Moreover, the linking disks
correspond to generating Lagrangians of the stops, and the stops are punctured
spheres. Note that in dimension $2$, the generation question of the wrapped
Fukaya categories of punctured spheres has been solved by Abouzaid-Auroux-
Efimov-Katzarkov-Orlov [3].

$\Rightarrow$ Figure 3: Second cut of the $P_{j}$’s into the standard pieces.

By Sectorial Descent (Theorem 2.18), we get a collection of generating
Lagrangians, which are depicted in Figure 4.

$2$$3$$1$$4$ (a) Linking Disks

(b) Standard Lagrangian $L_{1}^{+}$

(c) Standard Lagrangian $L_{2}^{+}$

(d) Standard Lagrangian $L_{3}^{+}$

Figure 4: Generating Lagrangians for Two-Dimensional Pairs of Pants.

The green line segments in Figure 4(a) represent the linking disks associated
to the corresponding generating Lagrangians of the stop, and the remaining
three pictures describe the three generating Lagrangians obtained from each
crossing point (after gluing together all the sectors). We label the three
standard Lagrangians by $L_{1}^{+},L_{2}^{+},L_{3}^{+}$, where the $+$
indicates that the Lagrangian is the one shifted upwards in the imaginary
direction of $\xi$ in $\mathbb{C}^{2}$. We denote the linking disks by
$D_{ij}$, where $1\leq i\leq 4$ is the index of the stop and $1\leq j\leq 3$
is the order of cocores, from left to right. It remains to show that the
Lagrangians associated to the bounded-feasible chambers $L_{++-}$, $L_{+--}$
and $L_{+-+}$, as depicted in Figure 7, generate all the standard Lagrangians
$L_{1}^{+},L_{2}^{+},L_{3}^{+}$ and the linking disks $D_{ij}$ in the
partially wraped Fukaya category $\mathcal{W}(M(\mathbb{V}),\xi)$. Applying
the surgery exact triangle (cf. Proposition 4.7) we have:

######  1.4 Lemma (cf. Section (4c)).

There are quasi-isomorphisms

$L_{1}^{+}\simeq\mathit{Cone}(L_{++-}\to L_{+--}),\quad
L_{2}^{+}\simeq\mathit{Con}e(L_{+--}\to L_{+-+}),\quad L_{3}^{+}\simeq
L_{+-+}$

between objects in the triangulated $A_{\infty}$-category
$\mathcal{W}(M(\mathbb{V}),\xi)^{\mathit{perf}}$ (cf. Section (2b)), where
$\mathit{Cone}(\bullet\rightarrow\bullet)$ stands for the mapping cone.

Thus it remains to show that the linking disks $D_{ij}$ are generated by the
chamber Lagrangians $L_{++-}$, $L_{+--}$ and $L_{+-+}$. To prove this, let us
first focus on the top sector $X_{12}$ in Figure 5.

$\mathfrak{s}_{1}$$\mathfrak{s}_{2}$$D_{11}$$D_{12}$$D_{13}$$D_{21}$$D_{22}$$D_{23}$$L_{1}^{+}$$\widetilde{L}_{1}^{+}$
Figure 5: Description of $X_{12}$. $\tilde{L}_{1}^{+}$ is a flipping of
$L_{1}^{+}$.  $L_{1}^{-}$$L_{1}^{+}$$D_{11}$ Figure 6: The Wrapping Exact
Triangle

Consider the Lagrangian $L_{1}^{+}$ in this sector. We have the wrapping exact
triangle for linking disks

$L_{1}^{-}\to D_{11}\to L_{1}^{+}\to L_{1}^{-}[1],$

where $L_{1}^{-}$ is the triangular Lagrangian shifted downwards in the
imaginary direction of $\xi$, see Figure 6. This implies that $D_{11}$ is
generated by $L_{1}^{-}$ and $L_{1}^{+}$. On the other hand, if we look at the
second sectorial hypersurface $\mathfrak{s}_{2}$, regarded as a stop in
$X_{12}$, there is a similar wrapping exact triangle

$\widetilde{L}_{1}^{-}\to
D_{21}\to\widetilde{L}_{1}^{+}\to\widetilde{L}_{1}^{-}[1],$

Here, $\widetilde{L}_{1}^{\pm}$ is a flipping of $L_{1}^{\pm}$ (cf. Section
(4e-2)). Since $\tilde{L}_{1}^{\pm}\cong L_{1}^{\pm}$ in
$\mathcal{W}(X_{12})$, we abuse notations and write both as $L_{1}^{\pm}$.
This shows that we could get $L_{1}^{-}$ from $D_{21}$, therefore $D_{11}$ is
generated by $L_{1}^{+}$ and $D_{21}$, hence it can be removed from the
collection of generating objects for $\mathcal{W}\left(X_{12}\right)$. For the
remaining linking disks in the first stop $\mathfrak{s}_{1}$, we can show via
Lagrangian surgery that

######  1.5 Lemma (cf. Section (4e)).

We have quasi-isomorphisms

$D_{22}\simeq\mathit{Cone}(D_{11}\to D_{12})\quad\text{and}\quad D_{13}\simeq
D_{23}$

in the Fukaya category $\mathcal{W}(X_{12})^{\mathit{perf}}$.

######  1.6 Corollary.

The partially wrapped Fukaya category $\mathcal{W}(X_{12})$ is generated by
the Lagrangian $L_{1}^{+}$ and the linking disks $D_{21},D_{22}$ and $D_{23}$.

Note that the quasi-isomorphisms in Lemma 1.5 also hold in the ambient
Liouville sector $(M(\mathbb{V}),\xi)$, by the existence of an inclusion
functor associated to the embedding
$X_{12}\hookrightarrow(M(\mathbb{V}),\xi)$. Under this inclusion, $L_{1}^{+}$
in $\mathcal{W}(X_{12})$ is sent to the standard Lagrangian in Figure 4 (b).

Similar arguments can be carried out for the Liouville sectors $X_{23}$ and
$X_{34}$. We can identify linking disks on the top and at the bottom via
surgery exact sequences, therefore inductively replacing linking disks from
the top to the bottom in the collection of generating objects. Finally, it
remains to deal with the linking disks $D_{41},D_{42}$ and $D_{43}$. To do
this, note that the sector $X_{45}$ at the bottom has a geometric
decomposition:

######  1.7 Lemma.

We have an isomorphism of Liouville sectors
$X_{45}\cong\mathfrak{s}_{4}\times\mathbb{C}_{\operatorname{\mathrm{Re}}\geq
0}$.

This implies that all the linking disks $D_{41},D_{42}$ and $D_{43}$ become
zero objects after pushing forward to the ambient sector
$(M(\mathbb{V}),\xi)$. They can thus be removed from the collection of
generating objects. We conclude from Lemma 1.4 and corollary 1.6 that

######  1.8 Proposition (cf. Theorem 4.12).

The partially wrapped Fukaya category $\mathcal{W}(M(\mathbb{V}),\xi)$ is
generated by the Lagrangians $L_{++-},L_{+--}$ and $L_{+-+}$ corresponding to
bounded and feasible chambers in the hyperplane arrangement $\mathbb{V}$.

Figure 7: The Bounded Feasible Chambers.
The red one is $L_{++-}$, the green one is $L_{+--}$ and the blue one is
$L_{+-+}$

### (1d) Outline.

The paper is organized as follows. In Section 2, we recall the basics of
Liouville sectors and Fukaya categories based on the works [22, 21]. Section 3
contains the main technical input of this paper. We first recall the notion of
a polarized hyperplane arrangement in Section (3a), and then establish some
basic facts about the symplectic geometry of the Weinstein manifold
$M(\mathbb{V})$ in Section (3b). The sectorial cut of
$\left(M(\mathbb{V}),\xi\right)$ is carried out rigorously in Sections (3d)
and (3e). This is followed by a discussion about the generation for each
subsector in Section (3f). In Section 4, we prove Theorem 1.1 using the
surgery exact triangle for Lagrangian submanifolds with clean intersections.
Using a computational technique similar to that of [7], Theorem 1.2 is proved
in Section 5. As an application, we study the functoriality of Fukaya
categories with respect to the deletion and restriction of hyperplane
arrangements in Section 6.

### Acknowledgements

We thank Sheel Ganatra for numerous discussions on his works [21], [22]. We
also thank Aaron Lauda and Nick Sheridan for helpful communications.

We thank Heilbronn focused research grants for the support on this project.
S.L. was supported by the Leverhulme Prize award from the Leverhulme Trust and
the Enhancement Award from the Royal Society University Research Fellowship,
both awarded to Nick Sheridan. Y.L. is partially supported by Simons grant
#385571. S.-Y. L. was partly supported by NSF Career Award # DMS-2048055,
awarded to Sheel Ganatra. C.M. was partly supported by the Royal Society
University Research Fellowship while working on this project.

## 2\. Recollement of Sector Theory

In this section, we recall the definitions and some important structural
results for Fukaya categories of Weinstein manifolds.

### (2a) Liouville Sectors.

Recall that a symplectic manifold is a pair $(X,\omega)$, where $X$ is an even
dimensional real manifold and $\omega\in\Omega^{2}(X)$ is a closed
nondegenerate $2$-form on $X$. In this paper, we will study a particular class
of non-compact symplectic manifolds.

######  2.1 Definition.

A symplectic manifold $(X,\omega)$ is a Liouville manifold if $\omega$ is
exact and there exists a vector field $Z$ on $X$ such that

  1. (i)

$Z$ is complete;

  2. (ii)

$\mathcal{L}_{Z}\omega=\omega$, or equivalently, $Z$ is dual to a primitive of
$\omega$;

  3. (iii)

there exists a (codimension $0$) compact symplectic submanifold with boundary
$(X_{0},\omega)\subset(X,\omega)$ such that $Z$ is outward pointing along
$\partial X_{0}$ and $Z$ has no critical points outside $X_{0}$.

We call this vector field $Z$ a Liouville vector field and the pair
$(\omega,Z)$ a Liouville structure on $X$.

The compact symplectic submanifold $(X_{0},\omega)$ in the item (iii) is
called a Liouville domain. We can also construct a Liouville manifold $X$ out
of the given Liouville domain $(X_{0},\omega)$ by taking its completion:

$X=X_{0}\cup_{\partial X_{0}}\partial X_{0}\times[1,+\infty),$

where the Liouville structure on $\partial X_{0}\times[1,+\infty)_{r}$ is
$\left(\omega=d(r\lambda|_{\partial X_{0}}),Z=r\partial_{r}\right)$.

######  2.2 Definition.

Let $(X,\omega,Z)$ be a Liouville manifold. We call $X$ Weinsten if there
exists an exhausting Morse function $\phi\colon X\to\mathbb{R}$ that is
Lyapunov for $Z$. The triple $(\omega,Z,\phi)$ is called a Weinstein structure
on $X$.

######  2.3 Definition.

A Stein manifold is a complex manifold $(X,J)$ which admits a proper
holomorphic embedding $i\colon X\to\mathbb{C}^{N}$ for some $N$.

It is proved in [16] that the category of Stein manifolds and the category of
Weinstein manifolds are equivalent. Given a Stein manifold $(X,J)$ equipped
with an embedding $i\colon X\hookrightarrow\mathbb{C}^{N}$, one can consider
the squared distance function
$\phi_{\mathbb{C}^{N}}(z)=\displaystyle\sum_{i=1}^{N}|z_{i}|^{2}$. The pull-
back function $\phi\coloneqq\phi_{\mathbb{C}^{N}}\circ i$ defines a pluri-
subharmonic function on $X$, hence also a symplectic form
$\omega_{\phi}=-dd^{\mathbb{C}}\phi$ and a Liouville vector field $\nabla\phi$
dual to $-d^{\mathbb{C}}\phi$.

A submanifold $S\subseteq X$ is coisotropic if $\dim S\geq\frac{1}{2}\dim X$
and for every $s\in S$, $\ker\omega|_{T_{s}S}$ has constant positive rank. It
follows directly that every real hypersurface of $X$ is coisotropic. If
$S\subseteq X$ is coisotropic, write
$\mathcal{D}_{S}\coloneqq\ker\omega|_{S}\subseteq TS$ for the characteristic
foliation of $S$.

######  2.4 Definition.

Let $(X,\omega,Z)$ be a Liouville manifold. A real oriented hypersurface
$S\subseteq X$ is called sectorial if the following conditions are satisfied:

  1. (i)

$Z$ is tangent to $S$ near infinity.

  2. (ii)

There exists a $Z$-invariant neighbourhood
$\operatorname{\mathrm{Nbd}}^{Z}(S)$ of $S\subset X$ and functions

$R,I\colon\operatorname{\mathrm{Nbd}}^{Z}(S)\to\mathbb{R}$

such that $R^{-1}(0)=S$ and

$(R,I)\colon\operatorname{\mathrm{Nbd}}^{Z}(S)\to\mathbb{C}_{-\varepsilon<\operatorname{\mathrm{Re}}<\varepsilon}$

is a trivial symplectic fibration onto the open strip in $\mathbb{C}$
consisting of complex numbers with real part constrained in
$(-\varepsilon,\varepsilon)$, where
$\mathbb{C}_{-\varepsilon<\operatorname{\mathrm{Re}}<\varepsilon}$ is equipped
with the standard symplectic form restricted from $\mathbb{C}$.

  3. (iii)

Under the above decomposition
$\operatorname{\mathrm{Nbd}}^{Z}(S)\xrightarrow{\simeq}(R,I)^{-1}(0)\times\mathbb{C}_{-\varepsilon<\operatorname{\mathrm{Re}}<\varepsilon}$,
the Liouville $1$-form $\lambda_{X}$ on $X$ is sent to
$\lambda_{X}|_{(R,I)^{-1}(0)}+\lambda_{\mathbb{C}}^{\alpha}+df$ for some
$0<\alpha<1$, where

$\displaystyle\lambda_{\mathbb{C}}^{\alpha}=-\alpha ydx+(1-\alpha)xdy,$ (3)

and $f$ is some real-valued function with compact support.

A Liouville sector $Y$ is the closure of one connected component of
$X\backslash\displaystyle\bigsqcup_{i=1}^{k}S_{i}$, where the $S_{i}$’s are
all disjoint sectorial hypersurfaces.

Alternatively, we can describe Liouville sectors in terms of pairs of
Liouville manifolds [18]. A Liouville pair is a pair $(X,F)$ consisting of a
Liouville domain $(X,\omega,Z)$ and a Liouville hypersurface
$(F,\omega|_{F},Z_{F})$ in $\partial X$. We say the pair is Weinstein, if in
addition $X$ is a Weinstein domain and $F\subseteq\partial X$ is a Weinstein
hypersurface. Ganatra-Pardon-Shende [21] showed that we have a canonical way
to construct a Liouville sector out of a Liouville pair, so that $F$ arises as
one of the fibers $(R,I)^{-1}(0)$ for some chosen $R$\- and $I$-functions.
Such a hypersurface $F\subseteq\partial X$, or more generally a boundary
component of a Liouville sector, is usually referred to as a stop of the
sector.

######  2.5 Remark.

This definition is different from the one in [21], but by Proposition 2.25 in
loc.cit., they are equivalent as definitions for Liouville sectors. In other
words, it is not hard to show that our definition gives a Liouville sector,
and conversely, for any given Liouville sector $X$ as in [21], there is a
convex completion of $X$ into a Liouville manifold $\widetilde{X}$, so that
$\partial X\hookrightarrow\widetilde{X}$ is a sectorial hypersurface in
$\widetilde{X}$.

######  2.6 Definition (cf. [22] Definition 1.32).

Let $S_{1},\dotsb,S_{k}$ be a collection of transversely intersecting
sectorial hypersurfaces in $X$. The collection $\\{S_{1},\dotsb,S_{k}\\}$ is
sectorial if there is a collection of pairs of real-valued functions
$\\{(R_{i},I_{i})\\}_{1\leq i\leq k}$ together with positive real numbers
$(\varepsilon_{i})_{1\leq i\leq k}$ such that $R_{i}^{-1}(0)=S_{i}$ and:

  1. (i)

The intersections $\displaystyle\bigcap_{i\in J}S_{i}\eqqcolon S_{J}$ for all
$J\subseteq\\{1,\dotsb,k\\}$ are either empty or coisotropic.

  2. (ii)

When $S_{J}$ is non-empty, there exists a $Z$-invariant neighbourhood
$\operatorname{\mathrm{Nbd}}^{Z}(S_{J})$ of $S_{J}$ so that the functions
$R_{i}$ and $I_{i}$ for $i\in J$ define a trivial symplectic fibration

$\left((R_{i},I_{i})_{i\in
J}\right):\operatorname{\mathrm{Nbd}}^{Z}(S_{J})\rightarrow\mathbb{C}_{-\varepsilon_{i}<\operatorname{\mathrm{Re}}<\varepsilon_{i}}^{|J|}.$
(4)

  3. (iii)

Under the decomposition (4), the Liouville $1$-form $\lambda_{X}$ of $X$
projects to

$\lambda_{X}|_{\left((R_{i},I_{i})_{i\in J}\right)^{-1}(0)}+\sum_{i\in
J}\lambda_{\mathbb{C}}^{\alpha_{i}}+df$

for some finite collection of positive numbers $0<\alpha_{i}<1$ and a real-
valued function $f$ with compact support.

Similar to Definition 2.4, we define a cornered Liouville sector to be the
closure of one connected component of the complement $\displaystyle
X\backslash\bigcup_{i=1}^{k}S_{i}$, where $\\{S_{i}\\}_{1\leq i\leq k}$ is a
transversely intersecting sectorial collection of real hypersurfaces in $X$.

Given a Liouville sector $(X,\omega,Z)$ and a submanifold $S\subseteq X$ which
is cylindrical near infinity, we define the boundary at infinity of $S$ to be
the manifold

$\partial_{\infty}S\coloneqq\left\\{[x]\middle|x\in S\setminus X_{0},\ x\sim
y\Leftrightarrow\exists t\in\mathbb{R},\ \phi^{t}_{Z}(x)=y\right\\},$

where $X_{0}\hookrightarrow X$ is some subdomain such that $Z$ has no zeroes
outside $X_{0}$ (c.f. Definition 2.1). In particular, $\partial_{\infty}X$ is
a contact manifold with boundary.

### (2b) Wrapped Fukaya Categories.

In this subsection, we briefly recall the definition of wrapped Fukaya
categories associated to a cornered Liouville sector, following [21] and [22].
Given a Liouville sector $(X,\omega,Z)$, recall that a Lagrangian submanifold
of $X$ is a submanifold $L\subseteq X$ of half dimension such that
$\omega|_{L}=0$.

######  2.7 Definition.

A Lagrangian submanifold $L\subseteq X$ is exact if the restrction of the
primitive $\lambda|_{L}$ is exact on $L$. It is cylindrical near infinity if
$Z$ is tangent to $L$ near infinity.

To any two transversely intersecting exact cylindrical Lagrangian submanifolds
$L,K\subseteq X$ equipped with $\mathit{Spin}$ structures, we can associate a
($\mathbb{Z}/2$)-graded free $\mathbb{Z}$-module called the Floer cochain
complex

$\mathit{CF}^{\ast}(L,K)\coloneqq\bigoplus_{p\in L\cap K}\mathfrak{o}_{p},$

where $\mathfrak{o}_{p}$ is the orientation line associated to $p\in L\cap K$.
$\mathit{CF}^{\ast}(L,K)$ admits a differential defined by an oriented count
of holomorphic strips with boundary on $L\cap K$, whose cohomology gives the
Lagrangian Floer homology group $\mathit{HF}^{\ast}(L,K)$.

######  2.8 Definition.

An isotopy $\\{L_{t}\\}$ of exact cylindrical Lagrangian submanifolds
$L_{t}\subseteq X$ is called positive if for some, equivalently any, contact
form $\alpha$ on $\partial_{\infty}X$, we have
$\displaystyle\alpha\left(\partial_{t}\partial_{\infty}L_{t}\right)>0$.

A positive isotopy $(L_{0}\leadsto L_{1})\coloneqq\\{L_{t}\\}_{0\leq t\leq 1}$
of Lagrangian submanifolds $L_{t}$ such that $L_{0},L_{1}$ are transversal to
$K$ induces a chain map

$\mathit{CF}^{\ast}(L_{0},K)\to\mathit{CF}^{\ast}(L_{1},K)$

between Floer cochain complexes. We define the wrapped Floer cohomology
associated to $L,K$ by the homotopy colimit

$\mathit{HW}^{\ast}(L,K)\coloneqq\varinjlim_{L\leadsto
L^{\prime}}\mathit{HF}^{\ast}(L^{\prime},K).$

The wrapped Fukaya category $\mathcal{W}(X)$ of a Liouville sector $X$ is an
$A_{\infty}$-category first introduced by Auroux [6], and later rigorously
defined in [22, Definition 2.20, 2.32] and [41]. The objects of
$\mathcal{W}(X)$ are exact cylindrical Lagrangian submanifolds $L\subseteq X$
equipped with $\mathit{Spin}$ structures and the morphism space between $L$
and $K$ is the localization of $\mathit{CF}^{\ast}(L,K)$ along continuation
elements. In particular, the cohomology of the morphism space is
$\mathit{HW}^{\ast}(L,K)$.

For a smooth function $H:[0,1]\times X\to\mathbb{R}$, its Hamiltonian vector
field $X_{H}$ is the unique vector field on $X$ so that
$\omega(-,X_{H})=dH(-)$. The wrapped Fukaya category $\mathcal{W}(X)$ can be
equivalently defined by perturbing the Lagrangians using an appropriate class
of Hamiltonians [41].

The $A_{\infty}$-category $\mathcal{W}(X)$ becomes easier to study from an
algebraic perspective if we pass to its enlargement
$\mathcal{W}(X)^{\mathit{perf}}$, the $A_{\infty}$-category of perfect modules
over $\mathcal{W}(X)$. $\mathcal{W}(X)^{\mathit{perf}}$ is a triangulated
$A_{\infty}$-category and there is a fully faithful embedding
$\mathcal{W}(X)\hookrightarrow\mathcal{W}(X)^{\mathit{perf}}$. We write
$D^{\mathit{perf}}\mathcal{W}(X)=H^{0}\left(\mathcal{W}(X)^{\mathit{perf}}\right)$
for the derived category of $\mathcal{W}(X)$. This is a genuine triangulated
category.

### (2c) Generation of Wrapped Fukaya Categories.

In this section we recall the generation results for wrapped Fukaya categories
of Weinstein sectors proved in [15, 22]. A Liouville sector $(X,\omega,Z)$ is
said to be Weinstein if the convex completion of $X$ is Weinstein and there
exists a symplectic decomposition
$\operatorname{\mathrm{Nbd}}^{Z}_{X}(\partial X)\simeq F\times
T^{\ast}(-\varepsilon,0]$ such that $F$ is a Weinstein hypersurface in
$\partial X$. Similarly, a cornered Liouville sector $(X,\omega,Z)$ is
Weinstein if each sectorial corner $\partial_{I}X$, which is the intersection
of $|I|=k$ sectorial boundaries, decomposes into a product of
$T^{\ast}(\varepsilon,0]^{k}$ and a Weinstein submanifold
$F_{I}\subset\partial_{I}X$.

The wrapped Fukaya categories of Weinstein manifolds behave nicely since the
Lagrangian cocores provide a natural collection of generating objects.

######  2.9 Proposition ([16], Proposition 11.9).

If $(X,\omega,Z,\phi)$ is a Weinstein manifold, then all the unstable
manifolds of the pair $(\phi,Z)$ are coisotropic. In particular, if the index
of a critical point of $\phi$ is $n=\frac{1}{2}\dim X$, then a corresponding
unstable manifold is Lagrangian.

######  2.10 Definition.

We call an unstable manifold of a Weinstein manifold $(X^{2n},\omega,Z,\phi)$
with index $n$ a cocore of $X$.

######  2.11 Theorem ([15, 22]).

The wrapped Fukaya category of a Weinstein manifold $(X,\omega,Z,\phi)$ is
generated by its cocores.

In particular, wrapped Fukaya categories of Weinstein manifolds are finitely
generated.

We then consider the more general case of a cornered Weinstein sector. In this
case, each corner stratum is a product
$\operatorname{\mathrm{Nbd}}^{Z}_{X}(\partial_{I}X)\simeq F_{I}\times
T^{\ast}[-\varepsilon,0]^{|I|}$ for some $I\subseteq\\{1,\dotsb,n\\}$, hence
for $L\subseteq F_{I}$ a cocore of $F_{I}$ and $\gamma\subseteq
T^{\ast}[-\varepsilon,0]^{|I|}$ a product of cotangent fibers, we get an
object $L\times\gamma$ in the wrapped Fukaya category of $X$, which we call a
linking disk associated to the corner $\partial_{I}X$.

######  2.12 Theorem ([15, 22]).

The wrapped Fukaya category of a cornered Weinstein sector $(X,\omega,Z,\phi)$
is generated by its cocores and linking disks associated to all its corner
strata.

We review some examples that will become the building blocks in the proof of
our generation result (cf. Proposition 3.24).

######  2.13 Example.

In [1], the auther proves that for the cotangent bundle $T^{\ast}Q$ over a
compact manifold $Q$ which is $\mathit{Spin}$, the wrapped Fukaya category
$\mathcal{W}(T^{\ast}Q)$ is generated by a single cotangent fiber. This can be
generalized to the case when the manifold $Q$ is compact with corners, in
which case the cotangent bundle of $Q$ is a cornered Weinstein sector. In
particular, consider the cotangent bundle of an interval, as drawn in Figure
8.

$0$$1$ Figure 8: The cotangent bundle of $[0,1]$.

The red boundary components are sectorial boundaries, and the wrapped Fukaya
category $\mathcal{W}(T^{\ast}[0,1])$ is generated by the cotangent fiber
$L\coloneqq T^{\ast}_{0}[0,1]$, which is colored in green in the picture. From
this figure, it is not hard to see that
$\mathit{CW}^{\ast}(L,L)\cong\mathbb{Z}(0)$, therefore
$\mathcal{W}(T^{\ast}[0,1])\simeq\operatorname{\mathrm{Mod}}_{\mathbb{Z}}$ is
quasi-equivalent to the category of modules over $\mathbb{Z}$. By Künneth
theorem, it follows that for any cornered Weinstein sector $X$, we have
$\mathcal{W}(X\times T^{\ast}[0,1])\simeq\mathcal{W}(X)$.

######  2.14 Example.

Let $X=\left(\mathbb{C}^{\ast}\right)^{d}$ be an algebraic torus, which we can
regard symplectically as the cotangent bundle $T^{\ast}T^{d}$ of the real
torus $T^{d}$. By Example 2.13, we know that $\mathcal{W}(T^{\ast}T^{d})$ is
generated by one cotangent fiber, which corresponds to one of the $2^{d}$
quadrants of the real locus of $(\mathbb{C}^{\ast})^{d}$. Cutting
$(\mathbb{C}^{\ast})^{d}$ by sectorial hypersurfaces simply adds linking disks
associated to these sectorial hypersurfaces (and their intersections) to the
generating set.

It is a convenient viewpoint to regard linking disks as generalizations of
cocores. To make it precise, we introduce the notion of a relative core.

######  2.15 Definition.

Let $(X,\omega,Z,\phi)$ be a Weinstein manifold, then the union of the stable
manifolds of all critical points of $\phi$ is called the core of $X$. For a
Weinstein pair $(X,F)$, the relative core of $(X,F)$ is the union
$\mathfrak{c}_{X}\cup(\mathfrak{c}_{F}\times\mathbb{R})$, where
$\mathfrak{c}_{F}$ is the core of $F$ and
$\mathfrak{c}_{F}\times\mathbb{R}\subset X$ consists of points whose positive
Liouville flow converge to $\mathfrak{c}_{F}$.

######  2.16 Definition ([22]).

Let $\mathfrak{c}_{X,F}$ be the relative core of a Weinstein sector $(X,F)$, a
generalized cocore of $(X,F)$ associated to a component of
$\mathfrak{c}_{X,F}$ is a cylindrical exact Lagrangian submanifold
intersecting $\mathfrak{c}_{X,F}$ transversely at a single point in this
component.

Theorem 2.12 remains valid for generalized cocores: the wrapped Fukaya
category of a cornered Weinstein sector $(X,\omega,Z,\phi)$ is generated by a
collection of generalized cocores, one for each component of the relative
core.

In particular, instead of picking the standard generator of $T^{\ast}[0,1]$,
we can pick any curve $\gamma$ in $T^{\ast}[0,1]$ with a single transverse
intersection with the zero section, and the product $\gamma\times L$ for any
generating Lagrangian $L$ in $\mathcal{W}(F)$ will give us a linking disk in
$\mathcal{W}(X)$ to associated to $F$. Figure 9 shows one possible choice of a
generalized cocore in $T^{\ast}[0,1]$.

$0$$1$ Figure 9: An example of generalized cocore.

### (2d) Stopped Inclusion and the Viterbo Restriction.

Let $(X,\lambda)$ and $(X^{\prime},\lambda^{\prime})$ be two Liouville
sectors. A (stopped) inclusion $i\colon X^{\prime}\hookrightarrow X$ is a
proper map that is a diffeomorphism onto its image, such that
$i^{\ast}\lambda=\lambda^{\prime}+df$ for some compactly supported function
$f:X^{\prime}\rightarrow\mathbb{R}$. Ganatra-Pardon-Shende [21] showed that an
inclusion of Liouville sectors induces an $A_{\infty}$-functor between the
wrapped Fukaya categories

$\mathcal{W}(X^{\prime})\to\mathcal{W}(X),$

which we call a stopped inclusion functor. On the other hand, given an
embedding of Liouville domains $X_{0}\hookrightarrow X_{1}$, one would expect
that there is a Viterbo restriction functor

$\mathcal{W}(X_{1})\to\mathcal{W}(X_{0})$

between wrapped Fukaya categories, which is contravariant with respect to
Liouville embedding. The geometric construction, under additional assumptions
on cylindrical exact Lagrangians, has been carried out by Abouzaid-Seidel [4].
Algebraically, the existence of such a restriction functor for $X_{0}$ a
Weinstein domain is proved by Ganatra-Pardon-Shende [22]. The proof in [22,
Section 11.1] suggests that such an $A_{\infty}$-functor also exists for
embeddings of Weinstein sectors $W_{0}\hookrightarrow W_{1}$, provided that
the Weinstein embedding is a diffeomorphism on connected components of
sectorial boundaries. We record this fact in the following lemma.

######  2.17 Lemma.

Let $X_{0}\hookrightarrow X_{1}$ be a Weinstein embedding of Weinstein sectors
that is a diffeomorphism on the boundary, then there exists a Viterbo
restriction functor $\mathcal{W}(X_{1})\to\mathcal{W}(X_{0})$.

### (2e) The Orlov Functor.

Recall from Definition 2.4 that given a Liouville sector $(X,\omega,Z)$, the
boundary $\partial X$ admits a $Z$-invariant neighbourhood
$\operatorname{\mathrm{Nbd}}^{Z}(\partial X)\cong F\times
T^{\ast}[-\varepsilon,0]$ for some $\varepsilon>0$, where $F\subset\partial X$
is a Liouville hypersurface. When $F$ happens to be Weinstein, Theorem 2.12
tells us that the wrapped Fukaya category $\mathcal{W}(F)$ is generated by its
cocores. Example 2.13 implies that the wrapped Fukaya category of
$\operatorname{\mathrm{Nbd}}^{Z}(\partial X)$, regarded as a Weinstein sector,
is quasi-equivalent to $\mathcal{W}(F)$. Together with the stopped inclusion
$\operatorname{\mathrm{Nbd}}^{Z}(F)\hookrightarrow X$, we obtain an
$A_{\infty}$-functor

$\mathsf{Or}:\mathcal{W}(F)\rightarrow\mathcal{W}(X),$ (5)

which is usually known as the Orlov functor. Note that the images of the Orlov
functor is exactly the linking disks associated to the stop $F$.

### (2f) Sectorial Descent.

Suppose a Liouville sector $X$ is covered by the other two Liouville sectors
$X_{1}$ and $X_{2}$, i.e. $X=X_{1}\cup X_{2}$, the stopped inclusion functor
gives us a diagram of wrapped Fukaya categories

${\mathcal{W}(X_{1})}$${\mathcal{W}(X_{12})}$${\mathcal{W}(X)}$${\mathcal{W}(X_{2})}$
(6)

where $X_{12}:=X_{1}\cap X_{2}$. In general, this diagram is an almost
homotopy pushout, and if all the sectors are Weinstein, then it is a genuine
homotopy pushout [22, Theorem 1.28]. More generally, suppose that $X$ can be
covered by a collection of cornered Liouville sectors, i.e. $\displaystyle
X=\bigcup_{i=1}^{k}X_{i}$, the intersection poset of the cornered Liouville
sectors $\\{X_{i}\\}_{i=1}^{k}$ gives a diagram of wrapped Fukaya categories,
leading to the following is a generalization of (6).

######  2.18 Theorem (Sectorial Descent [22], Theorem 1.35).

For any Weinstein sectorial covering $X_{1},\dotsb,X_{k}$ of a Liouville
sector $X$, the induced functor

$\operatorname*{\mathrm{hocolim}}_{\emptyset\neq
I\subseteq\\{1,\dotsb,k\\}}\mathcal{W}\left(\bigcap_{i\in
I}X_{i}\right)\xrightarrow{\simeq}\mathcal{W}(X)$

is a pre-triangulated equivalence.

## 3\. Symplectic Geometry of Hyperplane complements and Generating
Lagrangians

In this section, we first introduce our main object of study, a polarized
hyperplane arrangement $\mathbb{V}=(V,\eta,\xi)$ (see Definition 3.5) and
establish some basic geometric properties for the complement $M(\mathbb{V})$
of the complexified hyperplanes in $\mathbb{V}$. Then we apply Theorem 2.18
together with the generation results reviewed in Examples 2.13 and 2.14 to get
a generating set of Lagrangians for the partially wrapped Fukaya category
$\mathcal{W}(M(\mathbb{V}),\xi)$.

### (3a) Hyperplane Arrangements.

Let $(V,\eta)$ be a pair of $d$-dimensional subspace $V\subset\mathbb{R}^{n}$
and an element $\eta\in\mathbb{R}^{n}/V$. This determines a hyperplane
arrangement of $n$ hyperplanes in $V\cong\mathbb{R}^{d}$. If we present $V$ as
the image of a linear map $\mathbb{R}^{d}\to\mathbb{R}^{n}$ induced by an
$n\times d$ matrix $A=(a_{ij})$ and $w=(w_{1},\dots,w_{n})\in\mathbb{R}^{n}$
respresents $\eta$, then the hyperplanes of the arragement are given by

$H_{\mathbb{R},i}=\left\\{s=(s_{1},\dots,s_{d})\in\mathbb{R}^{d}\ \middle|\
\sum_{j=1}^{d}a_{ij}s_{j}+w_{i}=0\right\\},$ (7)

for $i=1,\dots,n$.

The positive half-spaces of $\mathbb{R}^{d}$ induce co-orientations on the
hyperplanes so that we can assign a region (possibly empty) of the arragement
a sequence $\alpha$ of signs $\\{+,-\\}$ of length $n$. More precisely, given
$\alpha=(\alpha_{1},\dots,\alpha_{n})\in\\{+,-\\}^{n}$, the corresponding
region (also called chamber) $\Delta_{\alpha}\subset V+\eta$ is the set of
points $s$ with

$\alpha_{i}\left(w_{i}+\sum_{j=1}^{d}a_{ij}s_{j}\right)\geq 0,\quad 1\leq
i\leq n.$ (8)

Such a sign sequence $\alpha$ is called feasible, if
$\Delta_{\alpha}\neq\emptyset$. We denote the set of feasible sign sequences
for $\mathbb{V}$ by $\mathscr{F}(\mathbb{V})$. Given two sign sequences
$\alpha$ and $\beta$, define

$d_{\alpha\beta}:=\\#\left\\{1\leq i\leq n|\alpha(i)\neq\beta(i)\right\\}$ (9)

to be the number of sign changes required to turn $\alpha$ into $\beta$. Due
to the bijective correspondence, sign sequences and chambers will often be
mentioned interchangeably.

######  3.1 Definition.

Let $\mathbb{V}=(V,\eta)$ be an arrangement of $n$ hyperplanes as above. A
polarization is an element $\xi\in
V^{\ast}\cong(\mathbb{R}^{n})^{\ast}/V^{\perp}$ such that each element of
$(\mathbb{R}^{n})^{\ast}$ representing $\xi$ has at least $d$ non-zero
entries. We call the triple $\mathbb{V}=(V,\eta,\xi)$ a polarized hyperplane
arrangement of $n$ hyperplanes in $V$.

######  3.2 Remark.

The condition that $\xi$ has at least $d$ non-zero entries is added here to
ensure that $\xi$ is generic enough for our later purposes. This is usually
not required in other literatures.

Choose a lift $\tilde{\xi}\in(\mathbb{R}^{n})^{\ast}$ of $\xi$. We say a sign
sequence $\alpha\in\\{+,-\\}^{n}$ is bounded if the affine linear functional
$\tilde{\xi}$ on $V+\eta$ is bounded above on $\Delta_{\alpha}$. Note that
this is independent of the choice of the lift $\tilde{\xi}$. Denote the set of
bounded sign sequences for $\mathbb{V}$ by $\mathscr{B}(\mathbb{V})$. Let
$\mathscr{P}(\mathbb{V}):=\mathscr{B}(\mathbb{V})\cap\mathscr{F}(\mathbb{V})$
be the set of bounded and feasible sign sequences.

######  3.3 Definition.

A hyperplane arrangement $(V,\eta)$ is called simple if for every subset of
$k$ hyperplanes $H_{\mathbb{R},i_{1}},\dots,H_{\mathbb{R},i_{k}}$, the
intersection $\bigcap_{l=1}^{k}H_{\mathbb{R},i_{l}}$ is either empty or has
codimension $k$.

In fact, this is equivalent to saying that the intersection of hyperplanes is
either transverse or empty (when they are parallel). The following
combinatorial lemma will play a crucial role in the proof of our main result.

######  3.4 Lemma.

Let $\mathbb{V}$ be a simple hyperplane arrangement equipped with a generic
polarization $\xi$. Let
$\mathcal{H}=\\{H_{\mathbb{R},1},\dots,H_{\mathbb{R},n}\\}$ be the
corresponding collection of the hyperplanes in $\mathbb{R}^{d}$. Then the
number of bounded and feasible chambers is equal to the number of
$0$-dimensional strata in the union of all hyperplanes
$\bigcup_{i=1}^{n}H_{\mathbb{R},i}$.

###### Proof.

The proof is based on the double induction on the pair
$(d,n)\in\mathbb{N}^{2}$. Denote $N_{\mathbb{V}}$ the number of bounded-
feasible chambers of $\mathbb{V}$. Due to the genericity of $\xi$, it is
straightforward that $N_{\mathbb{V}}=0$ if and only if there is no
$0$-dimensional strata in $\bigcup_{i=1}^{n}H_{\mathbb{R},i}$. Also, when
$d=1$, it is clear that $N_{\mathbb{V}}=n$.

We first perform induction on $d$. Suppose that the claim holds when $d=r$.
Consider a polarized arrangement $\mathbb{V}$ of $n\geq r$ hyperplanes in
$\mathbb{R}^{r+1}$ and apply the induction on $n$. The initial step $n=r$ is
obvious. Suppose that the claim holds for $n=k>r+1$. We add a new hyperplane
$H_{\mathbb{R}}$ to $\mathcal{H}$ which is in general position, so that the
restrictions of the hyperplanes in $\mathcal{H}$ to $H_{\mathbb{R}}$ is still
a hyperplane arrangement in $H_{\mathbb{R}}\cong\mathbb{R}^{r}$. By induction
hypothesis, we know that the number of bounded and feasible chambers on the
induced hyperplane arrangement in $H_{\mathbb{R}}$ is the same with the number
of 0-dimensional strata. Moreover, the simplicity assumption implies that each
such chamber in $H_{\mathbb{R}}$ either divides one of the bounded and
feasible chambers in $\mathbb{V}$ or adds a new bounded and feasible chamber
to $\mathbb{V}$. This implies that after adding $H_{\mathbb{R}}$, the number
of bounded and feasible chambers in the hyperplane arrangement is increased by
the number of 0-dimensional strata on $H_{\mathbb{R}}$ created by the
intersection
$H_{\mathbb{R}}\cap\left(\bigcup_{i=1}^{k}H_{\mathbb{R},i}\right)$. ∎

######  3.5 Definition.

Let $\mathbb{V}=(V,\eta,\xi)$ be a polarized hyperplane arrangement. A
complexified polarized hyperplane arrangement of $\mathbb{V}$ is a linear
extension of $\mathbb{V}$ over $\mathbb{C}$, denoted by
$\mathbb{V}_{\mathbb{C}}=(V_{\mathbb{C}},\eta_{\mathbb{C}},\xi_{\mathbb{C}})$.
If it is clear from the context, we simply write it as $\mathbb{V}$.

For example, if we write one of the hyperplanes
$H_{\mathbb{R},i}\subset\mathbb{R}^{d}$ as in (7), then its complexification
$H_{i}\subset\mathbb{C}^{d}$ is given by

$H_{i}=\left\\{z\in\mathbb{C}^{d}\ \middle|\
\sum_{j=1}^{d}a_{ij}z_{j}+w_{i}=0\right\\},$ (10)

where $a_{ij},w_{j}\in\mathbb{R}$.

### (3b) Essential Geometry.

In this subsection, we describe the Weinstein structure of $M(\mathbb{V})$ and
show that $M(\mathbb{V})$ is of finite type (Corollary 3.10).

Let $\ell_{i}(z)$ be the defining equation of the hyperplane $H_{i}$ in the
arrangement $\mathbb{V}$ and $\xi(z)=\xi\cdot z$. We assume that the
polarization $\xi$ satisfies the following properties:

  1. (i)

$\xi$ does not belong to the span of the normal vectors of $k$ hyperplanes in
$\mathbb{V}$ for $k\leq d-1$;

  2. (ii)

no two $0$-dimensional crossings lie in the same level set
$\mathrm{Re}(\xi)^{-1}(c)$ for some $c\in\mathbb{R}$.

The set of polarizations satisfying these two properties are generic in the
space of all linear forms on $V^{\ast}$. Consider the affine canonical
embedding $\iota:\mathbb{C}^{d}\to\mathbb{C}^{n}$ given by
$\iota(z)=(\ell_{1}(z),\dots,\ell_{n}(z))$. This induces an embedding of the
complement $M(\mathbb{V}):=\mathbb{C}^{d}\setminus\bigcup_{i=1}^{n}H_{i}$ into
$(\mathbb{C}^{\ast})^{n}$, which we will denote by
$\iota:M(\mathbb{V})\to(\mathbb{C}^{\ast})^{n}$.

Next, we equip $M(\mathbb{V})$ with a symplectic form $\omega$ as follows:
First, consider the symplectic form $\omega_{\phi}$ on
$(\mathbb{C}^{\ast})^{n}$ which is the pullback of the standard symplectic
form on $\mathbb{C}^{2n}$ via the embedding
$i\colon(\mathbb{C}^{\ast})^{n}\hookrightarrow\mathbb{C}^{2n}$ given by
$i(z_{1},\dots,z_{n})=(z_{1},1/z_{1},\dots,z_{n},1/z_{n})$. We have

$\displaystyle\omega_{\phi}$
$\displaystyle=2\sqrt{-1}\sum_{i=1}^{n}\left(dz_{i}\wedge
d\bar{z}_{i}+d\left(\frac{1}{z_{i}}\right)\wedge
d\left(\frac{1}{\bar{z}_{i}}\right)\right)=2\sqrt{-1}\sum_{i=1}^{n}\left(1+\frac{1}{|z_{i}|^{4}}\right)dz_{i}\wedge
d\bar{z}_{i}$
$\displaystyle=4\sum_{i=1}^{n}\left(1+\frac{1}{(x_{i}^{2}+y_{i}^{2})^{2}}\right)dx_{i}\wedge
dy_{i}.$ (11)

In other words, we equip $(\mathbb{C}^{\ast})^{n}$ with the potential function
$\phi=\sum_{i=1}^{n}\left(|z_{i}|^{2}+\frac{1}{|z_{i}|^{2}}\right)$ so that

$\displaystyle-d^{\mathbb{C}}\phi$
$\displaystyle=2\sum_{i=1}^{n}\left(1-\frac{1}{(x_{i}^{2}+y_{i}^{2})^{2}}\right)x_{i}dy_{i}-2\sum_{i=1}^{n}\left(1-\frac{1}{(x_{i}^{2}+y_{i}^{2})^{2}}\right)y_{i}dx_{i}.$
(12)

and $\omega_{\phi}=-dd^{\mathbb{C}}\phi$. Then we take $\omega$ to be the
pullback of the form $\omega_{\phi}$ under the embedding
$\iota:M(\mathbb{V})\hookrightarrow(\mathbb{C}^{\ast})^{n}$:, which gives

$\omega:=\iota^{*}\omega_{\phi}=4\sum_{i=1}^{n}\left(1+\frac{1}{|\ell_{i}|^{4}}\right)d\mathrm{Re}(\ell_{i})\wedge
d\mathrm{Im}(\ell_{i}).$

Notice that $\omega$ depends on $M(\mathbb{V})$ as well as the choice of the
embedding $\iota:\mathbb{C}^{d}\to\mathbb{C}^{n}$. However, different choices
yield deformation-equivalent Weinstein structures.

######  3.6 Lemma.

For some $\vec{c}=(c_{1},\dots,c_{n})\in\mathbb{R}_{>0}^{n}$, the rescaled
embedding
$\iota_{\vec{c}}:=\vec{c}\iota=(c_{1}\ell_{1},\dots,c_{n}\ell_{n}):M(\mathbb{V})\to(\mathbb{C}^{\ast})^{n}$
gives us a new Weinstein structure $(M(\mathbb{V}),\iota_{\vec{c}}^{*}\phi)$
that is Weinstein deformation equivalent to $(M(\mathbb{V}),\iota^{*}\phi)$.

###### Proof.

Consider an isotopy of Stein embeddings $\iota_{t}\colon
M(\mathbb{V})\to\mathbb{C}^{2n}$ via

$\iota_{t}(z)=t\vec{c}\iota(z)+(1-t)\iota(z)$

for $t\in[0,1]$. Then we have
$\iota_{t}^{\ast}\phi(z)=\phi(t\vec{c}\iota(z)+(1-t)\iota(z))$. It is
straightforward to compute the differential and easy to see that the critical
points of $d\phi_{t}(z)$, for all $0\leq t\leq 1$, lie in a compact subset (by
rescaling the previous critical points). Therefore
$(M(\mathbb{V}),\iota^{*}\phi)$ and
$\left(M(\mathbb{V}),\iota_{\vec{c}}^{*}\phi\right)$ are Weinstein deformation
equivalent. ∎

Let $J\subset\\{1,\dots,n\\}$ be an index set such that $\bigcap_{i\in
J}H_{i}\neq\emptyset$. Since all the equation $\ell_{i}$’s are linear, near
the intersection $\bigcap_{i\in J}H_{i}$, the symplectic form $\omega$ can be
written as the sum of the dominant terms and bounded terms:

$\omega=4\sum_{i\in
J}\left(1+\frac{1}{|\ell_{i}|^{4}}\right)d\mathrm{Re}(\ell_{i})\wedge
d\mathrm{Im}(\ell_{i})+4\sum_{i\notin
J}\left(1+\frac{1}{|\ell_{i}|^{4}}\right)d\mathrm{Re}(\ell_{i})\wedge
d\mathrm{Im}(\ell_{i}).$

Geometrically, the symplectic form $\omega$ asymptotically behaves like the
standard symplectic form near $\bigcap_{i\in J}H_{i}\neq\emptyset$. The
following lemma gives an explicit formula for this.

######  3.7 Lemma.

Let $J\subset\\{1,\dots,n\\}$ with $|J|=d$. Enumerate the elements in $J$ by
$1,\dots,d$. Consider the linear change of coordinates
$\Phi:\mathbb{C}^{d}\to\mathbb{C}^{d}$ which pulls $\ell_{i}$ back to $z_{i}$
for $i\in J$. Write

$\displaystyle\ell_{j}=\displaystyle\sum_{i\in J}a_{ij}z_{i},$ (13)

for some $a_{ij}\in\mathbb{R}$.

Then $M(\Phi^{-1}(\mathbb{V}))=\Phi^{-1}\left(M(\mathbb{V})\right)$ and we
have the following formulae:

$\begin{split}\Phi^{\ast}\omega&=4\sum_{i\in
J}\left(1+\frac{1}{|z_{i}|^{4}}+\sum_{j\not\in
J}a_{ij}^{2}\left(1+\frac{1}{|\ell_{j}|^{4}}\right)\right)dx_{i}\wedge
dy_{i}\\\ &+4\sum_{\begin{subarray}{c}i\neq j\\\ i,j\in
J\end{subarray}}\sum_{k\not\in
J}a_{ik}a_{jk}\left(1+\frac{1}{|\ell_{k}|^{4}}\right)dx_{i}\wedge
dy_{j}.\end{split}$ (14)

$\begin{split}\Phi^{*}d\phi&=\sum_{i=1}^{d}2\left(\left(1-\frac{1}{|z_{i}|^{4}}\right)x_{i}+\sum_{j\not\in
J}\left(1-\frac{1}{|\ell_{j}|^{4}}\right)\operatorname{\mathrm{Re}}(\ell_{j})a_{ij}\right)dx_{i}\\\
&+\sum_{i=1}^{d}2\left(\left(1-\frac{1}{|z_{i}|^{4}}\right)y_{i}+\sum_{j\not\in
J}\left(1-\frac{1}{|\ell_{j}|^{4}}\right)\operatorname{\mathrm{Im}}(\ell_{j})a_{ij}\right)dy_{i},\end{split}$
(15)

$\begin{split}\Phi^{*}d^{\mathbb{C}}\phi&=-\sum_{i=1}^{d}2\left(\left(1-\frac{1}{|z_{i}|^{4}}\right)x_{i}+\sum_{j\not\in
J}\left(1-\frac{1}{|\ell_{j}|^{4}}\right)\operatorname{\mathrm{Re}}(\ell_{j})a_{ij}\right)dy_{i}\\\
&+\sum_{i=1}^{d}2\left(\left(1-\frac{1}{|z_{i}|^{4}}\right)y_{i}+\sum_{j\not\in
J}\left(1-\frac{1}{|\ell_{j}|^{4}}\right)\operatorname{\mathrm{Im}}(\ell_{j})a_{ij}\right)dx_{i},\end{split}$
(16)

The proof of Lemma 3.7 is a straightforward computation. For simplicity, we
will drop $\Phi^{*}$ from the notations if it is clear from the context. We
will find it convenient to use Lemma 3.7 when doing computations.

To write the formulae above in a more compact form, let $A$ be the
$(n-d)\times d$-matrix whose $(i,j)$-th entry is
$\left(a_{ji}\sqrt{1+\dfrac{1}{|\ell_{i}|^{4}}}\right)$ and let
$D=\left(\left(1+\dfrac{1}{|z_{i}|^{4}}\right)\delta_{ij}\right)$ be the
diagonal matrix. Under the standard basis
$\left\\{\dfrac{\partial}{\partial\vec{x}},\dfrac{\partial}{\partial\vec{y}}\right\\}$,
we can write the symplectic form $\omega$ in the matrix form

$\displaystyle\omega=\begin{pmatrix}0&4(D+A^{T}A)\\\
-4(D+A^{T}A)&0\end{pmatrix},$ (17)

where $D+A^{T}A$ is invertible.

Similarly, consider the matrices

$\tilde{A}=\left(a_{ji}\sqrt{1-\frac{1}{|\ell_{i}|^{4}}}\right),\quad\tilde{D}=\left(\left(1-\dfrac{1}{|z_{i}|^{4}}\right)\delta_{ij}\right)$

and

$[d^{\mathbb{C}}\phi]_{x}=2\vec{x}^{T}(\tilde{D}+\tilde{A}^{T}\tilde{A}),\quad[d^{\mathbb{C}}\phi]_{y}=2\vec{y}^{T}\left(\tilde{D}+\tilde{A}^{T}\tilde{A}\right),$

then we have
$d^{\mathbb{C}}\phi=-[d^{\mathbb{C}}\phi]_{x}d\vec{y}+[d^{\mathbb{C}}\phi]_{y}d\vec{x}$.
We can also compute the Hamiltonian vector fields
$X_{\operatorname{\mathrm{Re}}(\xi)}$ and
$X_{\operatorname{\mathrm{Im}}(\xi)}$ as follows.

######  3.8 Lemma.

Write $\xi=\sum_{i=1}^{d}b_{i}z_{i}$, where $b_{i}\in\mathbb{R}$. Then
$X_{\operatorname{\mathrm{Re}}(\xi)}$ is a purely imaginary vector field given
by the matrix formula

$\displaystyle
X_{\operatorname{\mathrm{Re}}(\xi)}=\frac{1}{4}\left(\frac{\partial}{\partial\vec{y}}\right)^{T}(D+A^{T}A)^{-1}\vec{b},$
(18)

where $\vec{b}$ refers to the vector $(b_{1},\dots,b_{d})^{T}$. Similarly, the
formula for $X_{\operatorname{\mathrm{Im}}(\xi)}$ is given by

$\displaystyle
X_{\operatorname{\mathrm{Im}}(\xi)}=-\frac{1}{4}\left(\frac{\partial}{\partial\vec{x}}\right)^{T}\left(D+A^{T}A\right)^{-1}\vec{b}.$
(19)

As a result, the square of the length of the vector field
$X_{\operatorname{\mathrm{Re}}\xi}$ is

$\left\|X_{\operatorname{\mathrm{Re}}(\xi)}\right\|^{2}=\omega\left(X_{\operatorname{\mathrm{Re}}(\xi)},X_{\operatorname{\mathrm{Im}}(\xi)}\right)=\frac{1}{4}\left(\vec{b}\right)^{T}\left(D+A^{T}A\right)^{-1}\vec{b}.$

###### Proof.

Since
$d\operatorname{\mathrm{Re}}(\xi)=\displaystyle\sum_{i=1}^{d}b_{i}dx_{i}$ and
$d\operatorname{\mathrm{Im}}(\xi)=\displaystyle\sum_{i=1}^{d}b_{i}dy_{i}$, we
get

$\begin{split}b_{i}&=d\operatorname{\mathrm{Re}}(\xi)\left(\frac{\partial}{\partial
x_{i}}\right)=\omega\left(\frac{\partial}{\partial
x_{i}},X_{\operatorname{\mathrm{Re}}(\xi)}\right)\\\
&=4\sum_{j=1}^{d}\left(1+\frac{1}{|z_{j}|^{4}}+\sum_{k\not\in
I}\left(1+\frac{1}{|\ell_{k}|^{4}}\right)a_{ik}a_{jk}\right)dy_{j}\left(X_{\operatorname{\mathrm{Re}}(\xi)}\right),\end{split}$
$\begin{split}0&=d\operatorname{\mathrm{Re}}(\xi)\left(\frac{\partial}{\partial
y_{i}}\right)=\omega\left(\frac{\partial}{\partial
y_{i}},X_{\operatorname{\mathrm{Re}}(\xi)}\right)\\\
&=-4\sum_{j=1}^{d}\left(1+\frac{1}{|z_{j}|^{4}}+\sum_{k\not\in
I}\left(1+\frac{1}{|\ell_{k}|^{4}}\right)a_{ik}a_{jk}\right)dx_{j}\left(X_{\operatorname{\mathrm{Re}}(\xi)}\right),\end{split}$

then the equations above can be rewritten as

$\vec{b}=4\left(D+A^{T}A\right)d\vec{y}\left(X_{\operatorname{\mathrm{Re}}(\xi)}\right),\quad\vec{0}=4\left(D+A^{T}A\right)d\vec{x}\left(X_{\operatorname{\mathrm{Re}}(\xi)}\right).$

Using (17) and the non-degeneracy of the matrix $D+A^{T}A$, we conclude the
result.

∎

######  3.9 Lemma.

Suppose that

$\displaystyle\bigcap_{\begin{subarray}{c}j\in K\\\
|K|=d+1\end{subarray}}\left\\{|\ell_{j}|\leq 1\right\\}=\emptyset.$ (20)

Then there are functions $C_{I}(z),C_{R}(z)>0$ and two constants
$0<c_{1}<c_{2}$ such that for $|\operatorname{\mathrm{Im}}\xi|$ sufficiently
large, we have $c_{1}<C_{I}(z)<c_{2}$ and

$-d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Im}}(\xi)})=C_{I}(z)\operatorname{\mathrm{Im}}(\xi).$

Similarly, for $|\operatorname{\mathrm{Re}}(\xi)|$ sufficiently large, we have
$c_{1}<C_{R}(z)<c_{2}$ and

$-d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Re}}(\xi)})=C_{R}(z)\operatorname{\mathrm{Re}}(\xi).$

###### Proof.

For any $J\subset\\{1,\dots,n\\}$ such that $|J|=d$ and $\epsilon>0$, we
define

$\displaystyle M(\mathbb{V})_{J,\epsilon}=\\{z\in
M(\mathbb{V})||\ell_{j}(z)|>\epsilon\text{ for all }j\in J\\}.$ (21)

By assumption (20), the open sets $\\{M(\mathbb{V})_{J,1}||J|=d\\}$ form a
finite cover of $M(\mathbb{V})$. Consider a single open set
$M(\mathbb{V})_{J,1}$ and let $\\{\ell_{j}|j\in J\\}$ be the coordinate
functions $\\{z_{i}|i=1,\dots,d\\}$.

By (16), (18) and (19), we can write
$d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Re}}(\xi)})$ and
$d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Im}}(\xi)})$ as

$\displaystyle
d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Re}}(\xi)})=-\frac{1}{2}\vec{x}^{T}\left(\tilde{D}+\tilde{A}^{T}\tilde{A}\right)\left(D+A^{T}A\right)^{-1}\vec{b},$
$\displaystyle
d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Im}}(\xi)})=-\frac{1}{2}\vec{y}^{T}\left(\tilde{D}+\tilde{A}^{T}\tilde{A}\right)\left(D+A^{T}A\right)^{-1}\vec{b}.$

When $|\operatorname{\mathrm{Im}}(\xi)|\to\infty$, the inequality
$|\operatorname{\mathrm{Im}}(\xi)|=\displaystyle\left|\sum_{i=1}^{d}b_{i}y_{i}\right|\leq\left(\sum
b_{i}^{2}\right)^{1/2}\left(\sum y_{i}^{2}\right)^{1/2}$ implies that there
must be at least one coordinate $y_{i}$ with $y_{j}\to\infty$. Using the
decomposition

$\tilde{D}+\tilde{A}^{T}\tilde{A}=\left(D+A^{T}A\right)-\left(D-\tilde{D}+A^{T}A-\tilde{A}^{T}\tilde{A}\right),$

we can write $-d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Im}}(\xi)})$ as

$\displaystyle-d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Im}}\xi})$
$\displaystyle=\vec{y}^{T}\vec{b}-\underbrace{\vec{y}^{T}\left(D-\tilde{D}+A^{T}A-\tilde{A}^{T}\tilde{A}\right)\left(D+A^{T}A\right)^{-1}\vec{b}}_{\text{remainder
term }R}=\operatorname{\mathrm{Im}}(\xi)-R.$

Note that the term

$\vec{y}^{T}\left(D-\tilde{D}+A^{T}A-\tilde{A}^{T}\tilde{A}\right)=\left(\frac{2y_{i}}{|z_{i}|^{4}}+\sum_{j\not\in
J}\frac{2}{|\ell_{j}|^{4}}\operatorname{\mathrm{Im}}(\ell_{j})a_{ij}\right),$

in the remainder matrix $R$ is bounded when all the $|z_{i}|$’s are bounded
away from $0$ since $|\ell_{j}|\geq 1$ for all $j\not\in J$. Since the vector
$\left(D+A^{T}A\right)^{-1}\vec{b}$ is also bounded, it follows that the
remainder term is bounded as $|z_{i}|$ are bounded away from $0$. Let
$K\subset\\{1,\dots,d\\}$ be a subset such that $|z_{i}|\to 0$ for $i\in K$
and $|z_{j}|$ is bounded away from $0$ for $j\notin K$, we can write
$D+A^{T}A$ as a $2\times 2$ block matrix

$D+A^{T}A={\begin{pmatrix}E&F\\\ G&H\end{pmatrix}}$

where the first block $E$ records all the entries coming from $K$. By [31,
Theorem 2.1], the the inverse $(D+A^{T}A)^{-1}$ is given by

$\begin{pmatrix}E^{-1}+E^{-1}F(H-GE^{-1}F)^{-1}GE^{-1}&-E^{-1}F(H-GE^{-1}F)^{-1}\\\
-(H-GE^{-1}F)^{-1}GE^{-1}&(H-GE^{-1}F)^{-1}\end{pmatrix}$ (22)

When the coordinates $(z_{i})_{i\in K}$ are sufficiently small, $E^{-1}$ is
close to a zero matrix and the determinants $|F|,|G|,|H|$ are bounded so
$(D+A^{T}A)^{-1}$ is close to a $2\times 2$ block matrix where only the bottom
right block is non-zero (except when $K=\\{1,\dots,d\\}$, in which case the
whole matrix $(D+A^{T}A)^{-1}$ is close to the zero matrix). On the other
hand, when $(z_{i})_{i\in K}$ are sufficiently small, the blow-up terms in
$\left(D-\tilde{D}+A^{T}A-\tilde{A}^{T}\tilde{A}\right)$ are the diagonal
entries $\frac{1}{|z_{i}|^{4}}$ for $i\in K$, which go to infinity at the same
rate that $E^{-1}$ is going to $0$. As a result, the terms in
$\vec{y}^{T}\left(D-\tilde{D}+A^{T}A-\tilde{A}^{T}\tilde{A}\right)(D+A^{T}A)^{-1}$
are still bounded even when some $z_{i}$’s are approaching $0$. Therefore $R$
is bounded for all possible $|z_{i}|$’s, which implies that as
$|\operatorname{\mathrm{Im}}(\xi)|\to\infty$,
$-d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Im}}(\xi)})\approx\operatorname{\mathrm{Im}}(\xi)$.
This proves the claim for
$-d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Im}}(\xi)})$. The case of
$-d^{\mathbb{C}}\phi(X_{\operatorname{\mathrm{Re}}(\xi)})$ can be dealt with
in a similar way.

∎

######  3.10 Corollary.

The critical points of the function $\phi\colon M(\mathbb{V})\to\mathbb{R}$
lie in a compact subset of $M(\mathbb{V})$. As a consequence,
$(M(\mathbb{V}),\omega,\nabla\phi,\phi)$ is a Weinstein manifold of finite
type.

###### Proof.

By Lemma 3.6, at the cost of replacing $\ell_{i}$ by $c_{i}\ell_{i}$, we can
assume that (20) is satisfied without changing the Weinstein deformation type.
Then by Lemma 3.9, $d\phi$ is non-zero when
$|\operatorname{\mathrm{Re}}(\xi)|$ or $|\operatorname{\mathrm{Im}}(\xi)|$ is
large. This is true for any polarization $\xi$. Since the zeros of $d\phi$
does not depend on $\xi$, we conclude that they lie in a compact set. ∎

### (3c) Some Useful Lemmas.

Before discussing the symplectic geometry of $M(\mathbb{V})$, we collect here
some lemmas that will be used later. We start from a simple fact in linear
algebra.

######  3.11 Lemma.

The inverse matrix $\omega^{-1}$ of $\omega$, as a section of
$\operatorname{\mathrm{Hom}}(T^{*}M(\mathbb{V}),TM(\mathbb{V}))$ over
$M(\mathbb{V})$, extends smoothly over $\mathbb{C}^{d}$. Moreover, for a point
$p\in\bigcap_{i\in J}H_{i}$, the image of $\omega^{-1}|_{p}$ lies in
$\bigcap_{i\in J}T_{p}H_{i}$.

###### Proof.

Since the complex structure $J_{\mathbb{V}}$ on $M(\mathbb{V})$ is
$\omega$-compatible, we have $-g\circ J_{\mathbb{V}}=\omega:TM(\mathbb{V})\to
T^{*}M(\mathbb{V})$, where $g$ is the Riemannian metric. As a result,
$\omega^{-1}=J_{\mathbb{V}}\circ g^{-1}:T^{*}M(\mathbb{V})\to TM(\mathbb{V})$.
Since $J_{\mathbb{V}}$ extends smoothly over $\mathbb{C}^{d}$, it suffices to
check that $g^{-1}$ extends smoothly over $\mathbb{C}^{d}$.

Let $p\in\bigcap_{i\in J}H_{i}$ and $p\notin H_{i}$ if $i\notin J$. In a
neighborhood of $p$, we write $g=(g_{ij})$ as a $2\times 2$ block matrix

$R={\begin{pmatrix}E&F\\\ G&H\end{pmatrix}}$

such that $E=(g_{ij})_{i,j\in J}$. Note that $R$ is only well-defined on the
complement $M(\mathbb{V})$ of the hyperplanes. By [31, Theorem 2.1], $R^{-1}$
is given by (22).

On the other hand, we know that as we approach the point $p$, $E^{-1}$ goes to
the zero matrix and the determinants $|F|,|G|\text{ and }|H|$ are bounded.
Thus the inverse matrix $R^{-1}$ converges to $\begin{pmatrix}0&0\\\
0&H^{-1}\end{pmatrix}$, which clearly extends smoothly over $p$.

Moreover, the image of $R^{-1}$ at $p$ is spanned by
$\\{\partial_{x_{i}},\partial_{y_{i}}\\}_{i\notin J}$, which gives precisely
the subspace $\bigcap_{i\in J}T_{p}H_{i}$. Since $\bigcap_{i\in J}T_{p}H_{i}$
is $J_{\mathbb{V}}$-invariant, the image of $\omega^{-1}$ is contained in
$\bigcap_{i\in J}T_{p}H_{i}$. ∎

######  3.12 Corollary.

For any smooth function $H:[0,1]\times\mathbb{C}^{d}\to\mathbb{R}$, the
Hamiltonian vector field of $H|_{M(\mathbb{V})}$ on $M(\mathbb{V})$ extends to
a smooth vector field over $\mathbb{C}^{d}$ that is tangent to the strata in
the union $\bigcup_{i=1}^{n}H_{i}$.

###### Proof.

The $1$-form $dH_{t}$ is smooth over $\mathbb{C}^{d}$. The result follows from
Lemma 3.11. ∎

The following lemma deals with the completeness of vector fields away from
lower-dimensional strata.

######  3.13 Lemma.

Let $X\subseteq\bar{X}$ be an open submanifold of a smooth manifold $\bar{X}$
(possibly with boundary) so that $\bar{X}\setminus X$ is a stratified space of
lower dimension and $V$ a vector field on $X$. If $V$ can be extended to a
complete $C^{1}$ (e.g. smooth) vector field $\bar{V}$ on $\bar{X}$ such that
$\bar{V}|_{\bar{X}\setminus X}$ is tangent to the tangent space of the
respective stratum of $\bar{X}\setminus X$, then the flow of $V$ on $X$ will
not escape to $\bar{X}\setminus X$.

###### Proof.

Since $\bar{V}$ is complete in $\bar{X}$, for every $t>0$ the flow
$\psi^{t}\colon\bar{X}\to\bar{X}$ is a well-defined diffeomorphism sending the
stratified space $\bar{X}\setminus X$ to itself. Therefore $\psi^{t}$
restricts to a diffeomorphism on $X$, which is also the flow of $V$ on $X$.
This shows that the flow of $V$ on $X$ will not escape to $\bar{X}\setminus
X$. ∎

### (3d) Homotopy Sectoriality.

In Section (3b), we equipped $M(\mathbb{V})$ with a natural Weinstein
structure. In order to apply the Sectorial Descent (Theorem 2.18), we need to
find a sectorial collection of real hypersurfaces in $M(\mathbb{V})$. In this
subsection, we will prove a slightly more general result:

Given any collection of transversely intersecting complex hyperplanes
$\left\\{S_{i}=\xi_{i}^{-1}(c_{i})\right\\}_{1\leq i\leq k}$ in
$\mathbb{C}^{d}$, which are also transversal to the hyperplanes at infinity
i.e. the complex hyperplanes $H_{1},\cdots,H_{n}\subset\mathbb{C}^{d}$, there
exists a Liouville deformation of $M(\mathbb{V})$ such that the real
hyperplanes $\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i})$’s form a
sectorial collection after deformation.

The proof involves a lengthy calculation so it is helpful to explain the steps
before we proceed. Pick a subcollection of $\\{S_{i}\\}$ such that the
intersection $\cap_{j=1}^{s}S_{i_{j}}$ is non-empty. We need to explain how to
deform the Liouville structure near $\cap_{j=1}^{s}S_{i_{j}}$ in a way that is
compatible with all other subcollections with non-empty intersections. We
first construct a collection of commuting vector fields
$\\{X_{\operatorname{\mathrm{Re}}(\xi_{j})},V_{j}\\}_{j=1}^{s}$ near
$\cap_{j=1}^{s}S_{i_{j}}$ such that
$\omega(V_{j},X_{\operatorname{\mathrm{Re}}(\xi_{j^{\prime}})})=\delta_{j,j^{\prime}}$,
and
$\omega(X_{\operatorname{\mathrm{Re}}(\xi_{j})},X_{\operatorname{\mathrm{Re}}(\xi_{j^{\prime}})})=\omega(V_{j},V_{j^{\prime}})=0$
(cf. Lemma 3.15 and 3.21). By integrating along (the normalization of) these
vector fields, it gives the corresponding $s$ pairs of $R$ and $I$ functions
(in the sense of Definition 2.6), denoted by $(R_{j},I_{j})$, which in turn
produce a symplectic trivial fibration from a neighborhood of
$\cap_{j=1}^{s}S_{i_{j}}$ to
$\mathbb{C}_{\varepsilon}^{s}:=((-2\varepsilon,2\varepsilon)\times\mathbb{R})^{s}$
(cf. Corollary 3.17). We will also show that $I_{j}$ is close to
$\operatorname{\mathrm{Im}}(\xi_{j})$ when $\varepsilon$ is small (cf.
Corollary 3.18 and Lemma 3.19). We can therefore apply Lemma 3.9 to give us
control of $d^{c}\phi$ in terms of $I_{j}$ outside a compact set. We write
down an explicit deformation of the Liouville structure using the product
coordinates obtained from the trivial fibration to
$\mathbb{C}_{\varepsilon}^{s}$ (cf. (26) and (27)), after an explicit
calculation together with the estimates of $d^{c}\phi$, we show that the
deformation is a Liovuille deformation such that the skeleton lies in a
compact set for all time (cf. Proposition 3.14 and 3.22). The construction of
the commuting vector fields and the explicit deformation of Liouville
structure are canonical enough that they are compatible when we consider
different subcollections of $\\{S_{i}\\}$.

We start with the simplest case where there is only one hyperplane.

######  3.14 Proposition.

Let $c\in\mathbb{R}$ be a real number so that
$\operatorname{\mathrm{Re}}(\xi)^{-1}(c)$ intersects all hyperplanes
$H_{1},\cdots,H_{n}$ transversely. Then there is a Liouville deformation
$(M(\mathbb{V}),\omega,Z_{t})$ with $Z_{0}=\nabla\phi$ and $Z_{1}$ is tangent
to the sectorial hypersurface $\\{\operatorname{\mathrm{Re}}(\xi)=c\\}$
outside of a compact subset. Moreover, the skeleton
$\displaystyle\overline{\bigcup_{0\leq t\leq
1}\operatorname{\mathrm{Skel}}(M(\mathbb{V}),Z_{t})}$ is compact.

As explained earlier, the first step is to construct commuting vector fields
$X_{R}$ and $V$.

######  3.15 Lemma.

Let $c\in\mathbb{R}$ so that $\operatorname{\mathrm{Re}}(\xi)^{-1}(c)$
intersects all hyperplanes $H_{1},\cdots,H_{n}$ transversely. Then there is a
small neighborhood
$\operatorname{\mathrm{Nbd}}\left(\left\\{\operatorname{\mathrm{Re}}(\xi)=c\right\\}\right)$
of $\\{\operatorname{\mathrm{Re}}(\xi)=c\\}$ and a vector field $V$ in
$\operatorname{\mathrm{Nbd}}\left(\left\\{\operatorname{\mathrm{Re}}(\xi)=c\right\\}\right)$
such that $\omega(V,X_{\operatorname{\mathrm{Re}}(\xi)})=1$ and the flow of
$V$ is well-defined in
$\operatorname{\mathrm{Nbd}}\left(\left\\{\operatorname{\mathrm{Re}}(\xi)=c\right\\}\right)\setminus\left(\cup
H_{i}\right)$ for some positive and negative time.

###### Proof.

If we take
$V=-JX_{\operatorname{\mathrm{Re}}(\xi)}/\|X_{\operatorname{\mathrm{Re}}(\xi)}\|^{2}$,
then
$\omega(V,X_{\operatorname{\mathrm{Re}}(\xi)})=g(X_{\operatorname{\mathrm{Re}}(\xi)}/\|X_{\operatorname{\mathrm{Re}}(\xi)}\|^{2},X_{\operatorname{\mathrm{Re}}(\xi)})=1$.
Therefore, it suffices to show that $\|X_{\operatorname{\mathrm{Re}}(\xi)}\|$
is non-vanishing and the flow of $V$ in
$\operatorname{\mathrm{Nbd}}\left(\left\\{\operatorname{\mathrm{Re}}(\xi)=c\right\\}\right)\setminus\left(\cup
H_{i}\right)$ is well-defined for small time.

To show this, consider the open cover
$\left\\{M(\mathbb{V})_{J,\epsilon}\right\\}$ of $M(\mathbb{V})$ defined by
(21) and identify the linear forms in $\ell_{j}$, $j\in J$ with the coordinate
functions $z_{1},\dotsb,z_{d}$. Then the $1$-form $d^{\mathbb{C}}\phi$ on
$M(\mathbb{V})_{J,\epsilon}$ takes the form (16).

Looking at the entries, we see that there is a constant $c>0$ such that the
matrix $(D+A^{T}A)-c\operatorname{\mathrm{Id}}$ is positive definite. Recall
that we are working over the subset $M(\mathbb{V})_{J,\epsilon}$, where
$|\ell_{j}|>\varepsilon$ for $j\notin J$, therefore the matrix $D+A^{T}A$
blows up only when some of the $z_{i}$’s approaches $0$. We apply the same
trick used in the proof of Lemma 3.9.

Let $K\subset\\{1,\dots,d\\}$ and consider a region $\mathcal{D}_{K}$ where
$(z_{i})_{i\in K}$ can be arbitrarily close to $0$ but $(z_{i})_{i\notin K}$
are bounded away from $0$. Write $D+A^{T}A$ as a $2\times 2$ block matrix,
where the first block records all the with indices in $K$. As in the proof of
Lemma 3.11, we can apply [31, Theorem 2.1] to compute the inverse of
$D+A^{T}A$. When the coordinates $(z_{i})_{i\in K}$ are sufficiently small,
$(D+A^{T}A)^{-1}$ converges to a $2\times 2$ block matrix where only the
bottom right block, denoted by $P$, is non-zero (except when
$K=\\{1,\dots,d\\}$, in which case $(D+A^{T}A)^{-1}$ converges to the $0$
matrix). In fact, we also know that there are constants
$C^{\prime}>c^{\prime}>0$ (depending on $\mathcal{D}_{K}$) such that
$P-c^{\prime}\operatorname{\mathrm{Id}}$ and
$C^{\prime}\operatorname{\mathrm{Id}}-P$ are positive definite. The genericity
of our choice of $\xi$ (see Definition 3.1) implies that all the $b_{i}$ are
non-zero. It follows that we have

$0<(\min_{i}|b_{i}|)\sqrt{c^{\prime}}\leq\|X_{\operatorname{\mathrm{Re}}(\xi)}\|\leq(\max_{i}|b_{i}|)\sqrt{C^{\prime}}$
(23)

in $\mathcal{D}_{K}$ as long as $K\neq\\{1,\dots,d\\}$. The transversality
assumption on $\\{\operatorname{\mathrm{Re}}(\xi)=c\\}$ implies that
$\\{\operatorname{\mathrm{Re}}(\xi)=c\\}\subset\mathbb{C}^{d}$ is away from
$0$-dimensional crossings (i.e $\bigcap_{i\in J^{\prime}}H_{i}$ for some
$|J^{\prime}|=d$). As a consequence, we can cover a small neighborhood of
$\\{\operatorname{\mathrm{Re}}(\xi)=c\\}$ using the regions $\mathcal{D}_{K}$
with $K\neq\\{1,\dots,d\\}$. Altogether, they give us a uniform upper bound
and a uniform lower bound on the norm
$\|X_{\operatorname{\mathrm{Re}}(\xi)}\|$. This proves the existence of an
open neighborhood of $\\{\operatorname{\mathrm{Re}}(\xi)=c\\}$ satisfying the
first property.

By Corollary 3.12, we know that $JX_{\operatorname{\mathrm{Re}}(\xi)}$ extends
to a vector field on $\mathbb{C}^{d}$. By abuse of notations, we denote its
extension by $JX_{\operatorname{\mathrm{Re}}(\xi)}$. Moreover, by (23) and the
$2\times 2$ block matrix argument above, we see that
$\|X_{\operatorname{\mathrm{Re}}(\xi)}\|$ also extends to $\mathbb{C}^{d}$ and
it is $0$ only at the $0$-dimensional crossings. As a result,
$V=\frac{-JX_{\operatorname{\mathrm{Re}}(\xi)}}{\|X_{\operatorname{\mathrm{Re}}(\xi)}\|^{2}}$
extends to a well-defined vector field in $\mathbb{C}^{d}$ except possibly at
the $0$-dimensional crossings, and it is tangent to the respective strata. By
Lemma 3.13, to show the completeness of $V$ on $M(\mathbb{V})$, it suffices to
exclude the possibility that its flowlines escape away from
$M(\mathbb{V})\subset\mathbb{C}^{d}$ in a finite time. Since in a neighborhood
of $\\{\operatorname{\mathrm{Re}}(\xi)=c\\}$, the length of the vector field
$\|V\|=\frac{1}{\|X_{\operatorname{\mathrm{Re}}(\xi)}\|}$ is uniformly bounded
above by a constant, the flow of $V$ on $M(\mathbb{V})$ is complete by the
Gronwall’s inequality [39, Theorem 2.8].

∎

######  3.16 Remark.

The proof of Lemma 3.15 shows that there are positive constants $c_{R}$ and
$C_{R}$ such that $c_{R}<\|X_{\operatorname{\mathrm{Re}}(\xi)}\|<C_{R}$ in
$\operatorname{\mathrm{Nbd}}\left(\left\\{\operatorname{\mathrm{Re}}(\xi)=c\right\\}\right)$
(see (23)). Moreover, if we choose
$\operatorname{\mathrm{Nbd}}\left(\left\\{\operatorname{\mathrm{Re}}(\xi)=c\right\\}\right)$
such that it is a union of $X_{\operatorname{\mathrm{Re}}(\xi)}$-integral
curves, then the flow of $X_{\operatorname{\mathrm{Re}}(\xi)}$ in
$\operatorname{\mathrm{Nbd}}\left(\left\\{\operatorname{\mathrm{Re}}(\xi)=c\right\\}\right)$
exist for all positive and negative time.

######  3.17 Corollary.

Under the assumption of Lemma 3.15, there is a constant $\varepsilon>0$ and a
function

$\displaystyle(R,I)\colon\operatorname{\mathrm{Nbd}}\left(\left\\{\operatorname{\mathrm{Re}}(\xi)=c\right\\}\right)\to\mathbb{C}_{\varepsilon}:=(-2\varepsilon,2\varepsilon)\times\mathbb{R}$
(24)

such that $(R,I)^{-1}(0)=\\{\xi=c\\}$, $\nabla
I=X_{\operatorname{\mathrm{Re}}(\xi)}$ and $\nabla R=V$.

In particular, $R$ and $I$ Poisson communte and hence (24) defines a trivial
symplectic fibration.

###### Proof.

Since $\omega(V,X_{\operatorname{\mathrm{Re}}(\xi)})=1$ is a constant, the
vector field $V$ and $X_{\operatorname{\mathrm{Re}}(\xi)}$ commute and define
an integrable 2-plane bundle transverse to $\\{\xi=c\\}$. By Lemma 3.15, the
flow of $V$ is defined for some positive and negative time. On the other hand,
the flow of $X_{\operatorname{\mathrm{Re}}(\xi)}$ is defined for all time (see
Remark 3.16). Therefore, by integrating the vector fields $V$ and
$X_{\operatorname{\mathrm{Re}}(\xi)}$, we get the functions $R$ and $I$
respectively.

Note that $\omega(X_{R},X_{I})=\omega(JX_{R},JX_{I})=\omega(-\nabla R,-\nabla
I)=\omega(V,X_{\operatorname{\mathrm{Re}}(\xi)})=1$ so $R$ and $I$ Poisson
communte and (24) defines a trivial symplectic fibration. ∎

######  3.18 Corollary.

The function $I$ in Corollary 3.17 agrees with
$\operatorname{\mathrm{Im}}(\xi)$.

###### Proof.

The gradient vector field of $\operatorname{\mathrm{Im}}(\xi)$ is given by
$\nabla\operatorname{\mathrm{Im}}(\xi)=-JX_{\operatorname{\mathrm{Im}}(\xi)}=X_{\operatorname{\mathrm{Re}}(\xi)}$,
which is the same as the gradient vector field of $I$.

Recall from Lemma 3.15 that
$V=-JX_{\operatorname{\mathrm{Re}}(\xi)}/\|X_{\operatorname{\mathrm{Re}}(\xi)}\|^{2}=-X_{\operatorname{\mathrm{Im}}(\xi)}/\|X_{\operatorname{\mathrm{Re}}(\xi)}\|^{2}$
which is tangent to $\\{\operatorname{\mathrm{Im}}(\xi)=0\\}$. As a result,
$I|_{\\{\operatorname{\mathrm{Im}}(\xi)=0\\}}=0$.

The uniqueness of integral curve with the same initial condition implies that
$I=\operatorname{\mathrm{Im}}(\xi)$. ∎

######  3.19 Lemma.

Let $R$ and $I$ be as in Lemma 3.15 and Corollary 3.17. Suppose also that

$\displaystyle\bigcap_{\begin{subarray}{c}j\in K\\\
|K|=d+1\end{subarray}}\left\\{|\ell_{j}|\leq 1\right\\}=\emptyset.$ (25)

Then inside the open neighborhood
$\operatorname{\mathrm{Nbd}}\left(\\{\operatorname{\mathrm{Re}}(\xi)=c\\}\right)\cong
F\times\mathbb{C}_{\varepsilon}$, we have a function $C(z)>0$ and two positive
constants $c_{1},c_{2}$ such that $c_{1}<C(z)<c_{2}$ and

$-d^{\mathbb{C}}\phi(X_{I})=C(z)I$

for $|I|$ sufficiently large.

###### Proof.

This is a consequence of Lemma 3.9 and Corollary 3.18. ∎

We will now start to prove Proposition 3.14, the main result of this
subsection, using Lemma 3.19.

###### Proof of Proposition 3.14.

[Step 0: preparation] Pick $c^{\prime}>0$ large enough so that
$(M(\mathbb{V}),c^{\prime}\phi)$ is covered by the open subsets
$M(\mathbb{V})_{J,1}$ (i.e. the assumption (20) holds). By Lemma 3.6, working
with $(M(\mathbb{V}),c^{\prime}\phi)$ does not change the Weinstein
deformation type, so it suffices to prove the proposition for
$(M(\mathbb{V}),c^{\prime}\phi)$ instead.

Using the cover $\left\\{M(\mathbb{V})_{J,1}\right\\}$, and under the
convention of Lemma 3.15, we write
$\xi=\displaystyle\sum_{i=1}^{d}b_{i}\ell_{i}=\displaystyle\sum_{i=1}^{d}b_{i}z_{i}$.
Recall $R$ and $I$ from Lemma 3.15 and Corollary 3.17. Let $F=(R,I)^{-1}(0)$,
we have a symplectic product decomposition

$\operatorname{\mathrm{Nbd}}\left(\\{\mathrm{Re}(\xi)=c\\}\right)\cong
F\times(-\varepsilon,\varepsilon)\times\mathbb{R},$

which is given by symplectic parallel transports along the second and the
third factors. Denote the Liouville $1$-form on $M(\mathbb{V})$ by
$\lambda_{\mathbb{V}}$. Let
$\lambda_{F}=\lambda_{\mathbb{V}}|_{F\times\\{0\\}}$ and recall the definition
of the $1$-form $\lambda_{\mathbb{C}}^{\alpha}$ from (3). For any
$0<\alpha<1$, the difference between our Liouville $1$-form
$\lambda_{\mathbb{V}}$ and $\lambda_{F}+\lambda_{\mathbb{C}}^{\alpha}$ is
exact in $\operatorname{\mathrm{Nbd}}\left(\\{\mathrm{Re}(\xi)=c\\}\right)$,
so we can write

$\lambda_{\mathbb{V}}=\lambda_{F}+\lambda_{\mathbb{C}}^{\alpha}+df,$

where
$f\colon\operatorname{\mathrm{Nbd}}\left(\\{\mathrm{Re}(\xi)=c\\}\right)\to\mathbb{R}$
is a not necessarily compactly supported smooth function in
$\operatorname{\mathrm{Nbd}}\left(\\{\mathrm{Re}(\xi)=c\\}\right)$ which
depends on $\alpha$.

[Step 1: an explicit deformation] Let $\rho\colon\mathbb{R}\to\mathbb{R}$ be a
smooth function depicted on the right-hand side of Figure 10, which is zero
near the origin, $\rho(x)=x-c^{\prime\prime}$ for some $c^{\prime\prime}>0$
near $x=\varepsilon$, $\rho(x)=x+c^{\prime\prime}$ near $x=-\varepsilon$ and
$\rho(x)=x$ when $|x|\geq 2\varepsilon$. We assume that
$0\leq\rho^{\prime}\leq 1$ when $|x|\leq\varepsilon$ and
$1\leq\rho^{\prime}\leq 1+\delta_{c^{\prime\prime}}$ when
$|x|\geq\varepsilon$, where $\delta_{c^{\prime\prime}}>0$ is a small number
depending on $c^{\prime\prime}$. By possibly replacing $c^{\prime\prime}$ with
a smaller one, we can make $\delta_{c^{\prime\prime}}$ as close to $0$ as we
want. Set

$\displaystyle\tilde{f}(x,p)\coloneqq
f\left(x,\rho(R(p)),\rho^{\prime}(R(p))I(p)\right)$ (26)

for $|R(p)|\leq\varepsilon$ and

$\displaystyle\tilde{f}(x,p)\coloneqq f\left(x,\rho(R(p)),I(p)\right)$ (27)

for $\varepsilon\leq|R(p)|\leq 2\varepsilon$. This is well-defined because
$\rho^{\prime}=1$ near $|R|=\varepsilon$. Moreover, $\tilde{f}=f$ when
$|R|\geq 2\varepsilon$.

$x$$y$$\varepsilon$$-\varepsilon$$-2\varepsilon$$\varepsilon$$2\varepsilon$
(a) Derivative of $\rho$

$x$$y$$-\varepsilon$$-2\varepsilon$$\varepsilon$$2\varepsilon$ (b) The
Function $\rho$

Figure 10: The Bump Function $\rho$ and its Derivative.

Let $\lambda_{1}:=\lambda_{F}+\lambda_{\mathbb{C}}^{\alpha}+d\tilde{f}$. Since
$\tilde{f}=0$ near $R=0$, the symplectic dual of $\lambda_{1}$ is tangent to
the hypersurface $F\times\\{0\\}\times\mathbb{R}$ near infinity, such that
$F\times\\{0\\}\times\mathbb{R}$ is sectorial with defining function $I$. For
$0\leq t\leq 1$, we define $\lambda_{t}$ by

$\lambda_{t}=t\lambda_{1}+(1-t)\lambda_{\mathbb{V}}=\lambda_{F}+\lambda_{\mathbb{C}}^{\alpha}+d\left(t\tilde{f}+(1-t)f\right).$

This family of $1$-forms induces a Liouville homotopy from
$(\hat{X},\omega,\lambda_{\mathbb{V}})$ to $(\hat{X},\omega,\lambda_{1})$. Our
goal is to show that for an appropriate choice of $\alpha\in(0,1)$, the
skeleton of this family of Liouvile $1$-forms is compact.

Let $\mathbb{C}_{\varepsilon}=(-2\varepsilon,2\varepsilon)\times\mathbb{R}$.
To show that the skeleton is compact, our strategy is first to compute

$d\tilde{f}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}\quad\text{ and }\quad
df|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}$

which shows that for every $z$, the zeros of
$\lambda_{t}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}$ lie in a compact set.
Then we will argue that the zeros of $\lambda_{t}|_{F\times\\{p\\}}$ lie in a
compact set for every $p$ to conclude that the zeros of $\lambda_{t}$ lie in a
compact set.

[Step 2: Analyzing critical points by Lemma 3.19] In the below we implement
the strategy described above. We first consider the region where
$|R(p)|\leq\varepsilon$.

On $\\{z\\}\times\mathbb{C}_{\varepsilon}$, $\partial_{R}$ is by definition
the unique vector field in
$\operatorname{Span}\\{V,X_{\operatorname{\mathrm{Re}}(\xi)}\\}=\operatorname{Span}\\{X_{R},X_{I}\\}$
such that $dR(\partial_{R})=1$ and $dI(\partial_{R})=0$. In other words,
$\omega(X_{R},\partial_{R})=-1$ and $\omega(X_{I},\partial_{R})=0$. Therefore,
$\partial_{R}=-X_{I}$ and similarly, $\partial_{I}=X_{R}$. Since
$\partial_{R}\tilde{f}=\rho^{\prime}\partial_{R}f+\rho^{\prime\prime}I\partial_{R}f$
and $\partial_{I}\tilde{f}=\rho^{\prime}\partial_{I}f$, we have

$\displaystyle
d\tilde{f}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}=\rho^{\prime}\left(-df(X_{I})dR+df(X_{R})dI\right)-\rho^{\prime\prime}Idf(X_{I})dR,$
(28)

Recall that

$\displaystyle\lambda_{\mathbb{C}}^{\alpha}=-\alpha IdR+(1-\alpha)RdI,$ (29)

so $df$ is given by

$\displaystyle df|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}$
$\displaystyle=-\left.d^{c}\phi\right|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}-\lambda_{\mathbb{C}}^{\alpha}=\underbrace{\left(-d^{\mathbb{C}}\phi(X_{R})-(1-\alpha)R\right)}_{df(X_{R})}dI-\underbrace{\left(-d^{\mathbb{C}}\phi(X_{I})-\alpha
I\right)}_{df(X_{I})}dR.$ (30)

Consider the zeros of the restriction of the $1$-form $\lambda_{t}$ to
$\\{z\\}\times\mathbb{C}_{\varepsilon}$ for $0\leq t\leq 1$. By (29), (30) and
(28),

$\begin{split}&\lambda_{t}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}\\\
=&\lambda_{\mathbb{C}}^{\alpha}+td\tilde{f}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}+(1-t)df|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}\\\
=&\left((1-\alpha)R+(1+t(\rho^{\prime}-1))df(X_{R})\right)dI-\left(\alpha
I+t\rho^{\prime\prime}Idf(X_{R})+(1+t(\rho^{\prime}-1))df(X_{I})\right)dR.\end{split}$
(31)

It suffices to look at the zeroes of the coefficients of $dR$ and $dI$. For
$dI$, the coefficient being zero implies that

$df(X_{R})=-\frac{(1-\alpha)R}{1-t(1-\rho^{\prime})}.$

When $|I|$ is sufficiently large, it follows from Lemma 3.19 that the
coefficient of $dR$ in (31) being $0$ means

$\displaystyle 0$ $\displaystyle=\left(\alpha
I+t\rho^{\prime\prime}Idf(X_{R})+(1+t(\rho^{\prime}-1))df(X_{I})\right)$
$\displaystyle=\left(\alpha
I-t\rho^{\prime\prime}I\frac{(1-\alpha)R}{1-t(1-\rho^{\prime})}+(1+t(\rho^{\prime}-1))\left(C(z)I-\alpha
I\right)\right)$
$\displaystyle=\left(t(1-\rho^{\prime})\alpha-t\rho^{\prime\prime}\frac{(1-\alpha)R}{1-t(1-\rho^{\prime})}+(1-t(1-\rho^{\prime}))C(z)\right)I.$

The only possible solution is

$\alpha=\dfrac{t\rho^{\prime\prime}R-C(1-t(1-\rho^{\prime}))^{2}}{t\rho^{\prime\prime}R+t(1-\rho^{\prime})(1-t(1-\rho^{\prime}))}=1-\dfrac{C(1-t(1-\rho^{\prime}))^{2}+t(1-\rho^{\prime})(1-t(1-\rho^{\prime}))}{t\rho^{\prime\prime}R+t(1-\rho^{\prime})(1-t(1-\rho^{\prime}))}.$
(32)

[Step 3: Analyzing critical points by a direct calculation] Let
$A=t(1-\rho^{\prime})$, then the right-hand side of (32) reads

$1-\dfrac{C(1-A)^{2}+A(1-A)}{t\rho^{\prime\prime}R+A(1-A)}.$ (33)

Since we are considering the region where $|R|\leq\varepsilon$, we have
$0\leq\rho^{\prime}\leq 1$, so
$\dfrac{C(1-A)^{2}+A(1-A)}{t\rho^{\prime\prime}R+A(1-A)}\geq 0$.

When $A\to 0$, we have $\rho^{\prime}\to 1$ or $t\to 0$, hence
$\dfrac{C(1-A)^{2}+A(1-A)}{t\rho^{\prime\prime}R+A(1-A)}\to\infty$, while if
$A\to 1$, we must have $\rho^{\prime}\to 0$ and $t\to 1$, and a variation of
the formula for $\alpha$ gives

$\frac{1-\alpha}{1-A}=\frac{C(1-A)+A}{t\rho^{\prime\prime}R+A(1-A)},$

which approaches$\infty$ as $A\to 1$. Look back into the coefficient of $dI$,
we have

$(1-A)df(X_{R})=-(1-\alpha)R,$

which is equivalent to the equation
$\dfrac{A(1-\alpha)}{1-A}R=d^{\mathbb{C}}\phi(X_{R})$. Since
$|d^{\mathbb{C}}\phi(X_{R})|\leq C<\infty$ is bounded, $\dfrac{R}{1-A}$ must
also be bounded. Traveling back to the coefficient for $dR$, we have

$1=\frac{C(1-A)+A}{t\left(\dfrac{1-\alpha}{1-A}R\right)\rho^{\prime\prime}+A(1-\alpha)}.$

If $A\to 1$, the numerator converges to $1$, while
$t\dfrac{1-\alpha}{1-A}R\rho^{\prime\prime}\to 0$ and $A(1-\alpha)<1$, so we
derive that $1<1$ as $A\to 1$, a contradiction. Thus $A$ cannot converge to
$1$, so there is some $0<M<1$ with $0\leq A\leq M<1$, such that (33) is away
from $1$. Therefore if $\alpha$ is sufficiently close to $1$, the equation
(32) has no solution.

We conclude that there is an $\alpha\in(0,1)$ such that the Liouville $1$-form
$\lambda_{t}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}$ has no critical points
in the region $|R|\leq\varepsilon$ when $|I|$ is large. Note that, Lemma 3.19
and the estimates above are uniform in $z$ so there is an $\alpha\in(0,1)$
such that when $|I|$ is large,
$\lambda_{t}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}$ has no critical points
in the region $|R|\leq\varepsilon$ for all $z\in F$.

[Step 4: the easier region] Next we consider the region where
$|R(p)|\geq\varepsilon$. When $|R(p)|\geq\varepsilon$, we have

$d\tilde{f}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}=-\rho^{\prime}(R)df(X_{I})dR+df(X_{R})dI$

In this case,

$\displaystyle\lambda_{t}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}$
$\displaystyle=\lambda_{\mathbb{C}}^{\alpha}+td\tilde{f}|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}+(1-t)df|_{\\{z\\}\times\mathbb{C}_{\varepsilon}}$
$\displaystyle=\left((1-\alpha)R+df(X_{R})\right)dI-\left(\alpha
I+(1+t(\rho^{\prime}-1))df(X_{I})\right)dR.$

Applying Lemma 3.19 again, the coefficient of $dR$ being zero means, for $|I|$
sufficiently large,

$\displaystyle 0$
$\displaystyle=-d^{\mathbb{C}}\phi(X_{I})-t(\rho^{\prime}-1)d^{\mathbb{C}}\phi(X_{I})-t(\rho^{\prime}-1)\alpha
I$ $\displaystyle=\left((1+t(\rho^{\prime}-1))C-t(\rho^{\prime}-1)\right)I$
$\displaystyle=\left(1+t(\rho^{\prime}-1)(C-1)\right)I$

Since $C$ is uniformly bounded from above, we can choose $\rho^{\prime}$ to be
very close to $1$ such that the equation above has no solution.

[Step 5: concluding the proof] We conclude that there is an $\alpha\in(0,1)$
such that the critical points of the Liouville $1$-form $\lambda_{t}$ lie in a
compact subset of $\\{z\\}\times\mathbb{C}_{\varepsilon}$ for any $z\in F$,
since the previous discussion does not depend on the choice of $z$.
Compactness of the critical points of $\lambda_{t}$ along the fiber follows
from the fact that we are deforming the Liouville structure so that it remains
complete in the complement $M(\mathbb{V})$, which we know from Corollary 3.10
to have compact set of critical points. ∎

We now generalize the argument above to the case where there are transversely
intersecting hyperplanes $\xi_{i}^{-1}(c_{i}^{\pm})$ with
$c_{i}^{-}<c_{i}^{+}$.

######  3.20 Lemma.

For any subcollection of hypersurfaces in
$\\{\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})\\}$, their
intersection is either coisotropic or empty.

###### Proof.

This is a direct consequence of the fact that the vector fields
$X_{\mathrm{Re}(\xi_{i})}$ are purely imaginary and the characteristic
foliation of $\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})$ is
generated by $X_{\operatorname{\mathrm{Re}}(\xi_{i})}$. ∎

Lemma 3.20 verifies the first item of Definition 2.6. To proceed, we need to
show the orthogonality (4) in the definition of sectorial collections (cf.
Definition 2.6). In fact, we can show the following generalization of Lemma
3.15.

######  3.21 Lemma.

There exists a $d$-tuple of vector fields
$\\{V_{1}^{\pm},\dotsb,V_{d}^{\pm}\\}$, defined in the product neighbourhood
of $\left\\{\operatorname{\mathrm{Re}}(\xi_{i})=c_{i}^{\pm}\right\\}$,
satisfying the following properties:

  1. i)

$\omega(V_{i}^{\pm},V_{j}^{\pm})=0$ for all $i,j$;

  2. ii)

$\omega\left(V_{i}^{\pm},X_{\operatorname{\mathrm{Re}}(\xi_{j})}\right)=\delta_{ij}$;

  3. iii)

The flow of $V_{i}^{\pm}$ exists in some positive time.

###### Proof.

Using the same conventions as in Lemma 3.7, write
$\xi_{i}=\displaystyle\sum_{j=1}^{d}b_{ij}z_{j}$, then the Hamiltonian vector
field of $\operatorname{\mathrm{Re}}(\xi_{i})$ is given by

$X_{\operatorname{\mathrm{Re}}(\xi_{i})}=\frac{1}{4}\vec{b}_{i}^{T}(D+A^{T}A)^{-1}\frac{\partial}{\partial\vec{y}},$

and similarly

$X_{\operatorname{\mathrm{Im}}(\xi_{i})}=-\frac{1}{4}\vec{b}_{i}^{T}(D+A^{T}A)^{-1}\frac{\partial}{\partial\vec{x}},$

so that the symplectic pairing between them is given by

$\omega_{ij}\coloneqq\omega\left(X_{\operatorname{\mathrm{Re}}(\xi_{i})},X_{\operatorname{\mathrm{Im}}(\xi_{j})}\right)=4X_{\operatorname{\mathrm{Re}}(\xi_{i})}^{T}(D+A^{T}A)X_{\operatorname{\mathrm{Im}}(\xi_{j})}=\frac{1}{4}\vec{b}_{i}^{T}(D+A^{T}A)^{-1}\vec{b}_{j},$

which is bounded for all $z\in M(\mathbb{V})$. Given a non-empty subset
$J\subseteq\\{1,\dotsb,d\\}$ with $|J|=r<d$, without loss of generality we may
assume that $J=\\{1,\dotsb,r\\}$. The restriction of the symplectic form
$\omega$ to the subbundle of $T^{\ast}M(\mathbb{V})$ generated by
$\left\\{X_{\operatorname{\mathrm{Re}}(\xi_{i})},X_{\operatorname{\mathrm{Im}}(\xi_{i})}\right\\}_{1\leq
i,j\leq r}$ is given by the matrix

$\Omega=\begin{pmatrix}0&(\omega_{ij})\\\ (-\omega_{ij})&0\end{pmatrix}.$

Let $\\{V_{i}|1\leq i\leq r\\}$ be a family of vector fields defined as linear
combinations of $\\{X_{\operatorname{\mathrm{Im}}(\xi_{j})}\\}$ such that
$\omega\left(V_{i},X_{\operatorname{\mathrm{Re}}(\xi_{i})}\right)=-1$ and
$\omega\left(V_{i},X_{\operatorname{\mathrm{Re}}(\xi_{j})}\right)=0$ for
$i\neq j$. Such $V_{i}$’s can be found via taking the inverse matrix of
$\Omega$. Also, since the coefficients of $\Omega$ are polynomials in
$\omega_{ij}$, which are bounded for $z\in M(\mathbb{V})$, the coefficients of
the vector field $V_{i}$ are Laurant polynomials in $\omega_{ij}$. These
coefficients are bounded unless the determinant of $\Omega$ goes to $0$.
However, Lemma 3.11 prevents this to happen near $\bigcap_{j\in J}H_{j}$. This
allows us to apply the completeness criterion (Lemma 3.13) to conclude that
all such $V_{i}$’s admit flows in the complement of $\bigcup_{j=1}^{d}H_{j}$
such that they do not escape to $H_{j}$. Now we have obtained $V_{i}$’s for
each open neighborhood of the intersection $\displaystyle\bigcap_{j\in
J}\operatorname{\mathrm{Re}}(\xi_{j})^{-1}(c_{j}^{\pm})$, removing a
neighborhood of some deeper strata, and we can patch them together using a
partition of unity to get vector fields $V_{i}$ defined in a neighborhood of
$\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})$, which still have
well-defined flows in the complement of $\bigcup_{j=1}^{d}H_{j}$ such that
they do not escape to $H_{j}$. There is a uniform upper bound on $\|V_{i}\|$
so they are complete by Gronwall’s inequality. It follows from definition that
$\omega\left(V_{i}^{\pm},X_{\operatorname{\mathrm{Re}}(\xi_{j})}\right)=\delta_{ij}$.
This completes the proof. ∎

As a consequence, by the same reasoning in Corollary 3.17, we can integrate
the vector fields $V_{i}^{\pm}$ and $X_{\operatorname{\mathrm{Re}}(\xi_{i})}$
to obtain functions $\\{R_{i},I_{i}\\}$, and a symplectic product
decompositions of a neighborhood of the intersection $\displaystyle
S_{I}^{\sigma}\coloneqq\bigcap_{i\in
I}\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\sigma_{i}})$

$\left((R_{i},I_{i})_{i=1,\dots,d}\right):\operatorname{\mathrm{Nbd}}(S_{I}^{\sigma})\rightarrow\mathbb{C}_{\varepsilon}^{d}$

where $\sigma=(\sigma_{1},\dotsb,\sigma_{d})\in\\{\pm\\}^{d}$. This verifies
that Definition 2.6(ii) holds for the collection of hypersurfaces
$\left\\{\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})\right\\}_{i=1,\dots,d}$.
Note that both $I_{i}$ and $\operatorname{\mathrm{Im}}(\xi_{i})$ are obtained
by integrating $X_{R_{i}}$. This time,
$I_{i}|_{\operatorname{\mathrm{Im}}(\xi_{i})=0}$ is not necessarily $0$
because $X_{\operatorname{\mathrm{Im}}(\xi_{j})}$ is not necessarily tangent
to $\\{\operatorname{\mathrm{Im}}(\xi_{i})=0\\}$. However, $I_{i}$ and
$\operatorname{\mathrm{Im}}(\xi_{i})$ are arbitrarily $C^{\infty}$ close to
each other when $\varepsilon$ is small because
$I_{i}|_{S_{I}^{\sigma}}=0=\operatorname{\mathrm{Im}}(\xi_{i})|_{S_{I}^{\sigma}}$
and they are obtained by integrating the same vector field. The direct
generalization of Lemma 3.19 is therefore also true by replacing $\xi$ and $I$
by $\xi_{i}$ and $I_{i}$.

We can adapt the proof of Proposition 3.14 to the collection of hypersurfaces
because we can use an explicit product type deformation in Step 1 (with
respect to the product decomposition of $\mathbb{C}_{\varepsilon}^{d}$), apply
the generalization of Lemma 3.19 in Step 2, and the rest follows by the same
computations. It implies that:

######  3.22 Proposition.

There exists a Liouville deformation $\left(M(\mathbb{V}),\omega,Z_{t}\right)$
with compact skeleton such that $Z_{0}=\nabla\phi$ and the real hypersurfaces
$\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})\subset M(\mathbb{V})$
are all sectorial with respect to the deformed Liouville structure
$(M(\mathbb{V}),\omega,Z_{1})$.

We can actually decompose this Liouville deformation into finite steps, each
of which makes the hypersurface defined by the corresponding linear form
sectorial. By [16, Proposition 11.8], we know that there exists an exact
symplectomorphism

$f\colon(M(\mathbb{V}),\omega,\lambda_{\phi})\rightarrow(M(\mathbb{V}),\omega,\lambda_{1}),$

such that $f^{\ast}\lambda_{1}=\lambda_{\phi}+dg$ where $g\colon
X\to\mathbb{R}$ is a compactly supported function. This implies that the
wrapped Fukaya category is unchanged up to quasi-equivalence when we perform
this deformation.

### (3e) Sectorial Decomposition.

Given a polarized hyperplane arrangment $\mathbb{V}$, we introduce sectorial
cuts of the Weinstein manifold $M(\mathbb{V})$, so that it is divided into
standard pieces with known generating Lagrangians. Note that the existence of
sectorial cuts relies on the simplicity assumption of the polarized hyperplane
arrangement $\mathbb{V}$. The construction is divided into two steps.

#### Step 1:

Let $\\{\xi_{i}\\}_{1\leq i\leq d}$ be a basis of $V^{*}$ with $\xi_{1}=\xi$
which does not contain any of the hyperplanes $H_{\mathbb{R},i}$. Let
$N_{\mathbb{V}}$ be the number of $0$-dimensional strata in the hyperplane
arrangemtn $\mathbb{V}$. Pick real numbers $(q_{1j})_{1\leq j\leq
N_{\mathbb{V}}}$ and $(q_{ij})_{2\leq i\leq d,1\leq j\leq 2N_{\mathbb{V}}}$
with the following properties:

  1. i)

For each $1\leq j\leq N_{\mathbb{V}}$, there exists a unique $q_{1j}$ so that
$\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(q_{1j})$ intersects non-transversely
with hyperplanes in $\mathbb{V}$ at a unique crossing. We choose real numbers
$(c_{1j})$ so that $c_{1,j-1}<q_{1j}<c_{1j}$ holds (when $j=1$, we set
$c_{1,0}=-\infty$).

  2. ii)

For each $1\leq j\leq N_{\mathbb{V}}$, there exists a $q_{ij}$ so that
$\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(q_{ij})$ intersects non-transversely
with hyperplanes in $\mathbb{V}$. Pick real numbers $(c_{ij})$ with
$c_{i,2j-1}<q_{ij}<c_{i,2j}$, so that

$\left(\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i,2j-1})\cup\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i,2j})\right)\cap\operatorname{\mathrm{Re}}(\xi_{1})^{-1}\left([c_{1j},c_{1,j+1}]\right)$

intersects transversely with at most $d-2$ hyperplanes in $\mathbb{V}$.

By Proposition 3.22, after a compactly supported Liouville deformation, we can
arrange that all the real hypersurfaces
$\left\\{\mathrm{Re}(\xi_{i})=c_{ij}\right\\}_{1\leq i\leq d}$ form a
sectorial collection. Thus we can use them to cut the Weinstein manifold
$M(\mathbb{V})$ into subsectors.

#### Step 2:

We first cut $M(\mathbb{V})$ using the hypersurfaces defined by $\xi_{1}$,
which leads to the decomposition

$M(\mathbb{V})=\bigcup_{j=1}^{N_{\mathbb{V}}}\operatorname{\mathrm{Re}}(\xi_{1})^{-1}\left([c_{1,j-1},c_{1j}]\right)\eqqcolon\bigcup_{j=1}^{N_{\mathbb{V}}}P_{j},$

so that we get a total number of $N_{\mathbb{V}}$ sectors with similar
behaviors as depicted in Figure 11. See also the related discussions in the
introduction (Section (1c)).

$\xi$$\Rightarrow$ Figure 11: Decomposing $2$-dimensional pair-of-pants into
sectors

Next, we perform further cuts on each subsector $P_{j}$ using the real
hypersurfaces $\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i,2j-1})$ and
$\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i,2j})$, which gives the
following decomposition

$P_{j}=\bigcup_{\alpha\in\\{-,0,+\\}^{d-1}}\left(\bigcap_{i=2}^{d}\operatorname{\mathrm{Re}}(\xi_{i})(\alpha_{i})\right)\cap\operatorname{\mathrm{Re}}(\xi_{1})^{-1}([c_{1,j-1},c_{1j}])\eqqcolon\bigcup_{\alpha\in\\{-,0,+\\}^{d-1}}P_{j\alpha},$

where $\alpha=(\alpha_{2},\dotsb,\alpha_{d})$ and

$\begin{split}\operatorname{\mathrm{Re}}(\xi_{i})(-)&=\operatorname{\mathrm{Re}}(\xi_{i})^{-1}((-\infty,c_{i,2j-1}]),\\\
\operatorname{\mathrm{Re}}(\xi_{i})(0)&=\operatorname{\mathrm{Re}}(\xi_{i})^{-1}([c_{i,2j-1},c_{i,2j}]),\\\
\operatorname{\mathrm{Re}}(\xi_{i})(+)&=\operatorname{\mathrm{Re}}(\xi_{i})^{-1}([c_{i,2j},+\infty)).\\\
\end{split}$

An intuitive picture in dimension two is shown in Figure 12, where the green
lines represent the second family of sectorial hypersurfaces that we have
introduced, and the dashed parts indicate that we do not use the hypersurface
to cut other irrelevant pieces. Now we can apply the Sectorial Descent
(Theorem 2.18) with respect to the covering $\\{P_{j\alpha}\\}$, which gives
the following.

$\Rightarrow$ Figure 12: Second cut of $P_{j}$’s into standard pieces.

######  3.23 Proposition.

The partially wrapped Fukaya category $\mathcal{W}(M(\mathbb{V}),\xi)$ is
generated by that of its covering sectors $\mathcal{W}(P_{j\alpha})$, where
$\alpha\in\\{-,0,+\\}^{d-1}$.

This reduces the generation problem to the corresponding problem for each
individual piece $P_{j\alpha}$.

### (3f) Generation for Each Piece.

We prove generation results for the Weinstein sub-sectors $P_{j\alpha}\subset
M(\mathbb{V})$ defined in the last subsection. Notice that there are two types
of sub-sectors: those containing a crossing, denoted simply by $P_{0}$
(corresponding to $P_{j}$’s), and those do not contain a crossing, denoted
simply by $P_{\alpha}$.

######  3.24 Proposition.

  1. i)

The Liouville sector $P_{0}$ is symplectomorphic to

$\left(\left(\mathbb{C}^{\ast}\right)^{d},\bigcup_{i=1}^{d}\left(f_{i}\right)^{-1}(c_{i}^{\pm})\right),$

where $f_{i}$ is the pull-back of $\operatorname{\mathrm{Re}}(\xi_{i})$ under
the symplectomorphism $P_{0}\cong\left(\mathbb{C}^{\ast}\right)^{d}$.

  2. ii)

The other Liouville sectors $P_{\alpha}$, $\alpha\neq 0$, are symplectomorphic
to the stabilizations

$\left(P_{\alpha}\cap\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{+})\right)\times
T^{\ast}[c_{1}^{-},c_{1}^{+}].$

######  3.25 Remark.

It follows from (i) of the proposition above that the wrapped Fukaya category
$\mathcal{W}(P_{0})$ is generated by a cotangent fiber of
$\left(\mathbb{C}^{\ast}\right)^{d}\cong T^{\ast}{T}^{d}$, together with the
linking disks associated to the stops
$\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})$. The cotangent fibre
$L\subset P_{0}$ will be called a standard Lagrangian associated to $P_{0}$.
Order the crossing points from bottom to top, the sector $P_{0}$ corresponds
to a unique crossing point, say the $j$th crossing, and we will write the
corresponding standard Lagrangian as $L_{j}$.

Note that here we use the term “standard Lagrangian” so that it is consistent
with the term used in geometric representation theory, where the standard
objects in the hypertoric category $\mathcal{O}$ [13] correspond exactly (by
our main theorem) to these Lagrangians.

On the other hand, (ii) of the proposition above implies that the wrapped
Fukaya category
$\mathcal{W}(P_{\alpha})\cong\mathcal{W}\left(\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{+})\cap
P_{\alpha}\right)$ is generated by the linking disks associated to the stop
$P_{\alpha}\cap\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{+})$.

###### Proof of Proposition 3.24 i).

Let $J\subset\\{1,\cdots,n\\}$ to be the set of hyperplanes going through the
dimension $0$ crossing contained in $P_{0}$. We apply Lemma 3.7 to get a
corresponding symplectic structure on $P_{0}$, regarded as a (codimension $0$)
symplectic submanifold of $M(\mathbb{V}[J])$. Recall the symplectomorphism
$\Phi:M(\mathbb{V}[J])\rightarrow M(\mathbb{V})$. Consider the symplectic
isotopy

$\omega_{t}\coloneqq 4\begin{pmatrix}0&D+(1-t)A^{T}A\\\
-D-(1-t)A^{T}A&0\end{pmatrix},$

whose derivative is

$\begin{split}\dot{\omega}_{t}&=4\begin{pmatrix}0&-A^{T}A\\\
A^{T}A&0\end{pmatrix}\\\ &=-4d\left(\sum_{j\not\in
J}\left(1-\frac{1}{|\ell_{j}|^{4}}\right)\left(\operatorname{\mathrm{Re}}(\ell_{j})d\operatorname{\mathrm{Im}}(\ell_{j})-\operatorname{\mathrm{Im}}(\ell_{j})d\operatorname{\mathrm{Re}}(\ell_{j})\right)\right)=d\theta_{t}.\end{split}$

Let $V$ be the symplectic dual to the $1$-form $\theta_{t}$, we need to prove
that $V$ is complete.

Consider a partial compactification $M\left(\mathbb{V}_{J}\right)$ of
$M(\mathbb{V})$, where $\mathbb{V}_{J}$ is hyperplane arrangement obtained by
removing those with indices in $J$. The stratification of the complement
$M\left(\mathbb{V}_{J}\right)\setminus M(\mathbb{V})$ is determined by
equations $\\{\ell_{i}=0|\forall i\in J^{\prime}\\}_{J^{\prime}\subset J}$.
With respect to the symplectic structure $\omega_{t}$, the vector field $V$ is
given by the formula

$V=-[d\phi_{J}](D+(1-t)A^{T}A)^{-1}\begin{pmatrix}\frac{\partial}{\partial\vec{x}}\\\
\frac{\partial}{\partial\vec{y}}\end{pmatrix},$

where $\phi_{J}=\displaystyle\sum_{j\not\in
J}\left(|\ell_{j}|^{2}+\frac{1}{|\ell_{j}|^{2}}\right)$ is the squared sum and
$[d\phi_{J}]$ is the matrix corresponding to its differential, which is given
by

$d\phi_{J}=\sum_{j\not\in
J}\sum_{i=1}^{d}\left(1-\frac{1}{|\ell_{j}|^{4}}\right)\operatorname{\mathrm{Re}}(\ell_{j})a_{ij}dx_{i}.$

In the sector $P_{0}$, we have $|\ell_{j}|$ bounded away from zero and
$\operatorname{\mathrm{Re}}(\ell_{j})$ is bounded for all $j\notin J$. If
$|z_{i}|\to 0$ for some $1\leq i\leq d$, the denominator of the coefficients
of $\partial_{x_{i}}$ and $\partial_{y_{i}}$ go to $\infty$, so the vector
field $V$ extends to a vector field over the open strata $\\{z_{i}=0\\}$ and
is tangent to it. To show the completeness of $V$, it suffices to look at the
case when some $\operatorname{\mathrm{Im}}(\ell_{i})\to\infty$. Under this
limit, the coefficient of $\partial_{x_{i}}$ is always bounded. Since all
$|\ell_{j}|$’s are bounded away from $0$, for any sequence of points in
$\Phi(P_{0})$ converging to any strata at infinity,

$[d\phi_{J}](D+(1-t)A^{T}A)^{-1}$

converges. It follows that near infinity, we have

$\left\|[d\phi_{J}](D+(1-t)A^{T}A)^{-1}\right\|\leq M$

for some constant $M>0$, which then implies that the coefficient of
$\partial_{y_{i}}$ is controlled by $CM|z|$, where $C>0$ is another constant.
It is a consequence of Gronwall’s inequality [39, Theorem 2.8] that the vector
field $V$ is complete. Thus the time-$1$ flow of $V$ induces a symplectic
embedding
$\phi_{V}^{1}\colon\Phi(P_{0})\hookrightarrow\left(\mathbb{C}^{\ast}\right)^{d}$,
which can be extended to an exact symplectomorphism. It implies that

$\mathcal{W}(P_{0})\simeq\mathcal{W}(\Phi(P_{0}))\simeq\mathcal{W}\left(\left(\mathbb{C}^{\ast}\right)^{d},\bigcup_{i=1}^{d}(f_{i})^{-1}(c_{i}^{\pm})\right)$

is generated by a cotangent fiber of $\left(\mathbb{C}^{\ast}\right)^{d}$ and
linking disks associated to the stops
$\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})$. ∎

###### Proof of Proposition 3.24 ii).

Recall that in the proof of Lemma 3.21, we only use the fact that
$\left\\{\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{\pm})\right\\}$
intersects with the punctures transversely, and we cut the Liouville sector so
that $\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(r)$, for $c_{1}^{-}\leq r\leq
c_{1}^{+}$, intersect the punctures in $P_{\alpha}$ transversely, so we
conclude that for every $c_{1}^{-}\leq r\leq c_{1}^{+}$, there exists a
product decomposition of the $Z$-invariant neighbourhood

$\operatorname{\mathrm{Nbd}}^{Z}\left(\\{\operatorname{\mathrm{Re}}(\xi_{1})=r\\}\right)\cap
P_{\alpha}\xrightarrow{\simeq}\\{\xi_{1}=r\\}\times
T^{\ast}[-\varepsilon_{r},\varepsilon_{r}]\cap P_{\alpha}$ (34)

for some small $\varepsilon_{r}>0$. The product decomposition (34) for each
$r$ gives us a locally defined vector field $X_{I_{i}^{r}}$, for some function
$V^{r}_{1}\coloneqq I_{1}^{r}$ coming from Lemma 3.21. The open intervals
$(r-\varepsilon_{c},r+\varepsilon_{c})$ form a covering of $[c^{-},c^{+}]$, so
we can find a finite sub-collection of the
$(c-\varepsilon_{c},c+\varepsilon_{c})$’s covering the same interval. Let
$\\{\psi_{r}\\}$ be a partition of unity defined on
$P_{\alpha}=\bigcup_{r}\operatorname{\mathrm{Nbd}}^{Z}\left(\\{\operatorname{\mathrm{Re}}(\xi_{1})=r\\}\right)\cap
P_{\alpha}$ that is subordinated to the finite cover, then the vector field
$V=\sum_{r}\psi_{r}X_{V_{1}^{r}}$ satisfies
$d\operatorname{\mathrm{Re}}(\xi_{1})(V)=1$ everywhere and is locally complete
by Lemma 3.13. Therefore the vector fields
$\left(X_{\operatorname{\mathrm{Re}}(\xi_{1})},V\right)$ induce the required
product decomposition in the second part of the Proposition 3.24. ∎

Proposition 3.24 together with Sectorial Descent (Theorem 2.18) imply that the
union of all standard Lagrangians $\\{L_{j}\\}_{j=1}^{N_{\mathbb{V}}}$,
together with all the linking disks corresponding to all the cornered stops
$P_{j\alpha}$ generate the wrapped Fukaya category of
$(M(\mathbb{V}),\xi=\xi_{1})$. Here we want to rule out the contributions from
the stops $\xi_{i}^{-1}(c_{i}^{\pm})$ for $i\geq 2$. Then in the next section,
the contributions from the stops $\xi_{1}^{-1}(c_{i}^{\pm})$ will also be
ruled out via several surgery formulae so that the wrapped Fukaya category
$\mathcal{W}(M(\mathbb{V}),\xi)$ will be shown to be generated by the standard
Lagrangian submanifolds. In particular, the standard Lagrangians are
isomorphic to the direct sums of bounded-feasible chamber Lagrangians (Section
(4e-4)), which concludes the generation part of the Theorem 1.1.

Recall from [22] that under Sectorial Descent, the linking disks associated to
the stops $\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})$ for $i>1$
are identified with the linking disks associated to the sectorial corner
$\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})\cap\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{+})$,
which are then identified with the linking disks associated to the stop
$\operatorname{\mathrm{Re}}(\xi_{i})^{-1}(c_{i}^{\pm})\cap\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{+})$
in $\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{+})\cap P_{j0}$. Note that
$\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{+})$ is itself a complement
of some complex hyperplanes which are $1$-dimensional lower, so we can rule
out these linking disks by induction. More precisely, by induction hypothesis
we know that the wrapped Fukaya category
$\mathcal{W}\left(\xi_{1}^{-1}(c_{j1}^{\pm})\right)$ is generated by all the
standard Lagrangians $L_{j\alpha}$ corresponding to a chosen collection of
bounded chamber Lagrangians in $\xi_{1}^{-1}(c_{j1}^{\pm})$, and the linking
disks associated to all the stops in a given sector $P_{j}$ are Orlov functor
images of the generating Lagrangians for Liouville sectors in a sectorial
decomposition of $\xi_{1}^{-1}(c_{j1}^{\pm})$. Thus all linking disks
contributing to the generating objects of $\mathcal{W}(P_{j})$ are given as
Orlov functor images of a chosen set of bounded chamber Lagrangians of the
stop $\xi_{1}^{-1}(c_{j1}^{\pm})$. The case when
$\dim_{\mathbb{C}}M(\mathbb{V})=2$ can be done by hand, since the stop
$\operatorname{\mathrm{Re}}(\xi_{1})^{-1}(c_{1}^{+})$ has complex dimension
$1$ and the required exact triangles generating linking disks are established
in [3]. Therefore, we are able to rule out all the linking disks coming from
the stops $\xi_{i}^{-1}(c_{ij}^{\pm})$ for $i\geq 2$. We end this subsection
with the following theorem:

######  3.26 Theorem.

The wrapped Fukaya category $\mathcal{W}(M(\mathbb{V}),\xi)$ is generated by
the standard Lagrangians $\\{L_{j}\\}_{1\leq j\leq N_{\mathbb{V}}}$
constructed in Proposition 3.24 and the linking disks associated to the stops
$\xi_{1}^{-1}(c_{1j})$.

## 4\. Exact triangle, Surgery and Isomorphic Objects

In Theorem 3.26, we showed that the partially wrapped Fukaya category
$\mathcal{W}(M(\mathbb{V}),\xi)$ is generated by the standard Lagrangians
together with the linking disks associated to all the stops. In this section,
we will use surgery exact sequences to rule out the contributions of all the
linking disks and show that the standard Lagrangians are generated by
Lagrangians associated to bounded feasible chambers of $\mathbb{V}$. We start
with the discussion about McLean type neighborhoods, which is a slight
adaptation of [34].

### (4a) A McLean Type Neighborhood.

Consider a projective completion of $M(\mathbb{V})\subset\mathbb{C}^{d}$,
denoted by $\overline{M}(\mathbb{V})\subset\mathbb{CP}^{d}$. Let $D_{\infty}$
denote the complement
$\mathbb{CP}^{d}\setminus\mathbb{C}^{d}\cong\mathbb{CP}^{d-1}$. By abuse of
notations, we will still use $\\{H_{i}\\}_{i=1}^{n}$ to denote the collection
of the completions of the complexified hyperplanes in the arrangement
$\mathbb{V}$. The intersection loci $\bigcup_{i=1}^{n}H_{i}\cap D_{\infty}$
can be seen as a generic hyperplane arrangement $\mathbb{V}_{\infty}$ in
$\mathbb{CP}^{d-1}$. This contatins a singular locus of
$\overline{M}(\mathbb{V})$, denoted by $\mathbb{V}_{sing}$. For example, such
singular locus arise from two hyperplanes whose normal vector has parallel
components.

One can desingularize $\overline{M}(\mathbb{V})$ by iteratively blowing up
along $\mathbb{V}_{sing}$ from the deepest strata as follows. First, blow up
$\overline{M}(\mathbb{V})$ along all 0-dimensional intersections in
$D_{\infty}$. Then the proper transform of the hyperplanes can only intersect
along dimension $1$ or higher. Next, we blow up along 1-dimensional
intersections, and so on. As the blow-up loci do not intersect with each
other, this iterated blow-up is independent of the ordering. Performing this
procedure inductively until the $(d-2)$-th step, we obtain a non-singular
projective variety $X_{\mathbb{V}}$ with a simple normal crossing divisor
$D_{\mathbb{V}}$ such that $M(\mathbb{V})=X_{\mathbb{V}}\setminus
D_{\mathbb{V}}$. Moreover, $D_{\mathbb{V}}$ is linearly equivalent to

$H_{1}+\cdots+H_{n}-\sum{a_{i}E_{i}}$

for some positive integers $a_{i}>0$, where the $E_{i}$’s are the exceptional
divisors. This is because at each step, there are at least two proper
transforms of hyperplanes intersecting with the blow-up locus, hence the
coefficient of the corresponding exceptional divisor of the proper transform
of $H_{i}$ is always negative. It follows that $D_{\mathbb{V}}$ supports an
ample divisor, after a rescaling of the coefficients if necessary, and the
corresponding ample line bundle $\mathcal{L}\rightarrow X_{\mathbb{V}}$ has a
section $s$ with $s^{-1}(0)=D_{\mathbb{V}}$. We equip $X_{\mathbb{V}}$ with a
symplectic form $\omega_{X}:=-dd^{\mathbb{C}}\log||s||$ where $||\cdot||$ is a
metric on $\mathcal{L}$ whose associated curvature form $F$ has the property
that $iF$ is a positive $(1,1)$-form.

Let $\omega_{\log}$ be the restriction of $\omega_{X}$ to $M(\mathbb{V})$, and
write $M(\mathbb{V})_{\log}$ for $M(\mathbb{V})$ equipped with the $2$-form
$\omega_{\log}$. We remove a small regular neighborhood $N_{D}$ of
$D_{\mathbb{V}}$ from $X_{\mathbb{V}}$, so that its complement is a Liouville
domain with respect to the $1$-form
$\lambda_{\log}:=-d^{\mathbb{C}}\log||s||$. Denote its completion by
$\widehat{M}(\mathbb{V})_{\log}$, which is a Liouville manifold. Let $\tau$ be
the complex conjugation. We may assume that $\tau$ extends to the
compactification $X_{\mathbb{V}}$ so that $\tau(D)=D$ and
$\tau^{*}\omega_{X}=-\omega_{X}$. By choosing the neighborhood $N_{D}$ to be
$\tau$-invariant and using the cone coordinates, we can extend $\tau$ to an
anti-symplectic involution over $\widehat{M}(\mathbb{V})_{\log}$, which we
continue to denote by $\tau$ by abuse of notations.

######  4.1 Lemma.

For any two sufficiently small choices of the neighborhood $N_{D}$, the
resulting Liouville manifolds $\widehat{M}(\mathbb{V})_{\log}$ are
$\tau$-equivariantly symplectomorphic.

###### Proof.

Suppose that $N_{D,1}$ and $N_{D,2}$ are two different choices. There is an
even smaller $\tau$-invariant neighborhood $N_{D}^{\prime}\subset N_{D,1}\cap
N_{D,2}$, so by transitivity of an equivalence relation, it suffices to assume
that $N_{D,2}=N_{D}^{\prime}\subset N_{D,1}$. In this case, there is a
canonical isomorphism between the respective completions, given by the
identity outside of $N_{D,1}$, and matches with the Liouville flow starting
from the boundary of $N_{D,1}$. It is clearly $\tau$-equivariant. ∎

######  4.2 Lemma.

[34, Lemma 5.18] $\widehat{M}(\mathbb{V})_{\log}$ is $\tau$-equivariantly
convex deformation equivalent to $M(\mathbb{V})$, where the latter is equipped
with the Liouville structure specified in Section (3b).

###### Proof.

The fact that $\widehat{M}(\mathbb{V})_{\log}$ is convex deformation
equivalent to $M(\mathbb{V})$ is proved in [34, Lemma 5.18]. The proof
directly generalizes to the $\tau$-equivariant setting. ∎

We recall the McLean type neighborhood theorem.

######  4.3 Proposition ([34] Lemma 5.14).

Let $\widehat{M}(\mathbb{V})_{\log}$ be defined as above. For each
$i\in\\{1,\dots,n\\}$, there is an open neighborhood
$U_{i}\subset\mathbb{C}^{d}$ of $H_{i}$, together with a projection
$\pi_{i}:U_{i}\to H_{i}$, such that the following staements are true:

  1. (i)

For all $i,j$, we have $\pi_{i}\circ\pi_{j}=\pi_{j}\circ\pi_{i}$ in $U_{i}\cap
U_{j}$.

  2. (ii)

For any $I\subset\\{1,\dots,n\\}$, the compositions of $(\pi_{i})_{i}\in I$,
which we denote by $\pi_{I}:U_{I}:=\bigcap_{i\in I}U_{i}\to
H_{I}:=\bigcap_{i\in I}H_{i}$, has fibres diffeomorphic to

$\displaystyle\left(\prod_{i\in I}\mathbb{D},r_{i}dr_{i}\wedge
d\theta_{i}\right),$ (35)

and the intersection of a fibre with $U_{I}\cap H_{j}$ is mapped to
$\\{0\\}_{j}\times\prod_{i\in I\setminus\\{j\\}}\mathbb{D}$ under the
diffeomorphism.

  3. (iii)

For any $j\in I$, the restriction of the map $\pi_{j}$ to any fibre
$\prod_{i\in I}\mathbb{D}$ of $\pi_{I}$ is the natural projection

$\prod_{i\in I}\mathbb{D}_{\epsilon}\to\\{0\\}_{j}\times\prod_{i\in
I\setminus\\{j\\}}\mathbb{D}.$

  4. (iv)

The symplectic parallel transport map of $\pi_{I}:U_{I}\to H_{I}$ has holonomy
in $\prod_{i\in I}U(1)$.

  5. (v)

Let $\tau_{\mathbb{C}}:\mathbb{C}^{d}\to\mathbb{C}^{d}$ and
$\tau_{\mathbb{D}}:\mathbb{D}\to\mathbb{D}$ be complex conjugations, and let
$\tau_{I}:H_{I}\to H_{I}$ be the restriction of $\tau$. For any
$I\subset\\{1,\dots,n\\}$, there is an open $\tau_{I}$-invariant neighborhood
$B_{I}$ of
$L_{I}^{\circ}:=\operatorname{Fix}(\tau_{I})\setminus(\bigcup_{I\subset
J}U_{J}$) in $H_{I}$, together with a $(\tau_{I}\times\prod_{i\in
I}\tau_{\mathbb{D}},\tau)$-equivariant fibre-preserving diffeomorphism

$\displaystyle\left(B_{I}\times\prod_{i\in
I}\mathbb{D},\omega_{H_{I}}|_{B_{I}}+\sum_{i\in I}r_{i}dr_{i}\wedge
d\theta_{i}\right)\simeq\left(U_{I}|_{B_{I}},\omega_{M(\mathbb{V})}|_{U_{I}^{\circ}}\right),$
(36)

where $U_{I}|_{B_{I}}:=\pi_{I}^{-1}(B_{I})$. In particular, it implies that
for any sign sequence $\alpha\in\mathscr{F}(\mathbb{V})$ with $U_{I}\cap
L_{\alpha}\neq\emptyset$, after possibly rotating the disk factors by
$e^{\sqrt{-1}\pi}$, the $(\tau_{I}\times\prod_{i\in
I}\tau_{\mathbb{D}},\tau)$-equivariant diffeomorphism sends

$L_{I}^{\circ}\times\prod_{i\in
I}\left\\{(r_{i},\theta_{i})\in\mathbb{D}^{*}|\alpha(i)e^{\sqrt{-1}\theta_{i}}=1\right\\},$

which is an open subset of $\operatorname{Fix}(\tau_{I}\times\prod_{i\in
I}\tau_{\mathbb{D}})$, to $U_{I}|_{B_{I}}\cap L_{\alpha}$.

###### Proof.

The proposition can be proved by generalizing [34, Lemma 5.14] to incorporate
the anti-symplectic involutions. For the sake of completeness, we present here
the details of proof.

Define a total order for subsets of $\\{1,\dots,n\\}$ as follows: if
$|I_{1}|>|I_{2}|$, then $I_{1}<I_{2}$, and if $|I_{1}|=|I_{2}|$ and
$\min(I_{1}\setminus I_{2})<\min(I_{2}\setminus I_{1})$, then $I_{1}<I_{2}$.
We are going to construct $U_{i}$, $\pi_{i}$ and $B_{I}$ inductively, starting
from its description near $H_{I}$ for small $I$ and progressing to large $I$.

We will construct $U_{i}$ so that $U_{I}=\emptyset$ if $H_{I}=\emptyset$ so
the base case is when $|I|=d$ and $H_{I}$ is a point. Define $B_{I}=H_{I}$.
Using a local $\prod_{i\in I}\tau_{\mathbb{D}}$-invariant metric and the
corresponding exponential map, we can find a small neighborhood $U_{I}$ of the
point $H_{I}$ and a $(\prod_{i\in I}\tau_{\mathbb{D}},\tau)$-equivariant
diffeomorphism $\phi_{I}:\prod_{i\in I}\mathbb{D}\to U_{I}$ such that
$\phi_{I}(\\{0\\}_{j}\times\prod_{i\in I\setminus{j}}\mathbb{D})=U_{I}\cap
H_{j}$ for all $j$. Moreover, there are constants $a_{i}>0$ such that
$\phi_{I}^{*}\omega_{\log}=\sum_{i\in I}a_{i}r_{i}dr_{i}\wedge d\theta_{i}$.
By rescaling $\phi_{I}$ factor by factor, we may assume that $a_{i}=1$ for all
$i$. Since $\tau$ is an anti-symplectic involution, by possibly shrinking
$\prod_{i\in I}\mathbb{D}$, we can isotope $\phi_{I}$ to another $(\prod_{i\in
I}\tau_{\mathbb{D}},\tau)$-equivariant diffeomorphism $\phi_{I}:\prod_{i\in
I}\mathbb{D}\to U_{I}$ such that $\phi_{I}(\\{0\\}_{j}\times\prod_{i\in
I\setminus{j}}\mathbb{D})=U_{I}\cap H_{j}$ and $\phi_{I}^{*}\omega_{\log}$ is
of the form (35). We define $\pi_{i}:U_{I}\to H_{I}$ by $\phi_{I}\circ
p_{i}\circ\phi_{I}^{-1}$, where $p_{i}:\prod_{j\in I}\mathbb{D}\to\prod_{j\in
I\setminus\\{i\\}}\mathbb{D}$ is the natural projection. Since $\phi_{I}$ is
$(\prod_{i\in I}\tau_{\mathbb{D}},\tau)$-equivariant, it sends the fixed point
set of $\prod_{i\in I}\tau_{\mathbb{D}}$ to that of $\tau$. By possibly
rotating the disk factors by $e^{\sqrt{-1}\pi}$, it implies that
$\phi_{I}^{-1}(U_{I}\cap L_{\alpha})$ equals to $\prod_{i\in
I}\left\\{(r_{i},\theta_{i})\in\mathbb{D}^{*}|\alpha(i)e^{\sqrt{-1}\theta_{i}}=1\right\\}$.
This completes the base case of the induction.

Let $I\subset\\{1,\dots,n\\}$. Assume the induction hypothesis holds for all
$J<I$. More precisely, it means that we have constructed
$\\{\pi_{i}\\}_{i=1}^{n}$ and $\\{U_{i}\\}_{i=1}^{n}$ near $H_{J}$ such that
all the listed properties are satisfied. We need to extend the definitions
over a neighborhood of $H_{I}$ such that all the listed properties continue to
hold. In fact, we only need to extend $\pi_{i}$ and $U_{i}$ for $i\in I$ in a
neighborhood of $H_{I}$.

First note that, as a complete intersection, the structure group of the normal
bundle $N_{H_{I}}$ of $H_{I}\subset X_{\mathbb{V}}$ is $\prod_{i\in I}U(1)$.
Using partition of unity, exponential map and the induction hypothesis, we can
construct a diffeomorphism $\phi_{I}$ from a polydisk subbundle of $N_{H_{I}}$
to a neighborhood of $H_{I}$ such that near $\phi_{I}^{-1}(H_{I}\cap H_{J})$,
for $J<I$. $\phi_{I}^{*}\omega_{\log}$ and $\phi_{I}^{*}\pi_{i}$ already
satisfy all the listed properties and $\phi_{I}^{*}\pi_{I}$ coincides with the
projection $N_{H_{I}}\to H_{I}$. By an abuse of notation, we denote the
polydisk subbundle to be $N_{H_{I}}$. Moreover, since every connected
component of $L_{I}^{\circ}$ is contractible, we can choose a smooth
trivialization of $N_{H_{I}}\to H_{I}$ over a $\tau_{I}$-invariant
neighborhood $B_{I}$ of $L_{I}^{\circ}$, which is of the form
$B_{I}\times\prod_{i\in I}\mathbb{D}$ (where the structure group $\prod_{i\in
I}U(1)$ respects the product decomposition), and assume that $\phi_{I}$ is
$(\tau_{I}\times\prod_{i\in I}\tau_{\mathbb{D}},\tau)$-equivariant over
$B_{I}\times\prod_{i\in I}\mathbb{D}$.

On the other hand, we can construct a symplectic form $\omega^{\prime}$ on
$N_{H_{I}}$ such that

  * •

$\omega^{\prime}=\phi_{I}^{*}\omega_{\log}$ near $\phi_{I}^{-1}(H_{I}\cap
H_{J})$, for $J$ less than $I$,

  * •

over $H_{I}^{\circ}:=H_{I}\setminus\left(\bigcup_{J<I}\phi_{I}^{-1}(H_{I}\cap
H_{J})\right)$, the bundle $N_{H_{I}}|_{H_{I}^{\circ}}\to H_{I}^{\circ}$ has
fibres diffeomorphic to (35), with the structure group respects the product
structure,

  * •

the symplectic parallel transport map of $N_{H_{I}}|_{H_{I}^{\circ}}\to
H_{I}^{\circ}$ has holonomy in $\prod_{i\in I}U(1)$,

  * •

over the trivialization $B_{I}\times\prod_{i\in I}\mathbb{D}$,
$\omega^{\prime}$ is of the form (36).

The construction is called a ‘bundle compatible’ symplectic form in [34, Lemma
5.14]. Note that, in our setup, the action of $\tau_{I}\times\prod_{i\in
I}\tau_{\mathbb{D}}$ on $B_{I}\times\prod_{i\in I}\mathbb{D}$ is anti-
symplectic with respect to $\omega^{\prime}$.

Since $\omega^{\prime}|_{H_{I}^{\circ}}=\omega_{H_{I}^{\circ}}$ and
$\omega^{\prime}=\phi_{I}^{*}\omega_{\log}$ near $\phi_{I}^{-1}(H_{I}\cap
H_{J})$, for $J<I$, we can apply Moser’s trick to the linear interpolation of
$\omega^{\prime}$ and $\phi_{I}^{*}\omega_{\log}$. The upshot is that we are
able to modify $\phi_{I}$ so that $\phi_{I}^{*}\omega_{\log}=\omega^{\prime}$.
Inside $N_{H_{I}}$, there is a natural projection map corresponding to the
$i$-th factor of the structure group, denoted by $p_{i}$, which extends
$\phi_{I}^{*}\pi_{i}$ from an open neighborhood of $\phi_{I}^{-1}(H_{I}\cap
H_{J})$ to the entire bundle. We can therefore extend $U_{i}$, for $i\in I$,
to a neighborhood of $H_{I}$ such that $U_{I}=\phi_{I}(N_{H_{I}})$. The maps
$\pi_{i}$, $i\in I$ can also be extended to a neighborhood of $H_{I}$ by
$\phi_{I}\circ p_{i}\circ\phi_{I}^{-1}$. It is immediate to check that they
satisfy all the listed properties. ∎

### (4b) Clean Surgery Formula and Lagrangian Cobordisms.

In this subsection, we briefly review the work on clean surgery formulae for
closed Lagrangian submanifolds [32], and extend it to cylindrical exact
Lagrangian submanifolds. Combining this with the recent work of Bosshard [11],
we show that clean surgery of cylindrical exact Lagrangian submanifolds will
induce exact triangles in the wrapped Fukaya category. We begin with a review
of Lagrangian cobordisms and Lagrangian surgeries.

A Lagrangian cobordism $V\subseteq\mathbb{C}\times X$ is a conical exact
Lagrangian submanifold with respect to $\lambda+\lambda_{\mathbb{C}}$, where
$\lambda$ is the Liouville $1$-form on $(X,\omega,Z)$ dual to $Z$, and
$\lambda_{\mathbb{C}}$ is the standard Liouville $1$-form on $\mathbb{C}$,
with the following additional requirements: there exist (sectorial) Liouville
domains $\mathbb{C}^{\mathrm{int}}\subseteq\mathbb{C}$ and
$X^{\mathrm{int}}\subseteq X$ such that

  1. (a)

$V$ is conical outside $\mathbb{C}^{\mathrm{int}}\times X^{\mathrm{int}}$.

  2. (b)

$V\cap(\mathbb{C}\setminus\mathbb{C}^{\mathrm{int}})\times X$ consists only of
finitely many submanifolds that are conicalizations
$\gamma_{j}\tilde{\times}L_{j}$ of product Lagrangian submanifolds of conical
rays $\gamma_{j}$ in $\mathbb{C}$ and exact Lagrangian submanifolds
$L_{j}\subseteq X$. These conicalized products are called ends of the
cobordism.

  3. (c)