# Symplectic Criteria on Stratified Uniruledness of Affine Varieties and
Applications

Dahye Cho<EMAIL_ADDRESS>
###### Abstract.

We develop criteria for affine varieties to admit uniruled subvarieties of
certain dimensions. The measurements are from long exact sequences of versions
of symplectic cohomology, which is a Hamiltonian Floer theory for some open
symplectic manifolds including affine varieties. Symplectic cohomology is hard
to compute, in general. However, certain vanishing and invariance properties
of symplectic cohomology can be used to prove that our criteria for finding
uniruled subvarieties hold in some cases. We provide applications of the
criteria in birational geometry of log pairs in the direction of the Minimal
Model Program.

###### Contents

  1. 1 Introduction
  2. 2 Symplectic Cohomology
    1. 2.1 Symplectic Structure on an Affine Variety or on a Log Pair
      1. 2.1.1 Liouville Domain corresponds to Affine Variety
      2. 2.1.2 Symplectic Convex Lefschetz fibration Structure on Affine Varieties
      3. 2.1.3 Mapping Cylinder of Milnor Monodromy Map of Isolated Singularities
      4. 2.1.4 Legendrian Front Projection and Flexible Weinstein Cobordism
      5. 2.1.5 Symplectic structure of Weighted Blow-up of Symplectic Orbifolds
    2. 2.2 Symplectic Cohomology
      1. 2.2.1 Symplectic Cohomology of Liouville Domains
      2. 2.2.2 Symplectic Cohomology of Symplectic Convex Lefschetz Fibrations
      3. 2.2.3 Symplectic Cohomology of Non-Exact Convex Symplectic manifolds
    3. 2.3 Morse-Bott Symplectic Cohomology
    4. 2.4 Twisted/Bulk Deformed Symplectic Cohomology
    5. 2.5 Computing Symplectic Cohomology
      1. 2.5.1 Relative Symplectic Cohomology of Compact Subsets of a Completed Liouville Domain and a Mayer-Vietoris Sequence
      2. 2.5.2 Relative Symplectic Cohomology with respect to Symplectic Structures
      3. 2.5.3 Spectral Sequence Relating Different Choices of Polarization
    6. 2.6 Vanishing or Invariance Theorems of Symplectic Cohomology
      1. 2.6.1 Symplectic Cohomology under Attaching Weinstein Handles
      2. 2.6.2 Vanishing Symplectic Cohomology from $\mathbb{C}^{*}$-action
      3. 2.6.3 Invariance from Convex Symplectic Lefschetz fibration
      4. 2.6.4 Vanishing Symplectic Cohomology from Stably-Displaceability
  3. 3 Proof of the Main Theorem
    1. 3.1 $J$-holomorphic $(k,\mathcal{E},[\mho])$-uniruledness
    2. 3.2 $J$-holomorphic $(k,\mathcal{E},[\mho])$-uniruled $\underline{M}$ $\Rightarrow$ Such a $\overline{M}$
    3. 3.3 $J$-holomorphic $(k,\mathcal{E},[\mho])$-uniruled $\overline{M}$ $\Rightarrow$ Polynomial maps $(\mathbb{P}^{1}\setminus\Gamma)\to M$
      1. 3.3.1 McLean’s Degeneration to the Normal Cone
      2. 3.3.2 Fish’s Target-local Gromov Compactness
    4. 3.4 Family of Rational Curves $\Rightarrow$ Algebraic Variety
    5. 3.5 Proof of the Main Theorem
  4. 4 Applications
    1. 4.1 Main Application: $\ell(M)\Rightarrow$ Uniruled Subvarieties
      1. 4.1.1 Uniruled Subvarieties for Pairs $(X,D)$
    2. 4.2 Variance of $\ell(M)$ under Symplectic Surgeries
    3. 4.3 Examples of Affine Varieties with Nontrivial $\ell(M)$
    4. 4.4 Towards the Log Minimal Model Program
      1. 4.4.1 Cylindrical Affine Varieties
      2. 4.4.2 Log Minimal Model Program on Affine Threefolds
      3. 4.4.3 Log Kodaira Dimension
  5. 5 Further Questions and Concluding Remarks
    1. 5.0.1 Symplectic View on Birational Geometry: Cohomological or Categorical
  6. 6 Appendix
    1. 6.1 Several Conventions

##  1\. Introduction

A projective variety $X$ over complex numbers $\mathbb{C}$ is uniruled if for
a generic point $x\in X$, there is a rational curve $\mathbb{P}^{1}\to X$
passing through $x$, where $\mathbb{P}^{n}$ denote $\mathbb{C}P^{n}$. An
affine variety $M$ over $\mathbb{C}$ is called $k$-uniruled if for a generic
point $x\in M$, there is a polynomial map $(\mathbb{P}^{1}\setminus\Gamma)\to
M$ passing through $x$, where $\Gamma$ is a set of at most $k$ distinct
points. (An affine variety $M$ is also called $\mathbb{C}$-uniruled if it is
$1$-uniruled, i.e. for a generic point $x\in M$ there is a polynomial map
$\mathbb{C}\to M$ passing through $x$.) A Liouville domain $\underline{M}$ is
$J$-holomorphic $(k,\mathcal{E})$-uniruled if, for every convex almost complex
structure $J$ on $\underline{M}$ and for any point $x$ in the interior
$\text{int}\underline{M}$ of $\underline{M}$ where $J$ is integrable near $x$,
there is a $J$-holomorphic map $\Sigma\to\text{int}\underline{M}$ passing
through $x$, where $\Sigma$ is a Riemann sphere with the number boundary is at
most $k-1$, and the energy is at most $\mathcal{E}$. A surprising fact is that
both uniruledness of projective varieties and $k$-uniruledness of affine
varieties are symplectic invariant by Kollár, Ruan and McLean using Gromov-
Witten theory [69], [119], [86]. Moreover, uniruledness is a symplectic
birational invariant by Hu-Li-Ruan [59] (i.e. Uniruledness is invariant under
symplectic blow ups and blow downs). Biolley used Floer cohomology to detect
hyperbolicities [11]. In [81], McDuff proved that every closed symplectic
manifold with Hamiltonian $S^{1}$-action is uniruled by detecting nonzero
Gromov-Witten invariant using Seidel representation [122] of the fundamental
group of Hamiltonian diffeomorphism into the small quantum cohomology. McLean
proved that $J$-holomophic $(1,\mathcal{E})$-uniruledness implies
$\mathbb{C}$-uniruledness on affine varieties [86]. Based on [11], Albers-
Frauenfelder-Oancea showed that vanishing symplectic cohomology implies
$J$-holomophic disk-uniruledness by SFT-neck stretching [6]. Zhou proved that,
for an exact domain, the existence of a symplectic dilation implies
$(1,\mathcal{E})$-uniruled [141] based on [86], [29]. Li also considered it
using the cyclic dilation in [78].

In this paper, we develop a criterion for affine varieties or log pairs to be
stratified uniruled, which means a criterion to find uniruled subvarieties of
certain dimensions, inspired by [11], [86]. To define a measurement, we use
symplectic cohomology, that is a Hamiltonian Floer theory for some open
symplectic manifolds including affine varieties. The key observation is that
the connecting maps of symplectic cohomology are Floer cylinders which we
deform to non-constant pseudo-holomorphic curves. Moreover, we can deform the
pseudo-holomorphic curves to rational curves passing through cycles
representing cohomology classes. The deformation/degeneration techniques are
heavily based on [56], [11], [16], [86]. We show how to apply the criteria
effectively by applying certain vanishing and invariance properties of
symplectic cohomology or by computing spectral sequences that converges to
symplectic cohomology, [89], [87], [47], [90]. An application of the criterion
comes when two affine varieties have isomorphic symplectic cohomologies but
different cohomologies. For example, the Kaliman modification of affine
varieties of complex dimension higher than $2$, does not change symplectic
cohomology. Therefore, we can detect uniruled subvarieties of affine varieties
after Kaliman modification. Moreover, the same idea works on the fact that for
a Stein manifold, symplectic cohomology is invariant under attaching flexible
Weinstein $k$-handles. We explore some applications as corollaries. If the
homological mirror symmetry would convert symplectic cohomology more
computable, we expect to use this work to understand log birational geometry
and suggest some questions considering the minimal models.

The following is the main theorem of this paper. Let $M$ be a smooth affine
variety of complex dimension $n$ together with a trivialization of some power
of its canonical bundle. Let $X$ be a smooth projective variety compactifying
$M$ with an ample divisor $D$. There is a long exact sequence:

${\cdots\to
SH^{*-1}(M)}$${SH^{*-1}_{+}(M)}$${H^{*}(M)}$${SH^{*}(M)\to\cdots.}$$\scriptstyle{\delta}$$\scriptstyle{\iota}$

where, $\delta:SH^{*}_{+}(M)\to H^{*}(M)$ is the connecting map.

######  Theorem 1.1 (3.1).

If a cohomology class $[\mho]\in H^{m}(M)$ be in the image of the map $\delta$
for $m=2k$ or $2k+1$ for some $k\in\mathbb{N}$. Then there exists a subvariety
$\Xi_{\mho}\subset M$ of dimension at least $n-k$ satisfying the following
properties:

  1. (1)

$\Xi_{\mho}$ is $\mathbb{C}$-uniruled. In other words, for each
$p\in\Xi_{\mho}$, there exists a non-constant algebraic map
$\overline{v}_{p}:\mathbb{C}\to\Xi_{\mho}$ whose image contains $p$.

  2. (2)

For any exhausting finite type Morse function $f$ on $X\setminus D$,
$\Xi_{\mho}$ set-theoretically intersect with a unstable submanifold
$\mho_{f}$ of $f$ representing $[\mho]\in H^{m}(M)\cong HM^{m}(M,f)$.

As an application, we can define a measurement of the co-dimension of maximal
uniruled subvariety and have the main criterion as follow.

######  Definition 1.2.

$\ell(M):=$ min$\\{$deg$([\alpha]):[\alpha]\in H^{*}(M)$ with
$[\alpha]=\delta(\gamma)$ for some $0\neq\gamma\in SH^{*-1}_{+}(M)\\}$.

######  Remark 1.3.

The same definition holds for all Liouville domains.

######  Corollary 1.4 (4.1).

If $\ell(M)=2k$ or $2k+1$ $(0\leq k<n)$, then $M$ admits a $(n-k)$-dimensional
family of affine lines. Here, an affine line means the image of $\mathbb{C}$
in $M$ under a nonconstant polynomial map. Moreover, $M$ admits a uniruled
subvariety of dimension $n-k$.

######  Corollary 1.5 (4.2).

If $\ell(M)=0$ or $1$, then $M$ is $\mathbb{C}$-uniruled.

######  Corollary 1.6 (4.3).

If $SH^{*}(M)=0$, then $M$ is $\mathbb{C}$-uniruled. (See Theorem 5.4 in
[141])

######  Theorem 1.7 (4.7).

Let $\underline{M}$ be the associated Liouville domain obtained by
intersection of $M$ and a large $2n$-ball. Assume that $\underline{M}$ is of
complex dimension equal to or bigger than $3$. Suppose that $\underline{M}$ is
a connected Liouville domain with a Weinstein $1$-handle attached, then $M$ is
$\mathbb{C}$-uniruled.

More generally, in the category of Weinstein manifolds, we consider attaching
flexible Weinstein handles in [138], [95], [24], [25], which does not change
symplectic cohomology [Theorem 5.6, [15]]. An example of a flexible Weinstein
handle is a subcritical handle, [23].

######  Theorem 1.8 (4.8).

Let $W$ be a Weinstein manifold of dim${}_{\mathbb{R}}W=2n$ with
$l(W)=\infty$. Suppose that we have a Weinstein manifold $W_{\Supset k}$,
obtained by attaching flexible $k$-handles to $W$ so that $\text{rank
}H_{k}(W_{\Supset k})>\text{rank }H_{k}(W)$. Then $l(W_{\Supset k})=2n-k$.
Hence, if $W_{\Supset k}$ is symplectomorphic to an affine variety $M$, then
$M$ admits a $\mathbb{C}$-uniruled subvariety of complex dimension $n-k$.

######  Remark 1.9.

To apply the theorem above in practice, we need to understand the condition
when a Stein manifold $W_{\Supset k}$ is symplectomorphic to an affine
variety.

A Liouville domain corresponding to an affine variety has a structure of
symplectic convex Lefschetz fibration [Lemma 8.6, [83]]. The Kaliman
modification $M^{\vee}$ of an affine variety $M$ become an affine variety
having $M$ as a sub-Lefschetz fibration so that all the singular points are
contained in $M$. Therefore, using the fact that symplectic cohomology does
not change under Kaliman modification, we can construct affine varieties with
non-trivial $\ell(M)$.

######  Theorem 1.10 (4.11).

For each $n\geq 3$, there is an affine variety $M^{2n}_{2}$ of
dim${}_{\mathbb{C}}M=n$ with $\ell(M^{2n}_{2})=2$.

An affine variety $M$ is called cylindrical if it contains a dense principal
Zariski open subset $U=M\setminus(f=0)$, for some $f\in\mathcal{O}(X)$,
isomorphic to $\mathbb{C}\times M^{\prime}$ for an affine variety
$M^{\prime}$. We call such $U$ a cylinder [Definition 3.1.4 [65]]. Applying
the idea of [89], [134] to the case of compact subsets of the completion of a
Liouville domain,

######  Theorem 1.11 (4.12).

Let $M$ be an cylindrical affine variety having a dense affine open subset
$U=M\setminus(f=0)$ for some $f\in\mathcal{O}(X)$,
$SH^{*}(\underline{M}\subset M)=0$. Therefore $\widehat{\ell}(M)=0$ and $M$ is
$\mathbb{C}$-uniruled. (For the definitons of $SH^{*}(\underline{M}\subset M)$
and $\widehat{\ell}(M)$, we refer the subsection containing lemma 2.23.)

Applying the similar idea of proof of theorem 4.12 to special case of half-
point attachment of an affine threefold, we have the following theorem.

######  Theorem 1.12 (4.14).

Consider special case of Kishimoto’s $\\#$\- minimal model program on affine
threefolds that restrict half-point attachment at a point on a hypersurface.
Then

  1. (1)

$\widehat{\ell}(M^{i})$’s are not decreasing.

  2. (2)

Let $M$ be an affine threefold and $M^{\\#}$ be a terminal object of $M$ in
Kishimoto’s $\\#$-minimal model program. Suppose that $\pi_{1}(M^{\\#})=0$,
$M\ncong M^{\\#}$ and at least one of steps of half-point attachment is done
on a hypersurface divisor. Then $\widehat{\ell}(M)\geq 2$.

Given two Liouville domains $\underline{M}$ and $\underline{N}$, define the
end-connected sum $\underline{M}\\#_{e}\underline{N}$ by attaching a
$1$-handle to points $p\in\partial\underline{M}$, $q\in\partial\underline{N}$
in the boundary, respectivley. Then $\underline{M}\\#_{e}\underline{N}$ become
a Liouville domain [Theorem 2.10, [82]]. Moreover, $SH^{*}(M\\#_{e}N)\cong
SH^{*}(M)\times SH^{*}(N)$ as rings for complex dimension greater than 1
[Theorem, 2.17, [82]]. From the following diagram,

${0}$${SH^{*}_{+}(M)}$${SH^{*}_{+}(M\\#_{e}N)}$${SH^{*}_{+}(N)}$${0}$${0}$${H^{*+1}(M,\partial
M)}$${H^{*+1}(M\\#_{e}N,\partial(M\\#_{e}N))}$${H^{*+1}(N,\partial N)}$${0,}$

######  Proposition 1.13.

For the end-connected sum $\underline{M}\\#_{e}\underline{N}$ of two Liouville
domains $\underline{M}$ and $\underline{N}$,
$\ell(M\\#_{e}N)=\text{min}\\{\ell(M),\ell(N)\\}$.

By the K$\ddot{\text{u}}$nneth formula,

######  Proposition 1.14.

$\ell(M\times N)=\text{min}\\{\ell(M),\ell(N)\\}$

######  Example 1.15 (1.21).

Let $\phi:M\to M$ be a compact-supported symplectomorphism on a Liouville
domain $(M,\omega)$ whose support is contained in the interior of $M$. Assume
that $\ell(M)=\infty$. Let $M_{\Supset k}$ be a $k$-handle attaching on $M$.
$\Phi$ be an extension of $\phi$ to $M_{\Supset k}$ by identity on attached
locus. Denote
$\mathbb{C}^{*}\rtimes_{\phi}M:=(\mathbb{R}\times\mathbb{R}\times
M)/\mathbb{Z}\cdot(s,\theta,x)\sim(s+1,\theta,\phi(x))$ the symplectic mapping
cylinder of $\phi:M\to M$. Then for any $k<dim_{\mathbb{C}}M$,
$SH^{*}(\mathbb{C}^{*}\rtimes_{\phi}M)\cong
SH^{*}(\mathbb{C}^{*}\rtimes_{\Phi}(M_{\Supset k}))\cong
SH^{*}((\mathbb{C}^{*}\rtimes_{\phi}M)_{\Supset k+1})$ but
$\ell(\mathbb{C}^{*}\rtimes_{\phi}(M_{\Supset
k}))=\text{min}\\{k+2,\ell(\mathbb{C}^{*}\rtimes_{\phi}M)\\}$,
$\ell((\mathbb{C}^{*}\rtimes_{\phi}M)_{\Supset
k+1})=\text{min}\\{k+1,\ell(\mathbb{C}^{*}\rtimes_{\phi}M)\\}$.

We can extend the usual symplectic cohomology $SH^{*}(M)$ to the twisted
symplectic cohomology $SH^{*}(M;\underline{\Lambda}_{\beta})$: symplectic
complexes have the same generators with Novikov field $\Lambda$ as a
coefficient ring. But, the differential map is twisted by a closed 2-form
$\xi$ on $M$, where $\beta$ is a singular cocycle representing a class in
$H^{1}(\mathcal{L}M;\mathbb{R})$ for $\xi\in H^{2}(M;\mathbb{R})$. It means
that the differential map remember the ”area” of Floer cylinder with respect
to $\xi$ in [114]. We also consider symplectic cohomology bulk-deformed by
homology classes in [132], [126]. There exists a long exact sequence [Lemma
8.1, [116]], ${\cdots\to SH^{*-1}(M;\underline{\Lambda}_{\beta})\to
SH^{*-1}_{+}(M;\underline{\Lambda}_{\beta})\overset{\delta}{\longrightarrow}H^{*}(M;\underline{\Lambda}_{\beta})\overset{\iota}{\longrightarrow}SH^{*}(M;\underline{\Lambda}_{\beta})\to\cdots}$.

The cotangent bundle $(T^{*}\mathbb{P}^{1},\omega_{std})$ over
$\mathbb{P}^{1}$ is $\mathbb{C}$-uniruled but $\ell(T^{*}\mathbb{P}^{1})=2$.
On the other hand, twisted symplectic cohomology
$SH^{*}(T^{*}\mathbb{P}^{1},\Lambda_{\omega_{I}})$ vanishes, [115]. So, by
considering the minimum of $\ell(M,\omega)$ over all the twisted cohomolgy, we
get stronger measures as follows.

######  Definition 1.16.

  1. (1)

$\ell(M;\underline{\Lambda}_{\beta}):=\text{min}\\{\text{deg}([\alpha]):\text{
for }[\alpha]\in H^{*}(M;\underline{\Lambda}_{\beta})$ with
$[\alpha]=\delta(\gamma)$ for some $0\neq\gamma\in
SH^{*-1}_{+}(M;\underline{\Lambda}_{\beta})\\}$.

  2. (2)

$\ell(M;\Lambda_{2}):=$min$\\{\ell(M;\underline{\Lambda}_{\beta})$ : for all
$\beta\in$ image$(H^{1}(\mathcal{L}M)\leftarrow H^{2}(\mathcal{L}M\times
S^{1})\leftarrow H^{2}(M))\\}$

  3. (3)

$\ell(M;\Lambda_{\text{all}}):=$min$\\{\ell(M;\underline{\Lambda}_{\beta})$ :
for all bulk deformation by $\beta\in H^{*}(M)\\}$.

######  Corollary 1.17.

If $\ell(M;\underline{\Lambda}_{\beta})$or $\ell(M;\Lambda_{2})=2k$ or $2k+1$
$(0\leq k<n)$, then $M$ admits a uniruled subvariety of dimension $n-k$.

######  Corollary 1.18.

If $\ell(M,\Lambda_{2})=0$, then $M$ is $\mathbb{C}$-uniruled.

######  Example 1.19.

[115] $SH^{*}(T^{*}\mathbb{P}^{1},\Lambda_{\omega_{I}})=0$ implies
$\ell(T^{*}\mathbb{P}^{1},\Lambda_{2})=0$. Therefore, $\ell(M,\Lambda_{2})$ is
a stronger quantitative measure than $\ell(M)$ since
$\ell(T^{*}\mathbb{P}^{1},\omega_{std})=2$.

When we extend the coefficient ring to $\Lambda_{Mori}$ using the cone of
effective curves, the Mori cone, of $H_{2}(X)$, we could get more precise
results in log-birational geometry, that is a generalization of Pascaleff’s
work on log Calabi-Yau surfaces in [109]. We will explain how to compute
$\ell(M,\Lambda_{Mori})$ using homological mirror symmetry and cohomology from
tropicalization in a subsequent paper.

Let $\phi:M\to M$ be a Liouville diffeomorphism on a Liouvlle domain
$(M,\omega)$. Denote
$\mathbb{C}^{*}\rtimes_{\phi}M:=(\mathbb{R}\times\mathbb{R}\times
M)/\mathbb{Z}\cdot(s,\theta,x)\sim(s+1,\theta,\phi(x))$, the symplectic
mapping cylinder of $\phi$, [88]. Denote
$\Sigma\rtimes_{\phi}M:=\\{(\mathbb{R}\times
S^{1}\setminus\mathbb{Z}\times\\{1\\})\times
M\\}/(s,\theta,x)\sim(s+1,\theta,\phi(x))$ the symplectic mapping torus of
$\phi$, [(1-5), [61]]. We can define similar measurments $\ell$ for symplectic
mapping cylinder/tori.

######  Example 1.20.

For the identity map $\phi=id_{M}$ on $M$,

  1. (1)

$\mathbb{C}^{*}\rtimes_{id}M=\mathbb{C}^{*}\times M$, where
$\mathbb{C}^{*}:=\mathbb{C}\setminus\\{0\\}$. By the K$\ddot{\text{u}}$nneth
formula, $SH^{*}(\mathbb{C}^{*}\rtimes_{id}M)\cong
SH^{*}(\mathbb{C}^{*})\otimes SH^{*}(M)$. Therefore,
$\ell(\mathbb{C}^{*}\rtimes_{\textnormal{id}}M)=\text{min}\\{\ell(\mathbb{C}^{*})=\infty,\ell(M)\\}=\ell(M)$.

  2. (2)

$\Sigma\rtimes_{id}M=\Sigma_{1,1}\times M$, where
$\Sigma_{1,1}:=\mathbb{T}^{2}\setminus\\{\text{a point}\\}$, a once-punctured
torus. Similarly, $\ell(\Sigma\rtimes_{id}M)=\ell(M)$.

######  Example 1.21.

Let $\phi:M\to M$ be a Liouville diffeomorphism on a Liouville domain
$(M,\omega)$. Assume that $\ell(M)=\infty$. Let $M_{\Supset k}$ be a
$k$-handle attaching on $\partial M$. $\Phi$ be an extension of $\phi$ to
$M_{\Supset k}$ by identity on attached locus. Then for any
$k<dim_{\mathbb{C}}M$, $SH^{*}(\mathbb{C}^{*}\rtimes_{\phi}M)\cong
SH^{*}(\mathbb{C}^{*}\rtimes_{\Phi}(M_{\Supset k}))\cong
SH^{*}((\mathbb{C}^{*}\rtimes_{\phi}M)_{\Supset k+1})$ but
$\ell(\mathbb{C}^{*}\rtimes_{\phi}(M_{\Supset
k}))=\text{min}\\{k+2,\ell(\mathbb{C}^{*}\rtimes_{\phi}M)\\}$,
$\ell((\mathbb{C}^{*}\rtimes_{\phi}M)_{\Supset
k+1})=\text{min}\\{k+1,\ell(\mathbb{C}^{*}\rtimes_{\phi}M)\\}$.

Let $M$ be a Liouville domain. Recall that $SH^{*}(M)=\oplus_{g\in
H_{1}(M)}SH^{*}_{g}(M)$ is a $H_{1}(M)$-graded algebra with the pair-of-pants
product $\cdot_{p.p}$ : for $\gamma_{1}\in SH^{*}_{g_{1}}(M),\gamma_{2}\in
SH^{*}_{g_{2}}(M)$, $\gamma_{1}\cdot_{p.p}\gamma_{2}\in
SH^{*}_{g_{1}+g_{2}}(M)$. The pair-of-pants product counts three-punctured
spheres with two positive periodic orbits and one negative periodic orbit as
asymptotic orbits. So if the pair-of-pants product of two periodic orbits is
the unit $e$ of the symplectic cohomology, which is the
Poincar$\acute{\text{e}}$ dual of the fundamental class $[M]$, then the same
idea gives us the following.

######  Definition 1.22.

$\ell_{p.p}(M):=\text{min}\\{\text{deg }\alpha:\exists\alpha\neq 0\in
H^{*}(M)\text{ and }\exists g\in H_{1}(M)\text{ such that
}\alpha=\gamma_{1}\cdot_{p.p}\gamma_{2}\text{ for some }\gamma_{1}\in
SH^{*}_{g}(M),\gamma_{2}\in SH^{*}_{-g}(M)\\}$.

######  Example 1.23.

If $SH^{*}(M)$ has an element $\gamma\neq 0$ satisfying $\gamma^{k}=[M]$, then
$\ell_{p.p}(M)=0$.

######  Example 1.24.

For the mapping cylinder of the identity map,
$\ell_{p.p}(\mathbb{C}^{*}\rtimes_{id}M)=0$. Moreover, for a Liouville
diffeomorphism $\phi:M\to M$,
$\ell_{p.p}(\mathbb{C}^{*}\rtimes_{\phi}M)\geq\ell_{p.p}(\mathbb{C}^{*}\rtimes_{id}M)$.

######  Definition 1.25.

$\ell_{p.p}(\phi):=\text{min}\\{k:\ell_{p.p}(\mathbb{C}^{*}\rtimes_{\phi^{k}}M)=0\\}$,
where $\phi^{k}:=\phi\circ\cdots\circ\phi$, $k$th iteration of $\phi$.

######  Example 1.26.

If $\phi^{k}=id$ for some $k$, then $l_{p.p}(\phi)\leq k$.

######  Example 1.27.

Define $f:\mathbb{C}^{n+1}\to\mathbb{C}$ to be
$f(z_{1},z_{2},\cdots,z_{n+1)}:=z_{1}^{k_{1}}+\cdots+z_{n+1}^{k_{n+1}}$ with
an isolated singular point at $0$. Let $S_{\epsilon}$ be the sphere of radius
$\epsilon$ centered at $0$ and $L_{f}:=S_{\epsilon}\cap f^{-1}(0)$. Let
$\phi:f^{-1}(\delta)\to f^{-1}(\delta)$ be a Milnor monodromy map. Then
$\phi^{l.c.m(k_{i})}=id$. Therefore, $l_{p.p}(\phi)\leq l.c.m(k_{i})$.

From the proof of the main theorem,

######  Theorem 1.28.

If $\ell_{p.p}(M)=0$, then $M$ is $\mathbb{C}^{*}$-uniruled.

Let us briefly explain the main idea to prove the theorem 1.1. Assume that for
some $[\alpha]\in H^{\ell}(M)$, $\delta(\gamma)=[\alpha]$ for some
$0\neq\gamma\in SH^{\ell-1}_{+}(M)$. Then there exists a Floer cylinder
$u:\mathbb{R}\times S^{1}\to M$ with negative asymptotic periodic orbit
$\gamma$ and with positive asymptotic orbit which lands on homology class
represented by a pseudo-cycle of a unstable submanifold of index $2n-\ell$ in
Morse homology. Applying Bourgeois-Oancea’s correspondece theorem on the
Morse-Bott symplectic cohomology, we get a pseudo-holomorphic curve in $M$
passing through a point on a unstable submanifold representing a cycle in
$HM^{\ell}(M)$. McLean’s degeneration to the normal cone method in [86] let us
get a non-constant rational map $f:\mathbb{C}\to M$ passing through a point in
a cycle of index $2n-\ell$. Because of the set-theoretic intersection with
pseudo-cycle, we get a family of rational curves with evaluation. By
Grothendieck’s construction of Hilbert schemes (or Quot Schems), we get a
uniruled subvariety of dimension at least $\lfloor\ell/2\rfloor$ that
parametrizes the family of rational curves. For readers’ convenience, let us
provide the structure of this paper. To read the section 3 on proof of the
main theorem, the readers only need to check definitions in subsections 2.1.1
and 2.2.1. Other symplectic structures and properties of symplectic
cohomology, summarized in the section 1 and 2, will be used in the section 4
on applications.

Acknowledgments: This paper is a part of my PhD thesis. I would like to thank
my advisor Mark McLean for suggesting this project and for his kind guidance,
helpful advice, patience and warm encouragement. I would also like to thank
Jason Starr for his inspiring lectures, helpful advice and warm words. I would
like to thank my friends and professors in the Math department at Stony Brook
for their continued support and for providing a friendly environment. I have
also learned a lot by attending, seminars and classes at Stony Brook
University(Symplectic/Contact/Gauge theory seminar, Algebraic Geometry
seminar, Student AG seminar), British Isles Graduate Workshop in 2018, Kylerec
Workshop, Categorical Symplectic Topology Conference, MSRI summer school on
Current Trends in Symplectic Topology, Joint Math Meetings 2020 (AMS-AWM
Special Session for Women in Symplectic and Contact Geometry), Symplectic
Zoominar, and Western Hemisphere Virtual Symplectic Seminar. I appreciate all
the organizers and people whom I met there. During this project, I was
supported by John W. Milnor Endowed Graduate Fund Travel Award, Stony Brook
Math Department Summer Research Funding, and GSEU Professional Development
Award at Stony Brook.

##  2\. Symplectic Cohomology

We review known definitions and theorems about symplectic cohomology in this
section so experts who are familiar with the basics can jump to the section 3.
To read section 3, the subsections 2.1, 2.2, and 2.3 are needed only.
Symplectic cohomology is well-defined on the completion of Liouville domains
[[136], [56], [124], [82], [16], [2], [47], and [29]]. Every smooth affine
variety has a unique Liouville domain, up to isotopy through Liouville
domains, whose completion is symplectomorphic to the affine variety [85].
Since, on Liouville domains, symplectic form is exact, many assumptions,
computations related to the definition of symplectic cohomology become
simpler. Symplectic cohomology still be well-defined for non-exact convex
symplectic manifold with the weak monotonicity assupmtion, for example, the
blow-up of $\mathbb{P}^{1}$ at a point [115], [90]. To get a
$\mathbb{Z}$-grading on cohomology, we assume a Liouville domain with the
vanishing first Chern class, $c_{1}=0$. For simplicity, we mainly consider
simply-connected varieties, $\pi_{1}=0$, for example, complete intersections
of $\text{dim}_{\mathbb{C}}>3$ by the Lefschetz hyperplane theorem, then
consider the action by the fundamental group for $\pi_{1}\neq 0$. For
symplectic manifolds with $\pi_{1}\neq 0$, we use Novikov ring coefficient.

###  2.1. Symplectic Structure on an Affine Variety or on a Log Pair

A Stein manifold is a properly embedded complex submanifold of
$\mathbb{C}^{N}$ for some $N$, equivalently, a complex manifold with an
exhausting plurisubharmonic function. Smooth affine varieties are Stein
manifolds of finite type, which means a complex manifold $(M,J)$ that can be
properly embedded in $\mathbb{C}^{N}$ for some $N$ by admitting a function
$f:M\to\mathbb{R}$ satisfying the following conditions,

  * •

(Plurisubharmonic) $(-dd^{c}f)(v,Jv)>0$ for all non-zero vectors $v$ where
$d^{c}:=d\circ J$,

  * •

(Exhausting) $f:M\to\mathbb{R}$ is bounded below and the preimage of every
compact set is compact,

  * •

(Of finite type) $f$ has only finitely many singularities.

Since the space of exhausting plurisubharmonic functions on a given Stein
manifold is convex, so contractible and open in $C^{2}$, we can choose an
exhausting plurisubharmonic generalized Morse function $f:M\to\mathbb{R}$ on a
Stein manifold $(M,J)$.

A Weinstein structure on a symplectic manifold $(M,\omega)$ is a pair $(f,Z)$,
where $f:M\to\mathbb{R}$ is an exhausting generalized Morse function, and $Z$
is a complete Liouville vector field that is gradient-like for $f$. Given a
Stein manifold $(M,J,f:M\to\mathbb{R})$, one can find a symplectic form
$\omega_{f}$ so that $(f,Z_{f}:=\nabla_{g_{f}}f)$ become a Weinstein structure
on $(M,\omega_{f})$, where. Given a Weinstein structure $(Z,f)$ on
$(M,\omega)$, one can construct a Stein structure $(J,f)$ such that the
canonical Weinstein structure on $(M,J,f)$ is Weinstein homotopic to the given
one. We refer [24] for the detail. A Weinstein structure on a symplectic
manifold carry a Morse-theoretic handle decomposition, called Weinstein handle
decomposition: Weinstein cobordism is attaching a standard handle along a
neighborhood of an isotropic sphere in the contact level set with a Liouville
vector field transverse to contact level sets [Theorem 5.1, [138]].

####  2.1.1. Liouville Domain corresponds to Affine Variety

A Liouville domain is a compact manifold $\underline{M}$ with boundary
$\partial\underline{M}$ and a $1$-form $\theta_{\underline{M}}$ with
$d\theta_{\underline{M}}$ as a symplectic form and satisfying that for any
vector field $Z$ with
$\iota_{Z}d\theta_{\underline{M}}=\theta_{\underline{M}}$ is transverse to
$\partial\underline{M}$ and pointing outwards along $\partial\underline{M}$.
We call such a vector field $Z$, the Liouville vector field. Using the
Liouville flow of $Z$, we glue an half-infinite cylindrical end,
$([-\epsilon,\infty)\times\partial\underline{M},d(e^{r}\alpha))$, which is a
symplectization of the contact boundary, to $\underline{M}$ along
$((-\epsilon,0)\times\partial\underline{M}$ so that $Z$ extends to
$\partial_{r}$ on $((0,\epsilon)\times\partial\underline{M}$. Then we get a
completion of a Liouville domain,
$\widehat{\underline{M}}:=\underline{M}\cup_{\partial\underline{M}}([0,\infty)\times\partial\underline{M})$.

In [(4b), [124]], [Lemma 2.1, [85]], for any smooth affine variety $M$, there
is an associated Liouville domain $\underline{M}$, whose completion,
constructed by attaching a positive half of symplectization of contact
boundary $\partial\underline{M}$ of $M$ to $M$, is symplectomorphic to $M$,
which is unique up to Liouville isomorphism, i.e. an isotopy through Liouville
domains which let us consider the symplectic cohomology of affine varieties as
the completion of Liouville domains. The construction is the following: Let
$\iota:M\hookrightarrow\mathbb{C}^{N}$ be a algebraic embedding as a closed
subvariety. Let $(r_{i},\theta_{i})$ be polar coordinates for the $i$’th
factor in $\mathbb{C}^{N}$. Define $R:=\frac{1}{4}\sum_{i}r^{2}_{i}$ and
$\theta:=-d^{c}R=\frac{1}{2}\sum_{i}r_{i}^{2}d\mathcal{\theta}_{i}$,
$d\theta=\sum_{i}r_{i}dr_{i}d\mathcal{\theta}_{i}$, the standard symplectic
form on $\mathbb{C}^{N}$. Then there exists $C>0$ such that for all $c\geq C$,
$((R|_{M})^{-1}(-\infty,c],\theta_{M})$ is a Liouville domain whose completion
is symplectomorphic to $(M,d\theta|_{M})$. Such a number $C$ is the minimum of
numbers $c$ so that $R|_{M}$ has no singularities for
$(R|_{M})^{-1}(c,\infty)$, [Lemma 2.1, [85]].

By Hironaka’s resolution of singularities [57], we can consider an affine
variety $M$ as the complement of simple normal crossing divisors
$D=\\{D_{i}\\}^{l}_{i=1}$ in a projective variety $X$. We can consider a log
pair $(X,D)$ as an affine variety $M=X\setminus D$ with compactification data.
Let us recall the symplectic structure of a $X\setminus D$ from [Lemme 5.19,
Theorem 5.20, [85]], which has good properties that divisors are intersecting
positively and the wrapping numbers of them are negative and small tubular
neighborhood of them are fibrations with fibers, product of punctured disks.
Recall [Theorem 5.20, [85]]: Let $M$ be a smooth affine variety. Then $M$ is
convex deformation equivalent to a finite type convex symplectic manifold
$(W,\theta_{W})$ with the following properties:

  1. (1)

$W$ is symplectomorphic to $X\subset D$, where $D=\\{D_{i}\\}^{k}_{i=1}$ are
co-dimensionR $2$ symplectic submanifolds transversely intersecting.

  2. (2)

There are neighborhoods $UD_{i}$ of $D_{i}$ and fibration
$\pi_{i}:UD_{i}\twoheadrightarrow D_{i}$ such that

     * •

For $1\leq i_{1}<i_{2}<\cdots<i_{l}\leq k$,
$\pi_{i_{l}}\circ\cdots\circ\pi_{i_{1}}:\cap^{l}_{j=1}UD_{i_{j}}\twoheadrightarrow
D_{\\{i_{1},\cdots,i_{l}\\}}$
has fibers that are symplectomorphic to $\prod_{j=1}^{l}B_{\epsilon}$, where
$B_{\epsilon}$ is the disk of radius $\epsilon$.

     * •

On the fiber $\prod_{j=1}^{l}B_{\epsilon}$ of
$\pi_{i_{l}}\circ\cdots\circ\pi_{i_{1}}$, one of factor map $\pi_{i_{k}}$,
$1\geq k\geq l$, maps this fiber to itself.

     * •

The symplectic structure on $UD_{i}$ induces a natural connection for
$\prod_{j=1}^{l}B_{\epsilon}$ given by the $\omega$ orthogonal vector bundles
to the fibers.

  3. (3)

$\theta_{W}$ restricted to the fiber $B_{\epsilon}$ of $\pi_{i}$ is equal to
$(r_{i}^{2}+\kappa_{i})d\theta_{i}$ for some $\kappa<0$.

For any affine variety $M$, we can construct two Liouville domains
$\underline{M},\overline{M}$ with $\underline{M}\hookrightarrow
M\hookrightarrow\overline{M}$, where $\underline{M},\overline{M}$ are
Liouville deformation equivalent. Indeed, two inclusions are exact symplectic
embeddings of a codimension $0$ submanifold and a homotopy equivalence. This
fact will be used when we make holomorphic curves on $\underline{M}$ into
holomorphic curves on $\overline{M}$, then get rational curves on $M$. Let us
state the sandwich theorem for affine varieties [Lemma $4.3$, Lemma $4.4$,
[86]]:  Let $M$ be a smooth affine variety. Let $L$ be a line bundle
associated to a simple normal crossing compactification $(X,D)$ of $M$ and $s$
be a section with $s^{-1}(0)=D$. Define $f:=-log||s||$, $\theta=-d^{c}f$,
$\omega=-dd^{c}f$. Then

  * •

$f$ is a proper and bounded below function on $M$ with $df(X_{\theta})>0$ on
$f^{-1}([c,\infty))$ for some $c\gg 1$, and

  * •

there is an exact symplectic embedding
$\iota:(M,\theta)\hookrightarrow(\overline{M}:=f^{-1}((-\infty,c]),\nu\cdot\theta)$
for some $\nu$ which is a homotopy equivalence and
$(\overline{M},\nu\cdot\theta))$ is Liouville deformation equivalent to the
associated Liouville domain $\underline{M}$ of the affine variety $M$.

####  2.1.2. Symplectic Convex Lefschetz fibration Structure on Affine
Varieties

Symplectic geometric understanding on Lefschetz fibration was initiated by
Donaldson. Seidel has foundational works on categorification in symplectic
geometry of Picard-Lefschetz theory. McLean constructed symplectic convex
Lefschetz fibration structure on any (Kaliman modification of) affine
varieties [Theorem 2.32, [83]]. A symplectic convex Lefschetz fibration
structure with monodromy information is a complexified Morse function and is
useful for systematic understanding on symplectic structure, for example,
dimension deduction. So let us summarize the known results [Chap.III, [123],
[82], Chap.8, [83], [128]]. Here, systematic understanding means that we chop
the structure into smaller dimensional pieces or a complex of simpler objects
like Morse-theoretic building blocks and then glue them to get an homological
invariants, like spectral sequences, long exact sequences.

Let $X$ be a projective variety with an ample divisor $D$ and $M:=X\setminus
D$ be an affine variety. Let $\mathcal{L}$ be an ample line bundle on $X$ with
two holomorphic sections $s,t$ with $s^{-1}(0)=D$. Define a map
$p:=t/s:M\to\mathbb{C}$.

######  Definition 2.1.

An (open) algebraic Lefschetz fibration structure on a pair
$(M,p:M\to\mathbb{C})$ is following.

  1. (1)

$\overline{t^{-1}(0)}$ is smooth, reduced and intersects each stratum of $D$
transversally.

  2. (2)

$p$ has only nondegenerate critical points and there is at most one of these
points on each fiber.

The symplectic structure on $M$, $\omega=dd^{c}(-log||s||)$, is compatible
with $p$, in other words, $\omega$ restricted to the fiber is a symplectic
form away from singular part. But the horizontal boundary is not trivial. To
define symplectic cohomology on Lefschetz fibrations, we need compact convex
condition.

######  Definition 2.2.

A symplectic Lefschetz fibration is a pair $(M,p:M\to B)$, where
$(M,\omega=d\theta)$ is a compact exact symplectic manifold with corners whose
boundary is the union of horizontal $\partial_{h}M$ and vertical
$\partial_{v}M$ meeting in a codimension $2$ corner and $B$ is a surface with
boundary. The smooth map $p:M\to B$ satisfies the following.

  1. (1)

The map $p$ has only finitely many critical points.

  2. (2)

$\omega$ restricts to a symplectic form on smooth locus of each fiber
$p^{-1}(b)\setminus M^{\text{crit}}$, for every $b\in B$.

  3. (3)

There is an integrable complex structure $J_{0}$ (resp. $j_{0}$) defined on
some neighborhood of $M^{\text{crit}}$ (resp. $S^{\text{crit}}$) such that $p$
is $(J_{0},j_{0})$-holomorphic map near $M^{\text{crit}}$.

  4. (4)

There is at most one critical point in each fiber and at any critical point,
the Hessian $D^{2}p$ is nondegenerate as a complex quadratic form.

  5. (5)

$\omega$ is Kähler form for $J_{0}$ near $M^{\text{crit}}$.

######  Definition 2.3.

A symplectic Lefschetz fibration $(M,p:M\to B)$ is compact convex if

  1. (1)

(Exact) $p:M\to B$ is a proper map with $\partial_{v}M=p^{-1}(\partial B)$ and
such that $p|_{\partial_{v}M}$ is a smooth fiber bundle. Also, there is a
neighborhood $N$ of $\partial_{h}(M)$ such that $p|_{N}:N\to B$ is a product
fibration $B\times\text{nbhd}(\partial F)$ where exact symplectic structure is
pulled back from $\text{nbhd}(\partial F)$.

  2. (2)

(Compact) Fiber $(F,\theta|_{F})$ is a compact convex symplectic manifold.

Using well-defined exact parallel transport maps for an algebraic Lefschetz
fibration [Section 2, [45]],

######  Lemma 2.4 (Lemma 8.6, [83]).

Let $(M,p:M\to\mathbb{C},\omega:=-dd^{c}(-log||s||))$ be an open algebraic
Lefschetz fibration. Then there exists a convex symplectic structure
$\theta^{\prime}$ so that $(M,p,\theta^{\prime})$ is a symplectic convex
Lefschetz fibration.

Kaliman modification of a triple $(M,D,C)$ of an smooth affine variety $M$, an
irreducible divisor $D$, a closed subvariety $C$ contained in the smooth part
of $D$ is known to be an operation making a new affine variety that is
diffeomorphic to the original one but not biholomorphic. McLean constructed
symplectic convex Lefschetz fibration structure on Kaliman modification and
showed that they are convex deformation equivalent to each other [83].

######  Definition 2.5.

The Kaliman modification $M^{\vee}$ of $(M,D,C)$ is defined by
$M^{\vee}:=\mathcal{B}l_{C}M\setminus\tilde{D}$, where $\mathcal{B}l_{C}M$ is
the blow up of $M$ along $C$ and $\tilde{D}$ is the proper transformation of
$D$.

######  Lemma 2.6 (Lemma 8.7, [83]).

Let $M$ be an affine variety of $\text{dim}_{\mathbb{C}}M\geq 3$ with
compactifying projective variety $X$ by $D$. Choose a irreducible divisor $Z$
of $X$ so that $D\cup Z$ is ample. Choose a point $q\in Z\cap M$ smooth in
$Z$. Denote $M^{\vee}:=\text{Kaliman}(M,(Z\cap M),\\{q\\})$ and
$M^{\sqcup}:=M\setminus Z$. Then there exist algebraic Lefschetz fibrations
$p^{\vee}:M^{\vee}\to\mathbb{C}$, $p^{\sqcup}:M^{\sqcup}\to\mathbb{C}$ such
that $p^{\sqcup}$ is a subfibration of $p^{\vee}$ and the singularities of
$p^{\vee}$ are contained in $M^{\sqcup}$.

We apply the same idea of proof of Lemma 8.7 in [83] to a half-point
attachment from the weighted blow-up of a point with weights $(1,1,b)$.

######  Definition 2.7 (Definition 1.1, [63]).

Let $M$ be a normal quasi-projective complex threefold and $X$ be a normal
projective threefold compactifying $M$ with $D:=X\setminus M$. Choose a point
$q\in D$ where $X$ is smooth. Let $f:\tilde{X}\to X$ be the weighted blow-up
at $q$ with weights $(1,1,b)$, where $b\in\mathbb{N}$. Let
$E\cong\mathbb{P}^{2}_{(1,1,b)}$ be the exceptional divisor of $f$ and
$\tilde{D}:=f^{-1}(D)$. $\tilde{M}:=\tilde{X}\setminus\tilde{D}$ is called to
be a half-point attachment to $M$ of $(b,k)$-type if
$\tilde{D}|_{E}=\sum_{j=1}^{k}m_{j}l_{j}$, where $l_{j}$’s are the mutually
distinct rulings on $E$ and $m_{j}\in\mathbb{N}$ with $\sum_{j=1}^{k}m_{j}=b$.
$\tilde{M}\setminus M\cong\mathbb{C}^{(k-1)*}\times\mathbb{C}$.

We use the same idea of Lemma 8.7 in [83] to the following case. Let
$\tilde{M}$ be a half-point attachment to an affine threefold $M$ at a smooth
point on a hypersurface of $(b,1)$-type. We want to show that there exist
algebraic Lefschetz fibration $\tilde{p}:\tilde{M}\to\mathbb{C}$,
$p:M\to\mathbb{C}$ such that $p$ is a subfibration of $\tilde{p}$ and the
singularity of $\tilde{p}$ are contained in $M$. Let $X$ be a projective
variety with ample divisor $D$ with $X\setminus D=M$. Assume that $D$ is
effective and very ample so that we can choose a rational function $f$ on $X$
with $Y:=\overline{f^{-1}(0)}$ satisfying that $D\sim Y+D^{\prime}$ for an
ample divisor $D^{\prime}$ and $Y\cap(X\setminus D^{\prime})$ is reduced and
irreducible. Define $M^{\prime}:=X\setminus D^{\prime}$ an affine variety
containing $M$. Choose a smooth point $q\in Y\setminus D^{\prime}$.

Let $\pi:\tilde{X}\to X$ be a weighted blow-up at $q$ with weights $(1,1,b)$.
$\tilde{D}$ is a strict transformation of $D$ under $\pi$. Let $s$ be a
section of a line bundle $\mathcal{O}_{X}(D)$ with $s^{-1}(0)=Y+D^{\prime}$.
Choose another section $t$ of $\mathcal{O}_{X}(bD)$ satisfying the following,

  1. (1)

$p:=\frac{t}{s^{b}}:M\to\mathbb{C}$ is an algebraic Lefschetz fibration on
$M=M^{\prime}\setminus Y$, and

  2. (2)

$t^{-1}(0)$ is a smooth subvariety passing through $q$ and transverse to $Y$
at the point $q$.

Let $E$ be the exceptional divisor $\pi^{-1}(q)$. Then
$(\pi^{*}s)^{-1}(0)=E+D^{\prime\prime}$ for some divisor $D^{\prime\prime}$.
Choose an effective divisor $D^{\vee}$ on $\tilde{X}$ compactifying
$\tilde{M}:=\pi^{-1}(M^{\prime})\setminus\tilde{Y}$ such that $D^{\vee}-E$ is
ample. We can choose a meromorphic section $h$ of
$\mathcal{O}_{\tilde{X}}(bD^{\vee}-bE)$ such that $h^{-1}(0)\subset D^{\vee}$,
$\text{ord}_{E}(h)=-b$ and such that $h$ is holomorphic away from support of
$D^{\vee}$, $E$. Then $\pi^{*}s^{b}\otimes h\in
H^{0}(\mathcal{O}(bD^{\vee}-bE+b\pi^{*}D))$ is non-zero away from $D^{\vee}$.
Therefore, the map $\tilde{p}:=\frac{(\pi^{*}t)\otimes h}{(\pi^{*}s)\otimes
h}$ defines an algebraic Lefschetz fibration on $\tilde{M}$ and the
restriction of $\tilde{p}$ on $M$ is $p$. Choose a holomorphic coordinate
$(z_{1},z_{2},z_{3})$ around $q$ and a holomorphic trivialization of
$\mathcal{O}_{X}(Z+D^{\prime})$ so that $s(z_{1},z_{2},z_{3})=z_{1}^{b}$ and
$t(z_{2},z_{2},z_{3})=z_{2}^{b}$. Locally on $E$,
$\\{((z_{1},z_{2},z_{3}),[Z_{1}:Z_{2}:Z_{3}]_{(1,1,b)})\in\mathbb{C}^{3}\times\mathbb{P}^{2}_{(1,1,b)}:Z_{1}\cdot
z_{2}=Z_{2}\cdot z_{1},Z_{2}^{b}\cdot z_{3}=Z_{3}\cdot
z_{2}^{b},Z_{1}^{b}\cdot z_{3}=Z_{3}\cdot z_{1}^{b}\\}$. To show that all the
critical points of $\tilde{p}$ are in $M$, locally on a chart $(Z_{1}=1)$, we
can choose a trivialization of $\mathcal{O}_{\tilde{X}}(\tilde{D}-E)$ so that
$\pi^{*}s^{b}=z_{1}^{b}$, $\pi^{*}t=Z_{2}^{b}\cdot z_{1}^{b}$, and
$h=\frac{1}{z_{1}^{b}}$ from $\text{ord}_{E}(h)=-b$. Hence,
$\tilde{p}=Z_{2}^{b}$ has no singular points near $E\setminus\tilde{Y}$ where
$Z_{2}\neq 0$.

######  Remark 2.8.

It seems interesting to understand the Lefschetz fibration structure of a
half-point attachment of $(b,k)$-type in general. However, the author does not
know about it.

####  2.1.3. Mapping Cylinder of Milnor Monodromy Map of Isolated
Singularities

Let $f:\mathbb{C}^{n+1}\to\mathbb{C}$ be a polynomial with an isolated
singular point at $0$. Let $S^{2n+1}_{\epsilon}\subset\mathbb{C}^{n+1}$ be the
$(2n+1)$-sphere of radius $\epsilon$ centered at $0$ with the standard contact
structure $\eta:=TS^{2n+1}_{\epsilon}\cap J_{std}TS^{2n+1}_{\epsilon}$. The
link of $f$ at $0$ is a contact submanifold $L_{f}:=f^{-1}(0)\cap
S^{2n+1}_{\epsilon}\subset(S^{2n+1}_{\epsilon},\eta)$ where $\epsilon>0$ is
sufficiently small. Let
$\mathcal{N}_{S^{2n+1}}L_{f}:=(TS^{2n+1}_{\epsilon}|_{L_{f}}/TL_{f})$ be the
normal bundle of $L_{f}$ with the pushforwarded symplectic form from
$T^{\perp}:=\\{v\in\eta|_{x}:x\in
L_{f},d\alpha_{S^{2n+1}_{\epsilon}}(v,w)=0,\forall
w\in\eta_{S^{2n+1}_{\epsilon}}|_{x}\cap TL_{f}|_{x}\\}$ with a trivialization
$\Phi_{f}:=(id_{L_{f}},\overline{df}):\mathcal{N}_{S^{2n+1}}L_{f}\to
L_{f}\times\mathbb{C}$. Given an isolated singularity
$(f:\mathbb{C}^{n+1}\to\mathbb{C},0)$, we can assoicate a contact open book,
called the Milnor open book of $f$, and a Liouville domain, called the mapping
cylinder of the Milnor monodromy map on the Milnor fibration.

Let $arg(f):\mathbb{C}^{n+1}\setminus f^{-1}(0)\to\mathbb{R}/2\pi\mathbb{Z}$
be the argument of $f$. The Milnor fibration associated to an isolated
singularity $(f,0)$ is a smooth fibration,

$arg(f):S^{2n+1}_{\epsilon}\setminus
f^{-1}(0)\to\mathbb{R}/2\pi\mathbb{Z},arg(f)(z):=arg(f(z)),$

with a fiber $M_{f}:=arg(f)^{-1}(0)$, called Milnor fiber. There is a
compactly supported diffeomorphism, called the Milnor monodromy map,
$\phi:M_{f}\to M_{f}$. Then a contact pair with normal bundle data
$(L_{f}\subset S^{2n+1}_{\epsilon},\eta,\Phi_{f})$ is supported an open book
$(S^{2n+1}_{\epsilon},arg(f)|_{S^{2n+1}_{\epsilon}})$. An abstract contact
open book is a triple $(M,\theta_{M},\phi)$ where $(M,\theta_{M})$ is a
Liouville domain and $\phi:M\to M$ is an exact symplectomorphism with support
in the interior of $M$. Given an abstract contact open book
$(M,\theta_{M},\phi)$, we can construct contact structure $\alpha_{T_{\phi}}$
on the mapping torus $\pi_{T_{\phi}}:T_{\phi}:=M\times[0,1]/\sim$ of
$\phi:M\to M$ by

  1. (1)

$(x,1)\sim(\phi(x),0)$,

  2. (2)

$\pi_{T_{\phi}}(x,t):=t$ for all $(x,t)\in T_{\phi}$, and

  3. (3)

$\alpha_{T_{\phi}}:=\theta_{M}+d(\rho(t)F_{\phi})+Cdt$, where
$F_{\phi}:M\to\mathbb{R}$ is a smooth function with support in the interior of
$M$ with $\phi^{*}\theta_{M}=\theta_{M}+dF_{\theta}$, $\rho:[0,1]\to[0,1]$ is
a smooth function equal to $0$ near $0$ and $1$ near $1$, $C>0$ is large
enough so that $\alpha_{T_{\phi}}$ is a contact form.

The above construction gives a one-to-one correspondence between graded
abstract contact open books up to isotopy and graded open books up to isotopy
[[48], [19], Theorem 3.15, [88]].

Let $(M,\theta_{M},\phi)$ be an abstract contact open book. Let $\check{\phi}$
be a small positive slope perturbation of $\phi$. The mapping cylinder of
$\phi$ (Definition B.1, [88]) is a triple
$(\mathbb{C}^{*}\rtimes_{\phi}M,\pi_{\check{\phi}},\theta_{\check{\phi}})$
where

  1. (1)

$\mathbb{C}^{*}\rtimes_{\phi}M:=(\mathbb{R}\times\mathbb{R}\times
M)/\mathbb{Z}$, where $(s,t,x)\sim(s,t-1,\check{\phi}(x))$,

  2. (2)

$\check{\phi}:\mathbb{C}^{*}\rtimes_{\phi}M\to\mathbb{R}\times(\mathbb{R}/\mathbb{Z})\cong\mathbb{C}^{*}$
defined by $\check{\phi}(s,t,x)=(s,t)$, and

  3. (3)

$\theta_{\check{\phi}}:=sdt+\kappa\theta_{M}+\kappa
d(\rho(t)F_{\check{\phi}})$ where

     * •

$F_{\check{\phi}}:M\to\mathbb{R}$ is a smooth function with support in the
interior of $M$ satisfying
$\check{\phi}^{*}\theta_{M}=\theta_{M}+dF_{\check{\phi}}$,

     * •

$\rho:[0,1]\to[0,1]$ is a smooth function $\rho(0)=0,\rho(1)=1$,

     * •

$\kappa>0$ is a constant small enough so that $d\theta_{\check{\phi}}$ is
symplectic.

####  2.1.4. Legendrian Front Projection and Flexible Weinstein Cobordism

In this subsection, we review definitions and theorems of Legendrian front
projection and flexible Weinstein cobordisms, [138], [24], [95]. During
fundamental lectures in 1966-1968 [Section 46, Appendix 16, [8]], Arnol’d
emphasized geometric optics as the fundamental notions of Hamiltonian
mechanics so that one keep in mind when working on symplectic/contact
structure. He provided one of examples of Lagrangian submanifolds is the
manifold of all oriented normals to a smooth submanifolds in $\mathbb{R}^{n}$
is a Lagrangian submanifold of the space of lines in $\mathbb{R}^{n}$ passing
through the origin, which is the cotangent bundle of the $(n-1)$-sphere. The
fibration from the space of lines in $\mathbb{R}^{n}$ to the unit sphere of
directions is a Lagrangian fibration. A projection of a Lagrangian submanifold
to the base of a Lagrangian fibration is called a ”Lagrangian mapping”, for
example, mapping each point of a transversely oriented hypersurface in
$\mathbb{R}^{n}$ to the unit vector at the origin in the direction of the
normal. A caustic of a Lagrangian mapping is defined as the set of critical
values of the mapping. The caustic of a Lagrangian mapping is the image of the
cuspidal edge of the front of a Legendrian mapping under a projection from the
space of $1$-jets onto the phase space.

Murphy introduced loose Legendrians, roughly speaking, smooth Legendrians look
like unfolding of wrinkled Legendrian(zig-zag, Figure 2 in [95]) along some
markings which let us wiggle them to get the h-principle [Figure 3, [95]].
Define a (front-projected) plane curve $\psi:\mathbb{R}\to\mathbb{R}^{2}$ and
its rescaling by

  * •

$\psi(u)=(\psi^{x}(u),\psi^{z}(u))=(u^{3}-u,\frac{9}{4}u^{5}-\frac{5}{2}u^{3}+\frac{5}{4}u).$

  * •

$\psi_{\delta}(u)=(\delta^{\frac{3}{2}}\psi^{x}(u/\sqrt{\delta}),\delta^{\frac{5}{2}}\psi^{z}(u/\sqrt{\delta}))=(u^{3}-\delta
u,\frac{9}{4}u^{5}-\frac{5\delta}{2}u^{3}+\frac{5\delta^{2}}{4}u).$

Define $\Psi:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ by
$\Psi(x_{1},\cdots,x_{n-1},u)=(x_{1},\cdots,x_{n-1},\psi_{x_{n-1}}(u)).$ The
singular set is $\\{x_{n-1}=3u^{2}\\}$. For $x_{n-1}>0$, the singularity is
cusp $\times$ $\mathbb{R}^{n}$ and, for $x_{n-1}=0$, singularity is called an
unfurled swallowtail.

######  Definition 2.9 (p. 7, [95]).

  1. (1)

A wrinkle is a map $w:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ defined by
$w(\boldsymbol{x},u):=(\boldsymbol{x},\psi_{1-|\boldsymbol{x}|^{2}}(u))$, for
$\boldsymbol{x}\in\mathbb{R}^{n-1}$.

  2. (2)

A wrinkled embedding of $V^{n}$ to $W^{n+1}$ is a topological embedding
$f:V^{n}\to W^{n+1}$ that is smooth away from some finite collection of
codimension 1 spheres $S^{n-1}_{j}\subset V^{n}$ near which the map is a
wrinkle.

  3. (3)

A wrinkled Legendrian embedding is a topological embedding
$f:\Lambda\to(N^{2n+1},\xi)$ satisfying the following,

    1. (a)

$\text{Image}(df)\subset\xi$.

    2. (b)

$df$ is full rank outside a subset of codimension $2$.

    3. (c)

The singular set is diffeomorphic to a disjoint union of Legendrian wrinkles
$\\{S^{n-1}_{j}\\}$, near which the front projection of the image of $f$ is a
wrinkled embedding.

Let $C\subset\mathbb{R}^{3}$ be the cube of side length $1$ and
$\Lambda_{0}\subset C$ be a properly embedded Legendrian arc with zig-zag
front which is $\\{y=z=0\\}$ near the boundary. Let
$Z_{\rho}:=(-\rho,\rho)\subset\mathbb{R}$ and
$V_{\rho}:=\\{(q,p):|q|<\rho,|p|<\rho\\}\subset T^{*}\mathbb{R}^{n-1}$. Then
$\Lambda_{0}\times Z_{\rho}$ is a Legendrian submanifold of $C\times
V_{\rho}$, called the stabilization of the Legendrian $\\{y=z=0\\}\times
Z_{\rho}\subset(C\times V_{\rho})$ along $Z_{\rho}$ [Section 4.3, [30],
Section 7.4, [24]].

######  Definition 2.10 (Section 4.2, [95]).

A Legendrian submanifold $\Lambda$ of a contact manifold $(N^{2n+1},\eta)$ is
called loose if for each connected component of $\Lambda$, there is an open
subset $U\subset N$ satisfying $(\Lambda\cap U,U)$ is contactomorphic to
$(\Lambda_{0}\times Z_{\rho},C\times V_{\rho})$.

######  Definition 2.11 (Definition 11.29, [24]).

An elementary $2n$-dimensional Weinstein cobordism $(W,\omega,X,\phi)$ is
called flexible if the attaching spheres of all index $n$ handles form a loose
Legendrian link in $\partial_{-}W$. A Weinstein cobordism or manifold is
called flexible if it can be decomposed into elementary Weinstein cobordisms.

######  Example 2.12.

  1. (1)

A subcritical handle attaching is a flexible Weinstein cobordism [Remark
11.30, [24]].

  2. (2)

Let $(W,\omega)$ be a convex symplectic manifold of dimension $2n\geq 4$ of
Morse type at most $n+1$. Then its 1-stabilization
$(W,\omega)\times(\mathbb{R}^{2},\omega_{st})$ is flexible Weinstein. [Theorem
1.1, [37]]

Casals-Murphy developed a recipe to construct a Legendrian front projection of
Weinstein Lefschetz (bi)fibration of affine varieties: given Weinstein
Lefschetz fibration structure with fiber $(F,\lambda,\phi)$ is a plumbing of
vanishing spheres along tree, one apply sequences of Seidel’s half-Dehn twist
on matching paths, draw a diagram of Legendrian front projection of the
Legendrian lifts of the vanishing cycles to the contact manifold
$\partial(F\times D^{2},\lambda+\lambda_{standard})$, and simplify the diagram
by Reidemeister moves and Legendrian handleslides [Recipe 3.3, [18]]. Using
the recipe, if the Legendrian front diagram contains a zig-zag, then the
Legendrian submanifold is loose and the Weinstein manifold is flexible. For
example, they explicitly apply the recipe to the Koras-Russell cubic
$\mathfrak{C}:=\\{(x,y,z,w)\in\mathbb{C}^{4}:x+x^{2}y+w^{3}+z^{2}=0\\}$, which
is known to be diffeomorphic to $\mathbb{C}^{3}$ but not algebraically
isomorphic to it. They proved that indeed $\mathfrak{C}$ is Weinstein-
equivalent to $\mathbb{C}^{3}$ by the h-principle of the loose Legendrian.

Acu-(Capovilla-Searle)-Gadbled-Marinković-Murphy-Starkston-Wu launched an
algorithm for constructing Weinstein handlebodies for complements of some
toric divisors in toric surfaces from their moment polytope data. We refer [5]
and the references therein.

####  2.1.5. Symplectic structure of Weighted Blow-up of Symplectic Orbifolds

Symplectic structure of blow-up of $\mathbb{C}^{n}$ at the origin is
constructed as removing the ball $\\{z\in\mathbb{C}^{n}:||z||^{2}<\epsilon\\}$
and collapsing the fibers of the Hopf fibration of
$\\{z\in\mathbb{C}^{n}:||z||^{2}=\epsilon\\}$ by McDuff in [80]. The
symplectic cut is a construction of a symplectic structure on the reduced
spaces of symplectic manifolds with Hamiltonian circle action, which is a
generalization of the symplectic blow-up, by Lerman [74]. Godinho explained
symplectic structure of weighted blow-up [49], [50].

###  2.2. Symplectic Cohomology

([115], [85], [90]) Hamiltonian Floer cohomology of a symplectic manifold $M$
with a Hamiltonian function on it is a Morse cohomology on the space of loops
on $M$. The corresponding Morse function is the action functional on the loop
space, which is defined by integration of symplectic form over a surface
bounding the loop and evaluation of the Hamiltonian function on the loop. The
critical loops are the perioid orbits for Hamiltonian vector fields and the
gradient flow trajectories are the Floer cylinders which satisfy elliptic PDE,
called the Floer equation, connecting two critical loops. Symplectic
cohomology is the direct limit of the Hamiltonian Floer cohomology under the
system of admissible Hamiltonian functions on it by the Viterbo transfer maps.
Exactness of Liouville domain may not be preserved under the symplectic blow-
ups or resolution. For non-exact convex symplectic manifolds with weak
monotonicity assumptions, the action functional is multivalued but the
differential map is well-defined and the maximum principle still holds,
therefore symplectic cohomology is well-defined.

####  2.2.1. Symplectic Cohomology of Liouville Domains

Let $M$ be a Liouville domain, $\widehat{M}$ be its completion. A family of
Hamiltonians $H:S^{1}\times\widehat{M}\to\mathbb{R}$ is called to be
admissible if $H(t,x)=\lambda r_{M}(x)$ near infinity, where $r_{M}$ is the
cylindrical coordinate of $\widehat{M}$ and $\lambda$ is some positive
constant. We choose positive constant $\lambda$ away from the set of lengths
of Reeb orbits of $\partial M$, $\theta_{M}$ (The Reeb vector field $R$ on
$\partial M$ is defined by $\iota_{R}d\theta_{M}=0$ and $\theta_{M}(R)=1$).
Let $J_{t}$ be an $S^{1}$ family $\omega$-compatible almost complex structure
on $\widehat{M}$, (i.e., family of metrics, $g_{t}(u,v)=\omega(u,J_{t}v)$),
which is convex (or, of contact type on the collar;
$J_{t}^{*}\theta=e^{r}dr\iff J_{t}\partial_{r}=R$). For each $t\in S^{1}$, we
define the Hamiltonian vector field $X_{H_{t}}$ by
$\omega(X_{H_{t}},\cdot)=-dH_{t}$, where $\omega$ is the symplectic form on
$M$. A $1$-periodic Hamiltonian orbit is a map $x:S^{1}\to\widehat{M}$ with
$\dot{x}(t)=X_{H_{t}}(x(t))$. We say a $1$-periodic Hamiltomian orbit $x$ is
non-degenerate if the linearized return map of $x$ has no eigenvalue equal to
1. By [Lemma $2.2$, [85]], near small neighborhood $U$ of $1$-periodic
Hamiltonian orbits, we can perturb $H$ by a $C^{\infty}$ small to $\tilde{H}$
so that all the $1$-periodic Hamiltonian orbits are non-degenerate and
$\tilde{H}=H$ outside $U$, (i.e., $\tilde{H}$ is still admissible). Choose a
trivialization of the canonical bundle of $M$ and then get a canonical
trivialization of $TM|_{x}$. Under the trivialization along the orbit, the
linearlization of Hamiltonian flow become a smooth path of symplectic
matrices. We embed such a path of symplectic matrices to a path of Lagrangians
in the product symplectic vector space. We can define the Conley-Zehnder index
of the $1$-periodic Hamiltonian orbit $x$, denote $\mu_{CZ}(x)$, by the Maslov
index (roughly speaking, the winding number) of the path of Lagrangians with
respect to the diagonal Lagrangian in the product symplectic vector space
[113]. Then we grade the Floer cohomology by the index
$\mu(x)=\text{dim}_{\mathbb{R}}M-\mu_{CZ}(x)$. The choice of index is based on
the fact that it agrees with the Morse index of the critical points $x$ when
$H$ is a $C^{2}$-small Morse Hamiltonian and that the degree of the identity
element and of pants product is zero, therefore $SH^{*}(M)$ becomes a graded
ring. For a $1$-periodic orbit, we define the action functional by,

$\mathcal{A}_{H}(x):=-\int_{S^{1}}x^{*}\theta+\int_{0}^{1}H(x(t))dt=-\int_{\Sigma^{2}}u^{*}d\theta++\int_{0}^{1}H(x(t))dt.$

Let $\mathcal{L}\widehat{M}=C^{\infty}(S^{1},\widehat{M})$ be the space of
free loops in $\widehat{M}$. The differential of $\mathcal{A}_{H}$ at
$x\in\mathcal{L}\widehat{M}$ in the direction $\eta\in
T_{x}\mathcal{L}\widehat{M}=C^{\infty}(S^{1},x^{*}T\widehat{M})$ is,

$d\mathcal{A}_{H}\cdot\eta=-\int_{0}^{1}\omega(\eta,\dot{x}-X_{H}).$

So, the critical points of the action functional $\mathcal{A}_{H}$ are the
$1$-periodic Hamiltonian orbits. By,

$\int_{0}^{1}g_{t}(\eta,(\nabla\mathcal{A}_{H})_{x})=(d\mathcal{A})_{x}(\eta)=-\int_{0}^{1}\omega(\eta,\dot{x}-X_{H_{t}}),$

The gradient vector of $\mathcal{A}_{H}$ with respect to $L^{2}$-metric,
$\int_{0}^{1}g_{t}(\cdot,\cdot)dt$, is
$(\nabla^{g_{t}}\mathcal{A}_{H})_{x}=J_{t}(\dot{x}-X_{H})$. The negative
gradient flow is the map $u:\mathbb{R}\times S^{1}\to M$ satisfying,

$\partial_{s}u=-\nabla^{g_{t}}\mathcal{A}_{H_{t}}(u)\iff\partial_{s}u+J_{t}(\partial_{t}u-X_{H_{t}})=0,\text{
for }(s,t)\in\mathbb{R}\times S^{1}.$

The action functional $\mathcal{A}_{H_{t}}(u(s,t))$ decreases in $s$ along the
negative gradient flow, called Floer cylinder, since
$\partial_{s}(\mathcal{A}_{H_{t}}(u(s,t)))=d\mathcal{A}_{H_{t}}\cdot\partial_{s}u=-\int_{0}^{1}\omega(\partial_{s}u,\partial_{t}u-X_{H_{t}})dt=-\int_{0}^{1}|\partial_{s}u|_{g_{t}}^{2}dt<0.$

Let us define Hamiltonian Floer cohomology as a Morse cohomology. Choose a
coefficient field $\mathbb{K}$. Define the Floer chain complex for a pair
$(H,J)$ of an admissible Hamiltonian $H\in C^{\infty}(\widehat{M},\mathbb{R})$
and convex almost complex structure $J$, 111 • An almost complex structure $J$
on a Liouville domain $(M,\theta_{M})$ is called convex if there is a function
$f:M\to\mathbb{R}$ such that $\partial M$ is a regular level set of $f$, $f$
has its maximum on $\partial M$ and $\theta_{M}\circ J=df$. Every Liouville
domain $(\underline{M},\theta_{\underline{M}})$ has a convex
$d\theta_{\underline{M}}$-compatible almost complex structure [Lemma $2.1$,
[86]]. • An almost complex structure on an symplectic manifold $(M,\omega)$,
$J:TM\to TM$ with $J^{2}=-id$, is called $\omega$-tame if $\omega(v,Jv)>0$ for
$v\in TM$, $\omega$-compatible if $\omega(J\cdot,J\cdot)=\omega(\cdot,\cdot)$
and $\omega(v,Jv)>0$. Both the space of $\omega$-tamed almost complex
structures and the space of $\omega$-compatible almost complex structures are
contractible. But the tameness is an open condition.

$CF_{d<}^{k}(H,J):=\bigoplus_{x\in Crit(\mathcal{A}_{H})}\mathbb{K}\cdot
x=\bigoplus\\{\mathbb{K}x:x\in\mathcal{L}\widehat{M},\dot{x}(t)=X_{H}(x(t)),\text{nondeg},\mathcal{A}_{H}(x)>d,\mu(x)=k\\}.$

Define the differential map $\partial:CF_{d<}^{k}(H,J)\to CF_{d<}^{k+1}(H,J)$
by counting the negative gradient flow of $\mathcal{A}_{H}$, i.e., the Floer
cylinders connecting two Hamiltonian periodic orbits. Denote
$\mathcal{M}_{\mathbb{R}_{\text{param}}}(x_{-},x_{+}):=\\{u:\mathbb{R}\times
S^{1}\to
M:\partial_{s}u+J_{t}(\partial_{t}u-X_{H_{t}})=0,\lim_{s\to\pm\infty}u(s,t)=x_{\pm}(t)\\}$,
$\mathcal{M}(x_{-},x_{+}):=\mathcal{M}_{\mathbb{R}_{\text{param}}}(x_{-},x_{+})/\mathbb{R}$.

The energy of a Floer cylinder
$u\in\mathcal{M}_{\mathbb{R}_{\text{param}}}(x_{-},x_{+})$ is defined by

$E(u):=\int|\partial_{s}u|^{2}dsdt=\int\omega(\partial_{s}u,\partial_{t}u-X_{H})dsdt=-\int\partial(\mathcal{A}_{H}(u))ds=\mathcal{A}_{H}(x_{-})-\mathcal{A}_{H}(x_{+}).$

On the collar near the boundary of $M$, $H=h(e^{r})$,
$X_{H}=h^{\prime}(e^{r})R$ and $A_{H}(x)=-e^{r}h^{\prime}(e^{r})+h(e^{r})$. By
applying the maximum principle on the function $e^{r}\circ u$ on the collar,
if the limiting Hamiltonian periodic orbits $x_{\pm}$ sit inside a compact
subset $M\cup_{\partial M}([0,R]\times\partial M)$, then all the Floer
trajectoris connecting them sit in there. Therefore,
$\mathcal{M}(x_{-},x_{+})$ can be compactified to
$\overline{\mathcal{M}(x_{-},x_{+})}$ by the broken Floer cylinders. After
perturbing $(H_{t},J_{t})$, we get $\overline{\mathcal{M}(x_{-},x_{+})}$ a
smooth manifold with dimension $\mu(x_{-})-\mu(x_{+})$. If
$\mu(x_{-})-\mu(x_{+})=1$, $\overline{\mathcal{M}(x_{-},x_{+})}$ is a compact
zero dimensional manifold, so we can count the Floer cylinders, up to
orientation.

$\partial:CF_{d<}^{k}(H,J)\to CF_{d<}^{k+1}(H,J)\text{ by
}\partial(x_{-}):=\sum_{\mu(x_{-})-\mu({x_{+})=1}}\sharp\overline{\mathcal{M}(x_{-},x_{+})}x_{+}.$

$\partial^{2}=0$ is followed by considering compactifying broken trajectories
in ${\mathcal{M}(x_{-},x_{+})}$ for $\mu(x_{-})-\mu(x_{+})=2$.

######  Remark 2.13.

$CF_{d<}^{k}(H,J)$ is independent of the choice of $J$ but its boundary
operator does. The cohomology $HF_{d<}^{k}(H,J)$ depends on the choice of $H$
but not on $J$ up to canonical isomorphism.

Let us explain the action filtration. Increasing $d$ gives a subcomplex
because the action functional decreases along Floer trajectories and define
the quotient complex,

$CF_{(c,d]}^{k}(H,J):=CF_{c<}^{k}(H,J)/CF_{d<}^{k}(H,J)$

For $a<a^{\prime}<b$, $CF_{(a^{\prime},b]}^{k}(H,J)\hookrightarrow
CF_{(a,b]}^{k}(H,J)$ and, for $a<b^{\prime}<b$, $CF_{(a,b]}^{k}(H,J)\to
CF_{(a,b^{\prime}]}^{k}(H,J)$. Therefore, given $-\infty\leq a\leq b\leq
c<\infty$, there is a short exact sequence,

$0\to CF^{k}_{(b,c]}(H,J)\to CF^{k}_{(a,c]}(H,J)\to CF^{k}_{(a,b]}(H,J)\to 0,$
$0\to CF^{k}_{(-\epsilon,\epsilon]}(H,J)\to
CF^{k}_{(-\infty,\epsilon]}(H,J)\to CF^{k}_{(-\infty,-\epsilon]}(H,J)\to 0.$

Define $HF_{(c,d]}^{k}(H,J)$ as the cohomology of the chain complex
$CF_{(c,d]}^{k}(H,J)$.

Let us relate two Hamiltonian Floer cohomologies of $(M,H_{1})$ and
$(M,H_{2})$ with two non-degenerate admissible Hamiltonians $H_{1}<H_{2}$.
Define the Viterbo’s continuatin map,

$C:CF_{d<}^{k}(H_{1},J_{1})\to CF_{d<}^{k}(H_{d},J_{2})\text{ by
}\partial(x_{-}):=\sum_{\mu(x_{-})-\mu({x_{+})=0}}\sharp\mathcal{P}(x_{-},x_{+})x_{+},$

where $\mathcal{P}(x_{-},x_{+})$ is the set of solutions to the parametrized
Floer equations connecting $H_{1}$ and $H_{2}$ in the following way: Let
$K_{s}$ be a smooth non-degenerating family of admissible Hamiltonians equal
to $H_{1}$ for $s\ll 0$ and $H_{2}\gg 0$ and $J_{s}$ a smooth family of
admissible almost complex structures joining $J_{1}$ and $J_{2}$. Define

$\mathcal{P}(x_{-},x_{+}):=\\{u:\partial_{s}u+J_{s,t}(u(s,t))\partial_{t}u=\nabla^{g_{t}}K_{s,t},\lim_{s\to\pm\infty}u(s,\cdot)=x_{\pm}\\}.$

The action functional $\mathcal{A}_{H_{s}}(u(s,\cdot))$ along
$u\in\mathcal{P}(x_{-},x_{+})$,

$\partial_{s}(\mathcal{A}_{H_{s}}(u(s,\cdot))):=-\int_{0}^{1}|\partial_{s}u|^{2}dt+\int_{0}^{1}(\partial_{s}H_{s})(u)dt,$

The energy is

$E(u)=\int|\partial_{s}u|^{2}ds\wedge
dt=\mathcal{A}_{H_{1}}(x_{-})-\mathcal{A}_{H_{+}}(x_{+})+\int(\partial_{s}H_{s})(u)ds\wedge
dt.$

The action functional decreases if $\partial_{s}H_{s}\leq 0$ and the energy is
bounded if $\partial_{s}H_{s}\leq 0$ outside of a compact set in
$\widehat{M}$. Therefore, by the maximum principle, we can compactify
$\mathcal{P}(x_{-},x_{+})$.

We define the symplectic cohomology of $\widehat{M}$ by the direct limit of
$HF_{d<}^{k}(H_{I},J_{I})$ under the Viterbo’s continuation maps on admissible
Hamiltonians $(H_{I},<)$ such that $H|_{\text{interior }M}<0$ and the slopes
at cylindrical ends go to $\infty$ as the cylindrical coordinate $r$ goes to
$\infty$.

$SH^{*}(M):=\lim_{\longrightarrow}HF^{*}(H).$

The induced long exact sequence commutes with the Viterbo’s continuation maps,
so, after taking direct limits, we get, for small $\epsilon>0$,

${\cdots\to
SH^{*-1}_{(-\epsilon,\infty)}(M)}$${SH^{*-1}_{(\epsilon,\infty)}(M)}$${SH^{*}_{(-\epsilon,\epsilon]}(M)}$${SH^{*}_{(-\epsilon,\infty)}(M)\to\cdots}$${H^{*}(M)}$$\scriptstyle{\delta}$$\scriptstyle{\cong}$

Let us denote it by,

${\cdots\to
SH^{*-1}(M)}$${SH^{*-1}_{+}(M)}$${H^{*}(M)}$${SH^{*}(M)\to\cdots.}$$\scriptstyle{\delta}$$\scriptstyle{\iota}$

where, $\delta:SH^{*}_{+}(M)\to H^{*}(M)$ is the connecting map.

####  2.2.2. Symplectic Cohomology of Symplectic Convex Lefschetz Fibrations

Inspired by Oancea’s Künneth formula for symplectic cohomology [105], McLean
defined a Lefschetz-admissible Hamiltonian on a symplectic convex Lefschetz
fibration $M$ [Definition 2.21, [83]] and showed that Lefschetz-admissible
symplectic cohomology is isomorphic to the (original) symplectic cohomology
[Theorem 2.2, 2.4, [83]] and also isomorphic to Monodromy-Floer cohomology of
the fiber [Theorem 1.2, 1.3, [84]]. The chain complexes of Lefschetz-
admissible Hamiltonian Floer cohomology is generated by

  1. (1)

Critical points of Morse function on $M$, and

  2. (2)

Two copies of fixed points of iterates of the monodromy map around a large
circle on the base.

One of main results in [83] is isomorphism of symplectic cohomology of convex
Lefschetz fibration and that of certain subfibration.

######  Lemma 2.14 (Theorem 2.25, [83]).

Let $M^{\prime}$ be a convex symplectic Lefschetz fibration with fiber
$F^{\prime}$. Suppose that $M$ is a subfibration of $M$ with fiber $F$ over
the same base satisfying the following.

  1. (1)

The support of all the monodromy maps of $M^{\prime}$ are contained in the
interior of $M$.

  2. (2)

For any holomorphic curves $u$ in $F^{\prime}$ with boundary $\partial
u\subset F$, $u\subset F$.

Then $SH^{*}_{lef}(M^{\prime})\cong SH^{*}_{lef}(M)$. Therefore,
$SH^{*}(M^{\prime})\cong SH^{*}(M)$.

####  2.2.3. Symplectic Cohomology of Non-Exact Convex Symplectic manifolds

Many Kähler manifolds including the blow-up of $\mathbb{C}P^{1}$ at a point,
and the crepant resolutions of simple isolated singularities, have non-exact
symplectic structure. When open non-exact symplectic manifolds have convex
contact boundary condition, symplectic cohomology still can be defined by
Ritter [115]. Admissible Hamiltonians are radial at infinity and almost-
complex structure are of contact type on the cylindrical ends. Using one-to-
one correspondence between Hamiltonian orbits and Reeb orbits, we have the
maximum principle on the radial coordinate of the image of Floer cylinder so
that the moduli space of Floer cylinders with limiting orbits can be
compactified. Weak-monotonicity condition222$M$ is a $2n$-dimensional
symplectic manifold such that for every $A\in\pi_{2}(M)$, $3-n\leq c_{1}(A)<0$
implies $\omega(A)\leq 0$. Equivalently, one of the following holds  •
$\omega=\lambda\cdot c_{1}(A)$ for every $A\in\pi_{2}(M)$ for some
$\lambda\geq 0$. • $c_{1}(A)=0$ for every $A\in\pi_{2}(M)$. • The minimal
Chern number $N\geq 0$ defined by $c_{1}(\pi_{2}(M))=N\mathbb{Z}$ is greater
than or equal to $n-2$. is needed to get rid of bubbling when we compactify
the moduli spaces of Floer trajectories. Even though the action functional is
multi-valued, the cut-offed radial filtration works and gives long exact
sequence for non-exact convex symplectic manifolds [Theorem 6.2, [90]].

######  Remark 2.15.

Abouzaid-Blumberg constructed Floer homology with coefficients in Morava’s
K-theories [3]. Using it, they proved the Arnold conjecture in characteristic
$p$; the rank of the cohomology of a closed symplectic manifold with
coefficients in a field of characteristic $p$ is bounded above by the number
of non-degenerate Hamiltonian periodic orbits. Extending our symplectic
criteria of uniruledness in the setting of Morava K-theory seems interesting,
see [9] as well. The relation between fixed points of the Frobenius map and
the monodromy Floer cohomology of projective varieties would be also
interesting.

###  2.3. Morse-Bott Symplectic Cohomology

Morse-Bott approach to the Floer theory was developed by Fukaya [44],
Frauenfelder [43], Bourgeois-Oancea [16], and Diogo-Lisi[28]. The idea
relating Floer trajectories to $J$-holomorphic curves followed by Morse
trajectories was developed by Piunikhin-Salamon-Schwarz [111] and Oh-Zhu
[106], [107]. Let us review Bourgeois-Oancea’s correspondence theorem between
symplectic cohomology defined in the previous section and Morse-Bott
symplectic cohomology where the differential maps are defined by counting
cascades.

Let $H:M\to\mathbb{R}$ be a time-independent Hamiltonian on a symplectic
manifold $(M,\omega)$ satisfying that $H|_{\underline{M}}$ is $C^{2}$-small
and, for the cylindrical coordinate $r$ in the cylindrical end of $M$,
$H(\cdot,r)=h(r)=\lambda\cdot e^{r}+b$, for $\lambda,b\in\mathbb{R}$,
$\lambda\notin\text{Spec}(X_{H})$, $t>t_{0}$, otherwise
$h^{\prime\prime}-h^{\prime}>0$. The set of Hamiltonian $1$-periodic orbits
are constant orbits, corresponding to critical points of
$H|_{\text{int}(\underline{M})}$ and non-isolated, Morse-Bott nondegenerate
families $C$ of orbits. We can perturb time-independent Hamiltonian by perfect
Morse functions $f_{C}:C\to\mathbb{R}$ on each family $C$ of orbits so that
$C$ perturbed into nondegenerate orbits corresponding to critial points of
$f_{C}$. Let us construct such a time-dependent Hamiltonian
$H_{\delta}:S^{1}\times M\to\mathbb{R}$. For small $\delta>0$, define
$K_{\delta}:S^{1}\times M\to\mathbb{R}$ by
$K(\theta,\cdot):=H(\cdot)+\delta\cdot\sum_{C}\rho_{C}\cdot f_{C}$, where
$\\{\rho_{C}\\}$ is a smooth cutoff function supported in a small neighborhood
of $C$. Then $K_{\delta}$ is Hamiltonian whose non-constant Morse-Bott
families $G$ of orbits are of dimension 1. Denote $g_{G}:G\to\mathbb{R}$ the
perfect Morse function on $G$ with exactly one maximum and one minimum. Then
we perturb $K_{\delta}$ using $g$ to get $H_{\delta}$. Choose the maximal
positive integer $t_{G}$ so that $G(\theta+1/t_{G})=G(\theta)$ for all
$\theta\in S^{1}$. Choose a symplectic trivialization
$\psi_{G}:=(\psi_{G,1},\psi_{G,2}):\text{nbhd}(G)\to V\subset
S^{1}\times\mathbb{R}^{2n-1}$ of the open neighborhood of $G$ so that
$\psi_{G,1}(G(\theta))=t_{G}\cdot\theta$. Define $H_{\delta}:S^{1}\times
M\to\mathbb{R}$ by
$H_{\delta}(\theta,\cdot):=K+\delta\cdot\sum_{G}\rho_{G}(\psi_{G}(\cdot))\cdot
g_{G}(\psi_{G,1}(\cdot)-t_{G}\cdot\theta)$, where
$\rho_{G}:S^{1}\times\mathbb{R}^{2n-1}\to[0,1]$ is a smooth cutoff function
supported on a small neighborhood of $S^{1}\times\\{0\\}$ inside
$S^{1}\times\mathbb{R}^{2n-1}$ and $\rho|_{S^{1}\times\\{0\\}}\equiv 1$
[Equation (25), [16]]. Let us define some moduli spaces as follows. Let $J$ be
a generic time-dependent almost complex structure on $W$. Given
$p_{-}\in\text{Crit}(f_{C^{-}})$, $p_{+}\in\text{Crit}(f_{C^{+}})$,
$x_{-},x_{+}\in\text{Crit}(H)$, $A\in H_{2}(M;\mathbb{Z})$, define

  * •

$\mathcal{M}^{A}(p_{-},p_{+},;H_{\delta},J)$: the moduli space of Floer
trajectories for the pair $(H_{\delta},J)$ with negative asymptote $p$ and
positive asymptote $q$ having the same homology class with $A$ modulo
reparmetrization in $s$.

  * •

$\mathcal{M}^{A}(x_{-},C_{+},;H,J)$: the moduli space of Floer trajectories
for the pair $(H,J)$ with negative asymptote $x_{-}$ and positive asymptote in
$C_{+}$, having the same homology class with $A$ modulo reparmetrization in
$s$.

  * •

$\mathcal{M}^{A}(C_{-},C_{+};H,J)$: the moduli space of Floer trajectories for
the pair $(H,J)$ with negative asymptote in $C_{-}$ and positive asymptote in
$C_{+}$, having the same homology class with $A$ modulo reparametrization.

  * •

Define evaluation maps $\overline{ev}:\mathcal{M}^{A}(C_{-},C_{+};H,J)\to
C_{-}$ by $\overline{ev}([u]):=\lim_{s\to-\infty}u(s,\cdot)$,
$\underline{ev}:\mathcal{M}^{A}(C_{-},C_{+};H,J)\to C_{+}$ by
$\underline{ev}([u]):=\lim_{s\to\infty}u(s,\cdot)$, and let $\phi_{s}^{f_{C}}$
denote the gradient flow of $f_{C}$.

  * •

$\mathcal{M}^{A}_{m}(p_{-},p_{+};H,\\{f_{x}\\},J)=W^{u}(p_{-})\times_{\overline{ev}}(\mathcal{M}^{A_{1}}(C_{-},C_{1})\times\mathbb{R}^{+})_{\phi_{s}^{f_{C_{1}}}\circ\underline{ev}}\times_{\overline{ev}}(\mathcal{M}^{A_{2}}(C_{1},C_{2})\times\mathbb{R}^{+})\times\cdots_{\phi_{s}^{f_{C_{m-1}}}}\circ\underline{ev}\times_{\overline{ev}}\mathcal{M}^{A_{m}}(C_{m-1},C_{+})_{\underline{ev}}\times
W^{s}(p_{+})$: the moduli space of Floer trajectories for the pair $(H,J)$
with intermediate gradient fragments with $\sum_{i}A_{i}=A$.

  * •

$\mathcal{M}^{A}(p_{-},p_{+};H,\\{f_{x}\\},J):=\cup_{m\geq
0}\mathcal{M}^{A}_{m}(p_{-},p_{+};H,\\{f_{x}\\},J)$.

  * •

$\mathcal{M}^{A}_{]0,\delta_{0}[}(p_{-},q_{+};H,\\{f_{\gamma}\\},J):=\bigcup_{0<\delta<\delta_{0}}\\{\delta\\}\times\mathcal{M}^{A}(p_{-},q_{+};H_{\delta},J)$,
where $q_{+}$ is either $p_{+}$ or $x_{+}$.

  * •

A sequence $[u]\in\mathcal{M}^{A}(x^{-},x^{+};H,J)$ is called to converge to
the broken trajectories $([u_{k}],[u_{k-1}]$, $\cdots$, $[u_{1}])$,
$[u_{i}]\in\mathcal{M}^{A_{i}}(x_{i}^{-},x_{i}^{+};H,J)$, $\sum_{i}A_{i}=A$,
if there exist sequences $s_{i}\in\mathbb{R},1\geq i\geq k$, such that
$s_{i}\cdot u(s,\cdot)=u(s+s_{i},\cdot)$ converges uniformly on compact sets
to $u_{i}$, [16].

######  Lemma 2.16 (Theorem 3.7, [16]).

  1. (1)

Any sequence
$[v_{\delta_{n}}]\in\mathcal{M}(x^{-}_{p},x^{+}_{p};H_{\delta_{n}},J)$,
$\delta_{n}\to 0$, converges to an element $[\boldsymbol{\nu}]$ of
$\overline{\mathcal{M}(p_{-},q_{+};H,\\{f_{x}\\},J)}$.

  2. (2)

Any element of $\overline{\mathcal{M}(p_{-},q_{+};H,\\{f_{x}\\},J)}$ can be
obtained as such a limit in a unique way. Moreover, if
$dim\mathcal{M}(p_{-},q_{+};H,\\{f_{x}\\},J)=0$, then the intermediate
gradient fragments in $[\boldsymbol{\nu}]$ are non-constant.

  3. (3)

There is a bijective correspondence between elements of
$\mathcal{M}(p_{-},p_{+};H_{\delta_{n}},J)$ and elements of
$\overline{\mathcal{M}(p_{-},q_{+};H,\\{f_{x}\\},J)}$ if the moduli spaces
have dimension zero: More precisely quoting,

Let $H\in\mathcal{H}^{\prime}$ be fixed, and let
$\alpha:=\lim_{t\to\infty}e^{-t}H(p,t)$ be the maximal slope of $H$. Let
$J\in\mathcal{J}_{reg}(H)$, and let $\\{f_{x}\\}\in\mathcal{F}_{reg}(H,J)$.
There exists $\delta_{1}:=\delta_{1}(H,J)\in]0,\delta_{0}[$ such that for any
$p_{-}\in\text{Crit}(f_{C_{-}})$, $p_{+}\in\text{Crit}(f_{C_{+}})$, and
$x_{+}\in\text{Crit}(H)$, with $\mu(p_{+})-\mu(p_{-})=1$ or
$\mu(x_{+})-\mu(p_{-})=1$, and any $A\in H_{2}(W;\mathbb{Z})$, the following
hold ($q_{+}$ means $p_{+}$ or $x_{+}$):

    1. (a)

$J$ is regular for $\mathcal{M}^{A}(p_{-},q_{+};H_{\delta},J)$ for all
$\delta\in]0,\delta_{1}[$;

    2. (b)

The moduli space
$\mathcal{M}^{A}_{]0,\delta_{1}[}(p_{-},q_{+};H,\\{f_{x}\\},J)$ is a one-
dimensional manifold having a finite number of components that are graphs over
$]0,\delta_{1}[$;

    3. (c)

There is a bijective correspondence between points in
$\mathcal{M}^{A}(p_{-},q_{+};H,\\{f_{x}\\},J)$ and connected components of
$\mathcal{M}^{A}_{]0,\delta_{1}[}(p_{-},q_{+},H,\\{f_{x}\\},J)$.

For simplicity, assume that $c_{1}(TW)=0$ and refer [16] for definitions in
more general setting. Define the symplectic chain group $CF^{*}(H)$ by the
free $\mathbb{Z}$-module generated by Hamiltonian 1-periodic orbits $x$, with
grading $|x|=\mu_{CZ}$. Define the differential map $\partial:CF^{*}(H)\to
CF^{*+1}(H)$ by

$\partial x^{-}:=\sum_{\mu_{CZ}(x^{+})-\mu_{CZ}(x^{-})=1,}\sum_{A\in
H_{2}(M;\mathbb{Z})}\sharp\mathcal{M}^{A}(x^{-},x^{+};H,J)\cdot x^{+}.$

Since $\partial^{2}=0$, define $HF^{*}(H,J):=H^{*}(CF^{*}(H),\partial)$ and
$SH^{*}(M):=\varinjlim_{H}HF^{*}(H)$. Define the Morse-Bott Hamiltonian Floer
chain groups by

$CB^{*}(H):=\bigoplus\mathbb{Z}<\text{Crit}(f_{C})>\oplus\mathbb{Z}<\text{Crit}_{\text{nondeg}}(H)>.$

The grading is defined using the symplectic shear axiom for the Robbin-Salamon
index [16]. There are several conventions to define symplectic (co)homology
and we compare them in Appendix 6.1 below. Define the Morse-Bott Hamiltonian
Floer differential $\partial:CB^{*}(H)\to CB^{*+1}(H)$ by,

  * •

$\partial
x_{-}:=\sum_{x_{+}\in\text{Crit}(H),|x_{+}|-|x_{-}|=1}\sharp\mathcal{M}^{0}(x_{-},x_{+};H,\\{f_{\gamma}\\},J)\cdot
x_{+}$, for $x_{+}\in\text{Crit}(H)$.

  * •

$\partial
p_{-}:=\sum_{p_{+}\in\text{Crit}(f_{C_{+}}),|p_{+}|-|p_{-}|=1}\sum_{A\in
H_{2}(M;\mathbb{Z})}\sharp\mathcal{M}^{A}(p_{-},p_{+};H,\\{f_{\lambda}\\},J)\cdot
p_{+}$,
$+\sum_{x_{+}\in\text{Crit}(H),|\gamma_{x_{+}}|-|p_{-}|=1}\sum_{A\in
H_{2}(M;\mathbb{Z})}\sharp\mathcal{M}^{A}(p_{-},x_{+};H,\\{f_{\gamma}\\},J)\cdot
x_{+}$, for $p_{-}\in\text{Crit}(f_{C_{-}})$.

where, all the counting is considered with a sign of the corresponding
connected component. By definition, $CF^{*}(H_{\delta})\cong
CB^{*}(H),\delta\in]0,\delta_{1}[$ as free $\mathbb{Z}$-modules. And, by the
correspondence theorem 2.16 [Theorem 3.7, [16]], the differentials counts the
same. Therefore, $H^{*}(BC^{*}(H),\partial)=SH^{*}(H_{\delta},J)$. In this
paper, we take Hamiltonian to be constantly zero on the interior of the
Liouville domain $\underline{M}$, so that the Morse-Bott family of orbits is
codimension $0$, Bourgeois-Oancea’s argument on Morse-Bott moduli spaces still
applies. We note that the part of Floer cylinder which degenerates to Morse
trajectory are on the descending manifold of the critical point under the
Morse flow, the the degenerating locus is under the boundary of $M$.

###  2.4. Twisted/Bulk Deformed Symplectic Cohomology

We can extend the coefficient ring of the symplectic cohomology. Let us recall
the definitions by Ritter [Section 4, [115]]. Let $ev:\mathcal{L}M\times
S^{1}\to\mathcal{L}M$ be the evaluation map. Define $\tau:=\pi\circ
ev^{*}:H^{2}(M;\mathbb{R})\rightarrow H^{2}(\mathcal{L}M\times
S^{1};\mathbb{R})\to H^{1}(\mathcal{L}M;\mathbb{R})$. For $\xi\in
H^{2}(M;\mathbb{R})$, a smooth path $u$ in $\mathcal{L}M$,
$\tau\xi(u)=\int\xi(\partial_{s}u,\partial_{t}u)ds\wedge dt$. Let
$\beta=\tau(\xi)$ be a singular cocycle representing a class in
$H^{1}(\mathcal{L}M;\mathbb{R})$ for $\xi\in H^{2}(M;\mathbb{R})$. The twisted
symplectic chain complex for $(M,H)$ with twisted coefficients in $\Lambda$ is
the $\Lambda$-module generated by the periodic orbits of $X_{H}$, the
differential $\partial$ is twisted by
$t^{-\int_{[u]}\beta}=t^{-\int\eta(\partial_{s}u,\partial_{t}u)ds\wedge dt}$,
where $u$ is a Floer cylinder connecting two generators,

$\displaystyle\partial x_{-}:$
$\displaystyle=\sum_{u\in\mathcal{M}(x_{-},x_{+};H,J)}\epsilon(u)t^{\mathcal{E}_{H}(u)-\int_{[u]}\beta}x_{+}$
$\displaystyle=\sum_{u\in\mathcal{M}(x_{-},x_{+};H,J)}\epsilon(u)t^{\int_{\mathbb{R}\times\mathbb{R}/\mathbb{Z}}u^{*}\omega}t^{\int_{0}^{1}(H(x_{+}(t))-H(x_{-}(t)))dt}\cdot
t^{-\int\eta(\partial_{s}u,\partial_{t}u)ds\wedge dt}x_{+},$

All continuation maps are also twisted by
$t^{-\int_{[u]}\beta}=t^{-\int\eta(\partial_{s}u,\partial_{t}u)ds\wedge dt}$.
Denote $\beta=\tau(\xi)$-twisted symplectic cohomology of $M$ by
$SH^{*}(M;\underline{\Lambda}_{\beta})$.

###  2.5. Computing Symplectic Cohomology

Symplectic cohomology of an affine variety $X\setminus D$, where $X$ is a
projective variety and $D$ is simple normal crossing divisors, can be computed
in several ways: For admissible pairs $(X,D)$, spectral sequences which
converges to symplectic cohomology was developed by Seidel [124], McLean [87]
and Ganatra-Pomerleano [47]. Ritter computed symplectic cohomology of some
non-exact monotone convex symplectic manifold with Hamiltonian circle action
that generate Reeb flow on the boundary using Seidel representation [117],
[118]. Using the Legendrian surgery exact sequence by Bourgeois-Ekholm-
Eliashberg [15], symplectic cohomology of Weinstein manifold, constructed from
Legendrian surgery on a subcritical Weinstein manifold, is isomorphic to
Legendrian contact homology. For any $4$-dimensional Weinstein manifold,
symplectic cohomology can be computed combinatorially by Ekholm-Ng [32].
Mayer-Vietoris Sequence is developed by Cieliebak-Oancea for Liouville
cobordisms and by Varolgunes for compact submanifolds of a closed symplectic
manifold. Symplectic cohomology is also known to be isomorphic to Hochschild
cohomology of wrapped Fukaya category. Various mirror symmetries predict that
(wrapped) Fukaya categories are equivalent to derived category of coherent
sheaves, possibly with superpotentials(the Landau-Ginzburg model). The Gross-
Siebert program explains how to construct the mirrors by tropicalization.
Pascaleff computed symplectic cohomology of log Calabi-Yau surfaces using
tropicalization [109]. Lekili-Ueda computed symplectic cohomlogy of Milnor
fibers of ADE singularities by computing algebraic side of homological mirror
symmetry [75].

######  Example 2.17 ([87], See also Proposition 11.1, [29]).

Let $X$ be a projective variety with simple normal crossing divisor
$D:=\\{D_{j}\\}_{j\in S}$. Choose a meromorphic section $\kappa$ of the
canonical bundle of $X$ which is non-vanishing along $X\setminus D$. Define
the discrepancy $a_{i}$ of $D_{i}$ as
$a_{i}:=\text{ord}_{D_{i}}\kappa^{-1}(0)-\text{ord}_{D_{i}}\kappa^{-1}(\infty)$.
Choose an ample line bundle $L$ on $X$ and a holomorphic section $s$ of $L$
which is non-vanishing along $X\setminus D$. Define a winding number $w_{i}$
of $D_{i}$ as $w_{i}:=-\text{ord}_{D_{i}}s^{-1}(0)$. Choose $ND_{I}$ a small
tubular neighborhood of $D_{I}$ for $I\subset S$ so that
$\overset{\circ}{N}D_{I}:=ND_{I}\setminus\cup_{j\in S}D_{j}$ is a bundle over
$D_{I}\setminus\cup_{j\in S-I}D_{j}$ with fiber of a product of punctured
disks. Using the Morse-Bott spectral sequence, a spectral sequence
$(E_{r}^{p,q},d_{r}^{p,q}:=E_{r}^{p,q}\to E_{r}^{p+r,q-r+1})$ with
$E^{1}$-page,

$E_{1}^{p,q}:=\bigoplus_{(k_{i}\in\mathbb{N}^{S}:\sum_{i}k_{i}w_{i}=-p)}H^{p+q-2(\sum_{i}k_{i}(a_{i}+1))}(\overset{\circ}{N}D_{I_{(k_{i})}}),I_{(k_{i})}:=\\{i\in
S:k_{i}\neq 0\\},$

converges to $SH^{*}(X\setminus D)$.

Let $C_{(1,k)}$ be the complement of bidegree $(1,k)$-hypersufaces $D_{(1,k)}$
in $\mathbb{P}^{1}\times\mathbb{P}^{1}$.
$C_{(1,1)}=\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus
D_{(1,1)}=T^{*}\mathbb{P}^{1}$. $\overset{\circ}{N}D_{(1,1)}=\mathbb{R}P^{3}$,
considering the Hopf fibraion over $\mathbb{P}^{1}$ in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. Using $a=\frac{-2}{1}+\frac{-2}{1}=-4$,
$w=-2$,

$H^{*}(\mathbb{R}P^{3}):=\begin{cases}\mathbb{Z},&\text{for }*=3,\\\
\mathbb{Z}/2,&\text{for }*=2,\\\ 0,&\text{for }*=1,\\\ \mathbb{Z},&\text{for
}*=0,\end{cases}\quad\quad H^{*}(C_{(1,k)})=\begin{cases}\mathbb{Z},&\text{for
}*=2,\\\ \mathbb{Z}/2,&\text{for }*=1,\\\ \mathbb{Z},&\text{for
}*=0,\end{cases}$

we get the following $E^{1}$-page, $E^{\infty}$-page,

${8}$${\cdot}$${\cdot}$${\cdot}$${\mathbb{Z}\quad\quad}$${8}$${\cdot}$${\cdot}$${\cdot}$${\mathbb{Z}}$${7}$${\cdot}$${\cdot}$${\cdot}$${\cdot\quad\quad}$${7}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${6}$${\cdot}$${\cdot}$${\mathbb{Z}}$${\mathbb{Z}\quad\quad}$${6}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${5}$${\cdot}$${\cdot}$${\cdot}$${\mathbb{Z}\quad\quad}$${5}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${4}$${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}}$${\cdot\quad\quad}$${4}$${\cdot}$${\cdot}$${\mathbb{Z}}$${\cdot}$${3}$${\mathbb{Z}}$${\cdot}$${\mathbb{Z}}$${\cdot\quad\quad}$${3}$${\mathbb{Z}}$${\cdot}$${\mathbb{Z}}$${\cdot}$${2}$${\mathbb{Z}/2}$${\mathbb{Z}/2}$${\cdot}$${\cdot\quad\quad}$${2}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${1}$${\cdot}$${\mathbb{Z}}$${\cdot}$${\cdot\quad\quad}$${1}$${\cdot}$${\mathbb{Z}}$${\cdot}$${\cdot}$${0}$${\mathbb{Z}}$${\cdot}$${\cdot}$${\cdot\quad\quad}$${0}$${\mathbb{Z}}$${\cdot}$${\cdot}$${\cdot}$
${0}$${1}$${2}$${3\quad\quad}$${0}$${1}$${2}$${3}$

Therefore, we get $\ell(C_{(1,1)})=2$. However, $C_{(1,k)}$ is
$\mathbb{C}$-uniruled for all $k\geq 1$.

####  2.5.1. Relative Symplectic Cohomology of Compact Subsets of a Completed
Liouville Domain and a Mayer-Vietoris Sequence

We adapt two definitions of symplectic cohomology of a pair $(K\subset X)$ of
a projective variety $X$ and its compact subset $K$ in [Definition 2.64,
Definition 3.3 in [89], [134]] to a pair $(K\subset M)$ of a completion $M$ of
a Liouville domain $\underline{M}$ and a compact subset $K$ of $M$ following
[23], [89], [134]. Varolgunes’ Mayer-Vietoris property on relative symplectic
cohomology still works. There are some aspects that our setting is simpler
because we are given a Liouville domain with contact cylinder(exact symplectic
form) so we are not worried about the index-bounded contact cylinder for a
pair $(K\subset X)$ of a projective variety $X$ and its compact subset $K$.
And the symplectic form is exact compatible with the contact boundary.

Denote $\text{Cube}^{n}:=\\{(x_{1},\cdots,x_{n})|x_{j}\in[0,1]\\}$, the
standard unit cube. For $0\leq k\leq n$, a $k$-dimensional face of
$\text{Cube}^{n}$ is a subset of $\text{Cube}^{n}$ having that $n-k$ of the
coordinates are either $0$ or $1$. We call $0$-dimensional faces, vertices. A
vertex of a face is called the initial(resp. terminal) vertex if it has the
maximum number of $0$’s (resp. $1$’s) in its coordinates and denote
$\nu_{s(F)}$ (resp. $\nu_{t(F)}$). Two faces $F^{\prime}$ and
$F^{\prime\prime}$ are adjacent if
$\nu_{s(F^{\prime\prime})}=\nu_{t(F^{\prime})}$ and denote
$F^{\prime\prime}<F^{\prime}$. Let us summarize some notations as follows.

  * •

The faces of $Cube^{n}$ are in one-to-one correspondence with the set of
$n$-tuples of elements of a set $\\{0,1,-\\}$. Denote $\mu(F)$ the $n$-tuple
corresponding to a face $F$.

  * •

Given two adjacent faces $F^{\prime}>F^{\prime\prime}$, choose the smallest
face $F$ that contains both $F^{\prime}$ and $F^{\prime\prime}$. Denote
$v(F^{\prime},F)$ the subtuple of
$\mu(\nu_{t(F^{\prime})})-\mu(\nu_{s(F^{\prime})})$.

  * •

For tuples $w,v$ of elements of a set $A$, $\\#(w,v)$ is defined to be the
number of subtuples of $v$ that are equal to $w$.

  * •

$n(F):=\\#(0-,\mu(F))+\\#(0,\mu(F))$.

######  Definition 2.18 (Definition 2.1.1,[134]).

An $n$-cube of chain complexes over a commutative ring $R$ consists of
following data,

  1. (1)

To each vertex $\nu$ of $\text{Cube}^{n}$, we assign a $\mathbb{Z}/2$-graded
$R$-module $C^{\nu}$.

  2. (2)

To each $k$-dimensional face $F$, we assign maps $f_{F}:C^{\nu_{s(F)}}\to
C^{\nu_{t(F)}}$ from its initial vertex to its terminal vertex, of degree
$dim(F)+1$ modulo $2$.

  3. (3)

For any boundary $F^{\prime\prime}<F^{\prime}$ of a face $F$,
$\sum_{F\prime\prime<F^{\prime},\text{boundary
}F}(-1)^{*_{F^{\prime},F}}f_{F^{\prime\prime}}f_{F^{\prime}}=0$, where
$*_{F^{\prime},F}=n(1,v(F^{\prime},F))+n(01,v(F^{\prime},F))$.

When the signs $(-1)^{*_{F^{\prime},F}}$ are all positive, we call it
$n$-cubes with positive signs. There is a canonical way to make an $n$-cube
with positive signs [Lemma 2.1.5,[134]]. The cone of an $n$-cube with positive
signs $(\\{\mathcal{C}^{\nu}\\},\\{f_{F}\\})$ in direction $i$ is an
$(n-1)$-cube with positive signs constructed as:

  * •

For each vertex $w$ of
Cube${}^{n-1},\mathcal{C}^{w}:=\mathcal{C}^{(w,i,0)}[1]\oplus\mathcal{C}^{(w,i,1)}$,
where $(w,i,a)$ denotes the $n$-tuple with $a$ inserted as the $i$th entry to
$w$, for example, $((0,2,2),3,1)=(0,2,1,2)$.

  * •

For each face $F$, the map $f_{F}:\mathcal{C}^{s(F)}\to\mathcal{C}^{t(F)}$ is
given by the matrix

$\begin{pmatrix}f_{(\mu(F),i,0)}&0\\\
f_{(\mu(F),i,-)}&f_{(\mu(F),i,1)}\end{pmatrix}.$

The cone $cone^{i}(\mathcal{C})$ of an $n$-cube $C$ in direction $i$ is the
cone construction of $n$-cube with positive signs conjugated with canonical
maps to a general $n$-cubes. A map between two $n$-cubes
$\mathcal{C}\to\mathcal{C}^{\prime}$ is a filling of the partially defined
$(n+1)$-cube with the $n$-dimensional faces $\\{x_{n+1}=0\\}=\mathcal{C}$,
$\\{x_{n+1}=1\\}=\mathcal{C^{\prime}}$. A homotopy of two maps
$f_{0},f_{1}:\mathcal{C}\to\mathcal{C}^{\prime}$ of $n$-cubes is a filling of
the partially defined $(n+2)$-cube where the faces
$\\{x_{n+1}=0\\}=(f_{0}:\mathcal{C}\to\mathcal{C}^{\prime})$,
$\\{x_{n+1}=1\\}=(f_{1}:\mathcal{C}\to\mathcal{C}^{\prime})$ and the faces
$\\{x_{n+2}=0\\}$ and $\\{x_{n+2}=1\\}$ are the identity maps for the given
$n-cubes$.

${\mathcal{C}}$${\mathcal{C}^{\prime}}$${\mathcal{C}}$${\mathcal{C}^{\prime},}$$\scriptstyle{f_{0}}$$\scriptstyle{f_{1}}$
${\mathcal{C}}$${\mathcal{C}^{\prime}}$${\mathcal{C}}$${\mathcal{C}^{\prime}.}$$\scriptstyle{f_{0}}$$\scriptstyle{id}$$\scriptstyle{id}$$\scriptstyle{f_{1}}$
(2.1)

An $n$-ray is an infinite sequence of $n$-cubes
$\\{\mathcal{D_{i}}\\}_{i\in\mathbb{N}}$ satisfying that
$\mathcal{D}_{i},\mathcal{D}_{i+1}$ glued in the $n$-th direction for all $i$,
presented as a sequence of $(n-1)$-cubes
${\mathcal{C}=C_{1}}$${C_{2}}$${\cdots}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$
so that $\mathcal{D}_{i}:\mathcal{C}_{i}\to\mathcal{C}_{i+1}$ is a map of
$(n-1)$-cubes. Given a $1$-ray
${\mathcal{C}=C_{1}}$${C_{2}}$${\cdots}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{2}}$,
the (mapping) telescope $\text{tel}(\mathcal{C})$ of $\mathcal{C}$ is the
complex $\big{(}\bigoplus_{i\in\mathbb{N}}C_{i}[1]\oplus
C_{i},\delta:=(x[-1]-dx+f_{i}(x),dx)\big{)}$, where $x[-1]$ denotes the copy
of $x$ in $C_{i}$. The telescope $\text{tel}(\mathcal{C})$ of an $n$-ray
$\mathcal{C}$ is an $(n-1)$-cube constructed as follows.

  * •

At each vertex $w$ of Cuben-1 of tel$(\mathcal{C})$, we assign the $R$-module
$\bigoplus_{i\in\mathbb{N}}(\text{cone}^{n}(\mathcal{D}_{i}))^{w}=\bigoplus_{i\in\mathbb{N}}\mathcal{C}_{i}^{w}[1]\oplus\mathcal{C}_{i}^{w}$,
where $\text{cone}^{n}(\mathcal{D}_{i})$ is the cone in the last direction of
the $n$-cube $\mathcal{D}_{i}:\mathcal{C}_{i}\to\mathcal{C}_{i+1}$.

  * •

The induced maps are defined, for each face $F$ of Cuben-1,

$\delta_{F}(x)=\begin{cases*}f_{F}^{i}(x)&\text{if
$x\in\mathcal{C}^{s(F)}_{i}$},\\\ f_{F}^{i}(x)+x[-1]&\text{if
$x\in\mathcal{C}^{w}_{i}[1]$ and $F$ is of zero-dimension},\\\
f_{F}^{i}(x)&\text{if $x\in\mathcal{C}^{w}_{i}[1]$ and $F$ is of positive-
dimension}.\end{cases*}$

An $n$-cube of admissible Hamiltonians is a smooth map $H:\text{Cube}^{n}\to
C^{\infty}(M\times S^{1},\mathbb{R})$ which is locally constant near the
vertices and non-degenerate at the vertices. An $n$-cube family of admissible
Hamiltonians is called to be monotone if the Hamiltonians are non-decreasing
along all of the flow lines of $f$. By the energy inequality, a monotone
$n$-cube of admissible Hamiltonians gives an $n$-cube defined over the ring
$\Lambda_{\geq 0}$ [Definition 3.2.4, [134]]. Any two acceleration data for
$K\subset M$ are homotopic because any partially defined Hamiltonian-Floer-
homotopic (with monotonicity) $n$-cube has a filling to an Hamiltonian-Floer-
homotopic $n$-cube, [Proposition 3.2.18, [134], Lemma 3.5, [23]]. Here, an
$n$-cube is Hamiltonian-Floer-homotopic $n$-cube means that if for any two
points $x$ and $y$ in Cuben where Hamiltonians are defined and that there
exists a possibly broken negative gradient flow line of $f$ from $x$ to $y$,
then $H|_{x}\leq H|_{y}$.

######  Definition 2.19.

Let $(K\subset M)$ be a pair of a completion $M$ of a Liouville domain
$\underline{M}$ and a compact subset $K$ of $M$. We can assume that
$K\subset\underline{M}$. A Hamiltonian $H_{K\subset M}$ on $M$ is $(K\subset
M)$-admissible if

  1. (1)

$H_{K\subset M}|_{K}<0$ and $C^{2}$-small.

  2. (2)

$H_{K\subset M}\equiv c$ outside of a compact set whose interior containing
$K$ ($c\in\mathbb{R}_{>0}$).

######  Remark 2.20.

We refer p. 511-512 around the Figure 12 in [89].

Topological energy of $u:\mathbb{R}\times S^{1}\to(M,\omega,H:\mathbb{R}\times
S^{1}\times M\to\mathbb{R})$ is defined by,

$\mathcal{E}_{H}(u):=\int u^{*}\omega+\int\partial_{s}(H(s,t,u(s,t))).$

It defines the complex of Hamiltonian Floer cohomology as follows,

$CF^{*}(H,J):=\oplus_{x\in\mathcal{P}(H)}\mathbb{Z}\cdot x,\partial
x_{-}:=\sum_{\mu(x_{-})-\mu({x_{+})=1}}\sharp\overline{\mathcal{M}(x_{-},x_{+})}x_{+}.$

For two $(K\subset M)$-admissible Hamiltonians $H,\widetilde{H}$, define
$H\leq\widetilde{H}$ if $H(x)\leq\widetilde{H}(x)$ for any $x\in M$. We
consider a cofinal family of $(K\subset M)$-admissible Hamiltonians
$\\{H_{K\subset M,c_{i}},\leq\\}_{i\in\mathbb{N}}$ converging to
$\overline{H}_{K}$, defined by $\overline{H}_{K}|_{K}\equiv 0$,
$\overline{H}_{K}|_{M\setminus K}\equiv\infty$. An acceleration data for $K$
is a cofinal family of $(K\subset M)$-admissible Hamiltonians on a pair with a
choice of interpolating Hamiltonians $\\{H_{K\subset
M,c_{s}}\\}_{s\in[i,i+1]}$ for all $i$. Define a filtration
$\mathcal{F}_{\bullet}$ on $(CF^{*}(\mathcal{H}_{K\subset M}),\delta)$ by the
action, where $\mathcal{F}_{a}(CF^{*}(\mathcal{H}_{K\subset M}),\delta)$ is
the subcomplex generated by orbits with the action greater than $a$
($\partial\mathcal{F}_{a}\subset\mathcal{F}_{a}$). Denote that
$\mathcal{F}_{a,b}:=\mathcal{F}_{a}/\mathcal{F}_{b}$. Then for
$a_{1}<a_{2}<b_{1}<b_{2}$, $c_{1}<c_{2}$, we have a Hamiltonian-Floer-
homotopic $3$-cube, a $3$-cube where Hamiltonians are mononotonous along the
arrows of the following diagram with the connecting 1-parameter family of
Hamiltonians.

${\mathcal{F}_{a_{1},b_{2}}CF^{*}(H_{K\subset
M,c_{1}})}$${\mathcal{F}_{a_{1},b_{2}}CF^{*}(H_{K\subset
M,c_{2}})}$${\mathcal{F}_{a_{1},b_{1}}CF^{*}(H_{K\subset
M,c_{1}})}$${\mathcal{F}_{a_{1},b_{1}}CF^{*}(H_{K\subset
M,c_{2}})}$${\mathcal{F}_{a_{2},b_{2}}CF^{*}(H_{K\subset
M,c_{1}})}$${\mathcal{F}_{a_{2},b_{2}}CF^{*}(H_{K\subset
M,c_{2}})}$${\mathcal{F}_{a_{2},b_{1}}CF^{*}(H_{K\subset
M,c_{1}})}$${\mathcal{F}_{a_{2},b_{1}}CF^{*}(H_{K\subset M,c_{2}})}$

We also assume that for any two non-degenerate Hamiltonians $f$ and $g$ in the
cofinal family, $\overline{\\{f<g\\}}\cap\overline{\\{f>g\\}}=\emptyset$ so
that max$\\{f,g\\}$ and min$\\{f,g\\}$ are smooth [Proposition 4.1.1,[134]].
From the cofinal family of $(K\subset M)$-admissible Hamiltonians, we define
relative symplectic cohomology of a pair $(K,M)$ as follows.

######  Definition 2.21.

Let $M$ be the completion of a Liouville domain $\underline{M}$, and $K$ be a
compact subset of $\underline{M}$. Let $a,b\in[-\infty,\infty]$ with $a<b$.

$\displaystyle SC^{*}_{a,b}(K\subset M;\mathcal{H}_{K\subset M},\mathcal{J}):$
$\displaystyle=\varinjlim_{\begin{subarray}{c}a^{\prime}\searrow
a\end{subarray}}\circ\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow
b\end{subarray}}\circ\text{tel}_{c}[\mathcal{F}_{a^{\prime},b^{\prime}}CF^{\bullet}(\mathcal{H}_{K\subset
M,c})],$ $\displaystyle=\varinjlim_{\begin{subarray}{c}a^{\prime}\searrow
a\end{subarray}}\circ\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow
b\end{subarray}}\circ\varinjlim_{c\nearrow\infty}[\mathcal{F}_{a^{\prime},b^{\prime}}CF^{\bullet}(\mathcal{H}_{K\subset
M,c})],$

where
$\varinjlim_{\begin{subarray}{c}a\end{subarray}}\circ\varprojlim_{\begin{subarray}{c}b\end{subarray}}$
is completion with respect to action filtration, where $\text{tel}_{c}$
denotes telescope of a $3$-ray in the direction of the height $c$ of
admissible Hamiltonians, with choices of continuation map data. For a finite
number $a\in\mathbb{R}$, taking direct limit
$\varinjlim_{\begin{subarray}{c}a\leq\end{subarray}}$ does not necessary.

$SC^{*}_{a,b}(K\subset M;\mathcal{H}_{K\subset
M},\mathcal{J}):=\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow
b\end{subarray}}\circ\varinjlim_{c\nearrow\infty}[\mathcal{F}_{a,b^{\prime}}CF^{\bullet}(\mathcal{H}_{K\subset
M,c})].$

By [134], we can choose a homotopy interpolating two cofinal Hamiltonian data.

######  Lemma 2.22.

[134] For any two cofinal family of Hamiltonians $(\mathcal{H}_{K\subset
M},\mathcal{J}),(\tilde{\mathcal{H}}_{K\subset M},\widetilde{\mathcal{J}})$,
$SC^{*}_{a,b}(K\subset M;\mathcal{H}_{K\subset M},\mathcal{J})$ and
$SC^{*}_{a,b}(K\subset M;\tilde{\mathcal{H}}_{K\subset
M},\widetilde{\mathcal{J}})$ are quasi-isomorphic.

Therefore, we have well-defined cohomology,

$SH^{*}_{a,b}(K\subset
M):=H^{*}\Big{(}\varinjlim_{\begin{subarray}{c}a^{\prime}\searrow
a\end{subarray}}\circ\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow
b\end{subarray}}\circ\varinjlim_{c\nearrow\infty}[\mathcal{F}_{a,b^{\prime}}CF^{\bullet}(\mathcal{H}_{K\subset
M,c})]\Big{)}.$

Similar to the usual symplectic cohomology, we have the following lemma.

######  Lemma 2.23.

Let $M$ be the completion of a Liouville domain $\underline{M}$. Let
$SH^{*}(\underline{M}\subset M)$ denote
$SH^{*}_{-\epsilon,\infty}(\underline{M}\subset M)$ and
$SH^{*}_{+}(\underline{M}\subset M)$ denote
$SH^{*}_{\epsilon,\infty}(\underline{M}\subset M)$. Then

  1. (1)

$SH^{*}_{-\epsilon,\epsilon}(\underline{M}\subset M)\cong H^{*}(M)$.

  2. (2)

$\cdots\to SH^{*}_{+}(\underline{M}\subset
M)\xrightarrow{\hat{\delta}}H^{*}(M)\to SH^{*}(\underline{M}\subset M)\to
SH^{*}_{+}(\underline{M}\subset M)\to\cdots.$

  3. (3)

$SH^{*}(\underline{M}\subset M)=\widehat{SH^{*}}(M)$, where
$\widehat{SH^{*}}(M)$ is cohomology of action-completed $(a,b)$-truncated
symplectic chains, [135].

By the connecting map $\widehat{\delta}$ in the long exact sequence 2.23 (2),
we can define $\widehat{\ell}(M)$ as well.

######  Proposition 2.24.

[134] Let $K_{1},K_{2}$ be compact subsets of a completion $M$ of a Liouville
domain $\underline{M}$ satisfying that
$\partial(K_{1}\cap\underline{M})\cap\partial(K_{2}\cap\underline{M})\neq\emptyset$.
The Mayer-Vietoris sequence for such subsets $K_{1},K_{2}$ of $M$ holds,

$\overset{\delta}{\longrightarrow}SH^{*}(K_{1}\cup K_{2}\subset M)\to
SH^{*}(K_{1}\subset M)\oplus SH^{*}(K_{2}\subset M)\to SH^{*}(K_{1}\cap
K_{2}\subset M)\to.$

###### Proof.

Exactly the same idea in [134] works. Since
$\partial(K_{1}\cap\underline{M})\cap\partial(K_{2}\cap\underline{M})\neq\emptyset$,
the slices of any $3$-ray that is compatible with any acceleration datum for
$(K_{1}\subset M)$, $(K_{2}\subset M)$, $(K_{1}\cap K_{2}\subset M)$, and
$(K_{1}\cup K_{2}\subset M)$, we get an acyclic $2$-cube of the cofinal
families $\mathcal{H}_{1}$, $\mathcal{H}_{2}$,
$max\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}$, and
$min\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}$ respectively, as follows. Here,
$n$-cube $\mathcal{C}$ is called acyclic if $\text{Cone}^{n-1}\mathcal{C}$ is
an acyclic chain complex.

${C_{00}}$${C_{10}}$${C_{01}}$${C_{11}.}$

If a $2$-cube is acyclic, then the map $C_{00}\to Cone(C_{10}\oplus C_{01}\to
C_{11})$ is a quasi-isomorphism. Therefore, it induces an exact sequence

$\cdots\to H^{*}(C_{00})\to H^{*}(C_{10})\oplus H^{*}(C_{01})\to
H^{*}(C_{11})\to H^{*+1}(C_{00})\to\cdots.$

Hence, the acyclic $2$-cube

${CF^{*}(min\\{\mathcal{H}_{1},\mathcal{H}_{2}\\})}$${CF^{*}(\mathcal{H}_{1})}$${CF^{*}(\mathcal{H}_{2})}$${CF^{*}(max\\{\mathcal{H}_{1},\mathcal{H}_{2}\\}),}$

gives

${\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow\infty\end{subarray}}\circ\varinjlim_{c\nearrow\infty}[\mathcal{F}_{\epsilon,b^{\prime}}CF^{*}(min\\{\mathcal{H}_{1},\mathcal{H}_{2}\\})]}$${\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow\infty\end{subarray}}\circ\varinjlim_{c\nearrow\infty}[\mathcal{F}_{\epsilon,b^{\prime}}CF^{*}(\mathcal{H}_{1})]}$${\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow\infty\end{subarray}}\circ\varinjlim_{c\nearrow\infty}[\mathcal{F}_{\epsilon,b^{\prime}}CF^{*}(\mathcal{H}_{2})]}$${\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow\infty\end{subarray}}\circ\varinjlim_{c\nearrow\infty}[\mathcal{F}_{\epsilon,b^{\prime}}CF^{*}(max\\{\mathcal{H}_{1},\mathcal{H}_{2}\\})],}$

that is still an acyclic $2$-cube. Hence, it induces the Mayer-Vietoris
sequence. ∎

Given a Liouville domain $(M,d\theta)$, the skeleton or the core of the
Liouville structure is the attractor of the negative flow of the Liouville
vector field $Z$, which is a compact set [24]. For example, consider ribbon
graphs of certain punctured Riemann surfaces. The following theorem is a
direct corollary of Proposition 5.32 in [89]. We also refer Theorem D in [13].

######  Proposition 2.25.

For any compact subset $K$ of a Liouville domain $(\underline{M},d\theta)$
containing skeleton of the Liouville flow, the relative symplectic cohomology
$SH^{*}(K\subset M)$ is isomorphic to $SH^{*}(\underline{M}\subset M)$.

###### Proof.

The same idea of the proof of lemmas for proposition 5.32 in [89] works for
the completion of a Liouville domain, where symplectic form is exact and where
we are free from rearranging index-bounded contact cylinders. Let
$\phi^{-Z_{\theta}}_{t}:\underline{M}\to\underline{M}$ be the time $t$ flow of
the Liouville vector field $-Z_{\theta}$ with opposite direction. Since $K$
contains the skeleta of $\phi^{-Z_{\theta}}_{t}$, there exists $T>0$ such that
$\phi^{-Z_{\theta}}_{t}(\underline{M})\subset K$ for all $t>T$. By
independence of symplectomorphism and continuation maps, we get the
isomorphism,

$SH^{*}(\underline{M}\subset M)\hookrightarrow SH^{*}(K\subset
M)\hookrightarrow SH^{*}(\phi^{-Z_{\theta}}_{t}(\underline{M})\subset M)\cong
SH^{*}(\underline{M}\subset M).$ ∎

####  2.5.2. Relative Symplectic Cohomology with respect to Symplectic
Structures

Let $X$ be a projective variety with effective ample divisors $D_{1},D_{2}$.
Choose an effective ample divisor $D$ whose support is exactly $D_{1}\cup
D_{2}$ and which is linearly equivalent to a large multiple $m$ of $D_{1}$.
Let $s$ be a section of a line bundle $\mathcal{O}_{X}(D)$ with $s^{-1}(0)=D$
and $\omega_{D}:=dd^{c}(-log||s||)$ be the induced exact symplectic form on
$X\setminus D$. Then $\omega_{D}=m\cdot\omega_{D_{1}}$. Since the set of
critical point of $-log||s||$ is a compact subset of $X\setminus(D_{1}\cup
D_{2})$ by Lemma 4.3 in [124], $([X\setminus(D_{1}\cup
D_{2})]^{\wedge}_{\omega_{D}},\omega_{D})$ is a finite-type Weinstein
manifold, where $[X\setminus(D_{1}\cup D_{2})]^{\wedge}_{\omega_{D}}$ is the
union of $X\setminus(D_{1}\cup D_{2})$ and the trajectory of the Liouville
flow of $\omega_{D}$. Scaling by $\frac{1}{m}$, $([X\setminus(D_{1}\cup
D_{2})]^{\wedge}_{\omega_{D_{1}}},\omega_{D_{1}})$ is a finite-type Weinstein
manifold.

Let $K$ be a compact subset of $X\setminus D_{1}$. We define a relative
symplectic cohomology $SH^{*}_{D_{1}}(K\subset X\setminus(D_{1}\cup D_{2}))$
of a pair $(K\subset[X\setminus(D_{1}\cup D_{2})]^{\wedge}_{\omega_{D_{1}}})$
with respect to $\omega_{D_{1}}|_{[X\setminus(D_{1}\cup
D_{2})]^{\wedge}_{\omega_{D_{1}}}}$ by a cofinal family of
$(K\subset[X\setminus(D_{1}\cup D_{2})]^{\wedge}_{\omega_{D_{1}}})$-admissible
Hamiltonians converging to $\overline{H_{K}}$, where

  1. (1)

$H|_{K}<0$ and $C^{2}$-small.

  2. (2)

$H=$ constant outside of a compact set whose interior containing $K$

  3. (3)

$\overline{H}_{K}|_{K}\equiv 0$, and $\overline{H}_{K}|_{[X\setminus(D_{1}\cup
D_{2})]^{\wedge}_{\omega_{D_{1}}}\setminus K}\equiv\infty$

######  Proposition 2.26.

Let $X$ be a projective variety with effective ample divisors $D_{1},D_{2}$.
Let $K$ be a compact subset of $X\setminus(D_{1}\cup D_{2})$. Then the
following relative symplectic cohomologies are isomorphic,

$SH^{*}_{D_{1}\cup D_{2}}(K\subset X\setminus(D_{1}\cup D_{2}))\cong
SH^{*}_{D_{1}}(K\subset X\setminus(D_{1}\cup D_{2})).$

###### Proof.

of 2.26 We get the isomorphism after interpolating symplectic cohomologies of
different Käher structures on $X$ inducing exact symplectic structures on the
complement of ample divisors in [89] and by relabeling isomorphism [134].
Denote $(M,\omega_{M}):=(X\setminus(D_{1}\cup D_{2}),\omega_{D_{1}\cup
D_{2}})$, $(W,\omega_{W}):=\big{(}[X\setminus(D_{1}\cup
D_{2})]^{\wedge}_{\omega_{D_{1}}},\omega_{D_{1}}|_{[X\setminus(D_{1}\cup
D_{2})]^{\wedge}_{\omega_{D_{1}}}}\big{)}$.

Consider two pluri-subharmonic functions $f:=-log||s||$,
$f_{1}:=-log||s_{1}||$ on $M$, where $s,s_{1}$ are sections of line bundles
$\mathcal{O}_{X}(D)$, $\mathcal{O}_{X}(D_{1})$ with $s^{-1}(0)=D$,
$s_{1}^{-1}(0)=D_{1}$, respectively. Let $U_{1}$ be a small neighborhood of
$K$ and $U_{2}:=M\setminus K$. Consider a smooth partition of unity
$(\rho_{1},\rho_{2})$ subordinate to $(U_{1},U_{2})$. Since admissible
Hamiltonians are $\mathcal{C}^{2}$-small and approach to zero on $K$,

$SH^{*}_{\omega_{D_{1}\cup D_{2}}}(K\subset M)=SH^{*}_{dd^{c}f}(K\subset
M)\cong SH^{*}_{dd^{c}(f+\rho_{1}\cdot f_{1})}(K\subset M).$

Two exhausting plurisubharmonic functions $f_{0}:=f+\rho_{1}\cdot f_{1}$,
$f_{1}$ on a complex manifold $(M,J)$ give a finite type (noncomplete)
Weinstein homotopy $(M,dd^{c}f_{s},f_{s})_{s\in[0,1]}$ between
$(M,dd^{c}f_{0},f_{0})$, $(M,dd^{c}f_{1},f_{1})$ satisfying that
$\overline{\cup_{s\in[0,1]}\text{Skeleton}(M,dd^{c}f_{s})}$ is compact, by the
idea of proof of Proposition 11.22 in [24], and [34]. By the idea of proof of
Proposition 11.8 in [24], there exists a diffeotopy $h_{s}:M\to M$ so that
$h_{0}=Id_{M}$, $h_{s}^{*}df_{s}-df_{0}$ is exact for all $s\in[0,1]$ and that
$h_{s}^{*}df_{s}=df_{0}$ outside a compact set. After completion under the
Liouville flow, we get an exact symplectomorphism
$\phi:(M,dd^{c}(f+\rho_{1}\cdot f_{1}))\to(W,\omega_{W})$ between two finite
type Weinstein manifolds with $\phi(K)=K$. The relabeling all the choices by
the symplectomorphism $\phi$ gives an isomorphism of relative symplectic
cohomology, $SH^{*}_{dd^{c}(f+\rho_{1}\cdot f_{1})}(K\subset M)\cong
SH^{*}_{\omega_{D_{1}}}(K\subset W)$, [Proposition 3.3.3(3), [134]]. In sum,

$SH^{*}_{\omega_{D_{1}\cup D_{2}}}(K\subset
M)=SH^{*}_{\omega_{D_{1}}}(K\subset W).$

∎

####  2.5.3. Spectral Sequence Relating Different Choices of Polarization

Let us recall a spectral sequence associated to a filtered complex. Let
$(C^{*}_{\lambda},F^{\bullet})$ is a filtered complex of differential graded
module over $\Lambda$. There is a spectral sequence $(E_{r}=\oplus
E_{r}^{p,q},d_{r}^{p,q}:E_{r}^{p,q}\to E_{r}^{p+r,q-r+1})$ whose $E_{0}$-page
consists of an associated graded ring
$(gr^{p}(C^{p+q}),d_{0}^{p,q}=gr^{p}(d^{p+q}))$ and whose $E_{1}$-page
consists of $(E_{1}^{p,q}=H^{p+q}(gr^{p}(C^{*})))$, where
$gr^{p}(C^{*}):=F^{p}(C^{*})$ [Lemma 12.24.2, [129]]. If the filtration is
finite (i.e., there exist $n,m$ such that $F^{n}C^{*}=C^{*}$ and
$F^{m}C^{*}=0$), then the spectral sequence associated to
$(C^{*},F^{\bullet})$ converges to $H^{*}(C^{*})$, [Lemma 12.24.11(3), [129]].
Using the filtration by the winding numbers, we can construct a spectral
sequence explaining how relative symplectic cohomology changes when we delete
an ample divisor.

######  Proposition 2.27.

Let $X$ be a projective variety with ample divisors $D_{1},D_{2}$. Let $K$ be
a compact subset of $X\setminus D_{1}$ containing skeletons of both
$(X\setminus D_{1},\omega_{D_{1}})$ and $(X\setminus(D_{1}\cup
D_{2}),\omega_{D_{1}\cup D_{2}})$. Then there exists a spectral sequence
converging to $SH^{*}_{D_{1}}(K\subset X\setminus D_{1})$ whose $E_{1}$-page
is $SH^{*}_{D_{1}}(K\setminus\mathbb{D}_{2}\subset X\setminus(D_{1}\cup
D_{2}))$, where $\mathbb{D}_{2}$ is a small tubular neighborhood of $D_{2}$.

###### Proof.

of 2.27 By the stably-displaceability of a divisor $D_{2}$ and the vanishing
result in [89], we can choose a small neighborhood $\mathbb{D}_{2}$ of $D_{2}$
so that $SH^{*}_{D_{1}\cup D_{2}}(K\subset X\setminus D_{1})\cong
SH^{*}_{D_{1}\cup D_{2}}(K\setminus\mathbb{D}_{2}\subset X\setminus D_{1})$.
Therefore, we can consider $K$ as a compact subset of $X\setminus(D_{1}\cup
D_{2})$.

Define a filtration $W^{\bullet}$ on $(SC^{*}(K\setminus\mathbb{D}_{2}\subset
X\setminus D_{1}),\delta_{K\subset X\setminus D_{1}})$ (resp.,
$(SC^{*}(K\setminus\mathbb{D}_{2}\subset X\setminus(D_{1}\cup D_{2}))$,
$\delta_{K\subset X\setminus(D_{1}\cup D_{2})})$) by the winding number of
orbits along $D_{2}$ 333Since $D_{2}$ is effective ample, there exists
$f\in\mathcal{O}(X\setminus D_{1})$ so that $D_{2}\setminus D_{1}=f^{-1}(0)$.
Define $ev_{f}:X\setminus D_{1}\to\mathbb{C}$ by evaluating $f$. The winding
number of orbits is defined by the winding number of the image of each orbit
on $\mathbb{C}$ under the map $ev_{f}$ along the origin or the intersection
number of $D_{2}\mathbb{D}_{1}$ with a holomorphic capping of an orbit. is
greater than or equal to $w$. Since our admissible Hamiltonians are constant
near the divisors, Floer trajectories are holomorphic near the divisors. Since
holomorphic curves are positively intersect with any ample divisors, the
winding number of an orbit along $D_{2}$ is strictly increasing under the
differential map if the connecting Floer trajectory intersect with $D_{2}$,
otherwise it preserve the winding number.444In [13], the integrated maximum
principle in [4] was used to prove the non-negativity of a Floer trajectory
(perturbed $J$-holomorphic curve) and a divisor $D_{2}$ ($J$-holomorphic
hypersurface) and their special Hamiltonian is constant on the divisors
[Proposition 3.9, [13]; Section 2.9, [140]]. Therefore, $E_{1}$-page of the
spectral sequence associated to
$(SC^{*}_{D_{1}}(K\setminus\mathbb{D}_{2}\subset X\setminus
D_{1}),\delta_{K\subset X\setminus D_{1}})$ isomorphic to
$\bigoplus_{n}W^{n}(SH^{*}(K\setminus\mathbb{D}_{2}\subset
X\setminus(D_{1}\cup D_{2})))\cong SH^{*}(K\setminus\mathbb{D}_{2}\subset
X\setminus(D_{1}\cap D_{2}))$. Remark that the Floer trajectories of
$(SC^{*}(K\setminus\mathbb{D}_{2}\subset X\setminus D_{1}),\delta_{K\subset
X\setminus D_{1}})$ are allowed to pass through $D_{2}$, on the other hand,
those of $(SC^{*}(K\setminus\mathbb{D}_{2}\subset X\setminus(D_{1}\cup
D_{2})),\delta_{K\subset X\setminus(D_{1}\cup D_{2})})$ are not.

Since the symplectic structure on the completion of a Liouville domain is
exact, the spectral sequence associated to $(C^{*},W^{\bullet})$ is finite.
Hence the spectral sequence converges to
$H^{*}(SC^{*}(K\setminus\mathbb{D}_{2}\subset X\setminus
D_{1}),\delta_{K\subset X\setminus
D_{1}})=SH^{*}(K\setminus\mathbb{D}_{2}\subset X\setminus D_{1})\cong
SH^{*}(K\subset X\setminus D_{1})$ whose $E_{1}$-page isomorphic to
$\bigoplus_{n}W^{n}(SH^{*}(K\setminus\mathbb{D}_{2}\subset
X\setminus(D_{1}\cup D_{2})))\cong SH^{*}(K\setminus\mathbb{D}_{2}\subset
X\setminus(D_{1}\cap D_{2}))$. ∎

######  Remark 2.28.

When $D_{1}=\emptyset$, then the statement of proposition 2.27 recover that of
the Theorem C in [13] where a spectral sequence associated to the filtered
complex $\big{(}SC^{*},\partial,F^{*}\big{)}$ converging to
$H^{*}(SC^{*},\partial)=SH^{*}_{D}\big{(}\underline{X\setminus D}\subset
X\big{)}\cong QH^{*}(X,\Lambda)$ whose $E_{1}$-page consists of
$SH^{*}_{D}\big{(}\underline{X\setminus D}\subset X\setminus D\big{)}$. But
our proof is not enough to prove Theorem C so we refer [13].

Combining propositions 2.25, 2.26 and 2.27,

$SH^{*}_{D_{1}\cup D_{2}}\big{(}K\setminus\mathbb{D}_{2}\subset
X\setminus(D_{1}\cup D_{2})\big{)}\cong
SH^{*}_{D_{1}}(K\setminus\mathbb{D}_{2}\subset X\setminus(D_{1}\cup
D_{2}))\Rightarrow SH^{*}_{D_{1}}(K\subset X\setminus D_{1}),$
$\Big{(}\text{or }SH^{*}_{D_{1}\cup
D_{2}}\big{(}K\setminus\mathbb{D}_{2}\subset X\setminus(D_{1}\cup
D_{2})\big{)}\Rightarrow SH^{*}_{D_{1}\cup
D_{2}}\big{(}K\setminus\mathbb{D}_{2}\subset X\setminus D_{1}\big{)}\cong
SH^{*}_{D_{1}}(K\setminus\mathbb{D}_{2}\subset X\setminus D_{1}).\Big{)}$

######  Theorem 2.29.

Let $X$ be a projective variety with effective ample divisors $D_{1},D_{2}$.
Let $K$ be a compact subset of $X\setminus D_{1}$. Then there exists a
spectral sequence converging to $SH^{*}_{D_{1}}(K\subset X\setminus D_{1})$
whose $E_{1}$-page is $SH^{*}_{D_{1}\cup
D_{2}}\big{(}K\setminus\mathbb{D}_{2}\subset X\setminus(D_{1}\cup
D_{2})\big{)}$, where $\mathbb{D}_{2}$ is a small tubular neighborhood of
$D_{2}$.

The following corollary of theorem 2.29 will be used to prove theorem 4.12.

######  Corollary 2.30.

Let $M$ be an affine variety and $Y\subset M$ a smooth hypersurface. Choose a
small tubular neighborhood $\mathbb{Y}$ of $Y\cap\underline{M}$ inside
$\underline{M}$. Then there exists a spectral sequence converging to
$SH^{*}(\mathbb{Y}^{c}\subset M)\cong SH^{*}\big{(}\underline{M\setminus
Y}\subset M\big{)}$ whose $E_{1}$-page is $SH^{*}(\mathbb{Y}^{c}\subset
M\setminus Y)\cong SH^{*}\big{(}\underline{M\setminus Y}\subset M\setminus
Y\big{)}$.

###  2.6. Vanishing or Invariance Theorems of Symplectic Cohomology

In this section, we review some results on vanishing theorems and invariance
theorems of symplectic cohomology, which we use when we apply the main
theorems. First of all, a non-trivial fact is that $SH^{*}(\mathbb{C}^{n})=0$
for all $n$, [[42], Chapter 3, [104]] by computing Conley-Zehnder indices and
taking the direct limit.

####  2.6.1. Symplectic Cohomology under Attaching Weinstein Handles

As Morse theory let us understand topology of smooth manifolds and construct
them by attaching handles of critical points, pseudo-convex function theory
and h-principle let us understand symplectic topology of Liouville domains and
construct some of them by attaching Weinstein handles, [33], [54]. In [53],
Gromov developed the pseudo-holomorphic curve theory and showed that a
Liouville domain $\underline{M}$ whose boundary $\partial\underline{M}$ is
contactomorpic to the standard contact structure on $S^{3}$ is isomorphic to
$(\mathbb{R}^{4},\omega_{\text{standard}})$ as Liouville domains. The behavior
of symplectic cohomology under Weinstein handle attaching was studied by
Cieliebak in [23]. An important fact is that subcritical handle attaching does
not change the symplectic cohomology before and after. By attaching
subcritical handles and the h-cobordism, Seidel-Smith showed that Liouville
domains of dimension $2n\geq 4$, with sphere boundary with the standard
contact structure, should have vanishing symplectic cohomology in [124].
Moreover, Seidel-Smith and McLean found exotic Stein manifolds. Casals-Murphy
detected flexibility of affine varieties which leads vanishing symplectic
cohomology by detecting a zig-zag on Legendrian front projection [18].

Let us recall the Weinstein handle attachment as follows [138].

  * •

A $1$-form $\alpha$ on a manifold $E$ of dimension $2n+1$ is called to be
contact if $\alpha\wedge(d\alpha)^{n}\neq 0$. A hyperplane field $\xi$ of the
tangent bundle $TE$ is called to be a contact structure if it is a kernel of
some contact $1$-form (by definition, it is maximally non-integrable).

  * •

A submanifold $L$ of a contact manifold $(E,\xi)$ is called to be isotropic if
$T_{x}L\subset\xi_{x}$ for all $x\in L$ (i.e., $j^{*}_{L}\alpha=0\iff
j^{*}_{L}d\alpha=0\iff T_{x}L\subset(\xi_{x},d\alpha|_{\xi_{x}})$ isotropic).
A maximual isotropic submanifold is called a Legendrian submanifold.

Let $(E,\xi)$ be a contact manifold and $j:S^{k-1}\to E$ be a isotropic
sphere. Denote $(TS^{k-1})^{\perp}$ a sub-bundle of
$(\xi|_{L},d\alpha|_{\xi})$ that is orthogonal to $TS^{k-1}$ with respect to
$d\alpha|_{\xi}$. $TS^{k-1}\subset(TS^{k-1})^{\perp}$ (isotropic) and the
quotient bundle $(TS^{k-1})^{\perp}/TS^{k-1}\cong\mathbb{C}^{n-k}$ has a
structure of the conformal symplectic normal bundle of $S^{k}$ induced from
$d\alpha$. The standard chart is
$\mathbb{R}^{2n}=T^{*}\mathbb{R}^{k}\times\mathbb{C}^{n-k}$ with the
symplectic form $\omega=\sum_{i=0}^{k}dp_{i}\wedge
dq_{i}+\sum_{j=0}^{n-k}dx_{j}\wedge dy_{j}$ and with the Liouville vector
field $Z_{k}=\sum_{i=1}^{k}(2p_{i}\frac{\partial}{\partial
p_{i}}-q_{i}\frac{\partial}{\partial
q_{i}})+\frac{1}{2}(\sum_{j=1}^{n-k}x_{j}\frac{\partial}{\partial
x_{j}}+y_{j}\frac{\partial}{\partial y_{j}})$ and $Z_{k}$ is a gradient vector
field for the Morse function
$f_{k}:=\sum(p_{i}^{2}-\frac{1}{2}q_{i}^{2})+\sum(\frac{1}{4}x_{j}^{2}+\frac{1}{4}y_{j}^{2})$
with respect to the standard Euclidean metric on $\mathbb{R}^{n}$. Define a
$1$-form
$\alpha_{k}:=\iota_{Z_{k}}\omega=\sum_{i}(2p_{i}dp_{i}+q_{i}dq_{i})+\frac{1}{2}\sum_{j}(x_{j}dx_{j}-y_{j}dy_{j})$.
Along the flow of $Z_{k}$, the stable(resp. unstable) subspace is
$W^{s}_{k}=\\{q_{1}=\cdots=q_{k}=0\\}\cong\mathbb{R}^{2n-k}$ (resp.
$W^{u}_{k}=\\{p_{i}=0,x_{j}=y_{j}=0\text{ for all}i,j\\}\cong\mathbb{R}^{k}$).
Since the form $\alpha_{i}$ pulls back to $0$ on $W^{s}_{k}$ along the flow, a
sphere $S^{st}_{k}:=W^{s}_{k}\cap\\{f_{k}=1\\}$ is an isotropic submanifold of
$\\{f_{k}=1,\alpha_{k}\\}$, call it the ascending sphere of index $k$ (resp.
$S^{ut}_{k}:=W^{u}_{k}\cap\\{f_{k}=-1\\}$ is called the descending sphere). A
standard Weinstein $k$ handle in $\mathbb{R}^{2n}$ is a bounded neighborhood
of the descending sphere in $\\{f_{k=-1}\\}$ with a connectic manifold
diffeomorphic to $S^{2n-k-1}\times D^{k}$. It is known that every Weinstein
domain $M^{2n}$, $n\geq 3$, can be Weinstein homotoped to a subcritical domain
$M_{subcrit}$ with handles attached to the Legendrian (may not be loose) link
$\Lambda_{1}\sqcup\cdots\sqcup\Lambda_{k-1}\sqcup\Lambda_{k}\subset\partial
M_{subcrit}$ such that $\Lambda_{1}\sqcup\cdots\sqcup\Lambda_{k-1}$ is a loose
link and $\Lambda_{k}$ is a loose Legendrian where linking information is
symplectically nontrivial [73].

Cieliebak defined relative symplectic cohomology of a pair
$W\subset(M,\omega)$ of a codimension $0$ submanifold $W$ of $(M,\omega)$ with
$\omega$-convex boundary $\partial W$, using Hamiltonians that are
$C^{2}$-small Morse functions in the interior of $\underline{W}$ and
increasing sharply near $\partial\underline{W}$ [23]. Let $M_{\Supset k}$ be a
Liouville domain obtained by attaching a Weinstein $k$-handle to
$\partial\underline{M}$. Considering that the only generator of the complex of
relative symplectic cohomology of a pair $M\subset(M_{\Supset k},\omega)$ is a
critical point of the $k$-handle and that it disappears under the transfer
functor in the direct limit (the Conley-Zehnder index can be increased under
the transfer functor), he showed the following invariance property of
symplectic cohomology under subcritical handle attachment.

######  Proposition 2.31 (Theorem 1.11(1), [23]).

Let $(\underline{M},d\theta_{\underline{M}})$ be a Liouville domain of
dimension $2n$, and $S\subset\partial\underline{M}$ be a contact-
isotropic(i.e.,$\theta_{\underline{M}}|_{S}$=0) sphere of dimension $k-1<n-1$.
(The symplectic normal bundle of $S$ in $\underline{M}$ is trivial.) Let
$\underline{M}_{\Supset k}:=\underline{M}\cup_{S}H_{k}$ be a Liouville domain
obtained by attaching a Weinstein $k$-handle $H_{k}$ to
$\partial\underline{M}$ near $S$. Then $SH^{*}(M_{\Supset k})\cong SH^{*}(M)$.

A direct corollary is that for a subcritical Weinstein manifold $M$,
$SH^{*}(M)=0$.

######  Proposition 2.32 (Corollary 6.5, [124]).

Let $\underline{M}$ be a Liouville domain of dimension $2n\geq 4$, such that
$\partial\underline{M}$ is contact isomorphic to the standard contact
structure on $S^{2n-1}$. Then $SH^{*}(M)=0$.

Let $\underline{M}$ be a Liouville domain. Let
$\Lambda:=\cup_{i=1}^{k}\Lambda_{i}$ be a disjoint union of Legendrian spheres
in the boundary $\partial\underline{M}$ and denote $M_{\Supset\Lambda}$ a
convex Weinstein domain on which symplectic cohomology is well-defined,
obtained by attaching Weinstein-handles to $\underline{M}$ along $\Lambda$. In
the case that $\underline{M}$ is a Liouville subdomain of
$\underline{M}^{\prime}$ of dimension $2n$ and a Weinstein cobordism
$(\mathfrak{S}:=\overline{\underline{M}^{\prime}}\setminus\text{int}(\underline{M}),f:\mathfrak{S}\to\mathbb{R})$
contains only Morse critical points $p_{1},\cdots,p_{k}$ of index $n$, we
consider the Legendrian spheres $\Lambda_{j}$’s are the intersection of the
stable manifold $L_{j}$ of $p_{j}$ and the boundary
$\partial\underline{M}^{\prime}$ of $\underline{M}^{\prime}$. Then
$M^{\prime}\cong M_{\Supset\Lambda}$.

Let $LHA(\Lambda)$ be the Legendrian homology differential graded
algebra(DGA), whose complex is generated by Reeb chords connecting
$\Lambda_{j}$’s. Let $LHO(\Lambda)$ be the subalgebra of cyclically
compaosable monomials. Let $LH^{H_{0}}_{+}$ be a homology of a mapping
cylinder of the reduced Legendrian DGA $LHO^{+}(\Lambda)$ of $LHO(\Lambda)$.
Denote $LH^{H_{0}}(\Lambda)$ the full Legendrian homology DGA whose complex
consists of $LHO^{+}(\Lambda)$ and the Legendrian spheres $\Lambda$ [Section
4, [15]].

######  Lemma 2.33 (Theorem 5.4, 5.6, [15]).

There are exact sequences,

${\cdots\to
LH^{H_{0}}_{+}(\Lambda)}$${SH^{*}_{+}(M_{\Supset\Lambda})}$${SH^{*}_{+}(M)}$${LH^{H_{o}}_{+}(\Lambda)[1]\to\cdots.}$
${\cdots\to
LH^{H_{0}}(\Lambda)}$${SH^{*}(M_{\Supset\Lambda})}$${SH^{*}(M)}$${LH^{H_{o}}(\Lambda)[1]\to\cdots.}$

######  Lemma 2.34 ([31], [95]).

The Legendrian contact homology algebra of a loose Legendrian vanishes.

######  Proposition 2.35 (Theorem 3.2, [96]).

Let $M$ be a flexible Weinstein domain, then, for any closed two-form on $M$,
the twisted symplectic cohomology $SH^{*}(M,\Lambda_{2})=0$.

The following example is from Casals-Murphy’s recipe on the Legendrian front
projection of Weinstein Lefschetz bifibration of affine varieties in [18].
Define

$L_{a,b}^{n}:=\\{(x,y,z_{1},\cdots,z_{n-1}:x^{a}y^{b}+\sum_{i=1}^{n-1}z_{i}^{2}=1\\}\subset\mathbb{C}^{n+1},$

where $a,b\in\mathbb{N}$ are two coprime integers with $1\leq a<b$. By
[Theorem 1.1., [18]], $L_{1,b}^{n}$ are flexible for all $b\geq 2$, therefore,
$SH^{*}(L_{1,b}^{n})=0$. In particular, $L_{1,b}^{3}$ are diffeomorphic to
$S^{3}\times\mathbb{R}^{3}$, so we can see $\ell(L_{1,b}^{3})=0$ as well.
Inspired by it, we expect that finding flexible Weinstein cobordisms by
Casals-Murphy’s algorithm on the Lefschetz fibration of an affine variety $M$
would lead to compute $\ell(M)$ as well as to find uniruled subvarieties of
$M$.

####  2.6.2. Vanishing Symplectic Cohomology from $\mathbb{C}^{*}$-action

When a convex weak-monotone symplectic manifold has Hamiltonian circle action
which generates Hamiltonian/Reeb orbits near at the infinity, the circle
action make the Conley-Zehnder index of Hamiltonian orbits goes to infinity as
the slope of Hamiltonian goes to infinity. As a direct limit of Hamiltonian
Floer cohomology under the direct system by the increasing slope of
Hamiltonian, symplectic cohomology vanishes. Let us review on Ritter’s
vanishing result here.

######  Proposition 2.36.

  1. (1)

Let $M$ be the total space of the line bundle
$\mathcal{O}_{\mathbb{P}^{m}}(-n)$ over $\mathbb{P}^{m}$. Then,

     * •

for $n>2m$, $SH^{*}(M)=0$,

     * •

for $2+m\leq n\leq 2m$, the weak monotonicity condition does not hold.

     * •

for $n=1+m$, $SH^{*}(M)=0$,

     * •

for $1+\frac{m}{2}\leq n<1+m$, $SH^{*}(M)\neq 0$ has rank a multiple of
$1+m-n$,

     * •

for $1\leq n<1+\frac{m}{2}$,
$SH^{*}(M)=\Lambda[\omega_{Q}]/(\omega^{1+m-n}_{Q}-(-n)^{n}t)$ [Theorem 5,
[117]].

  2. (2)

If $c_{1}(TM)(\pi_{2}(M))=0$, then $SH^{*}(M)=0$ [Theorem 6, [117]].

  3. (3)

Let $M$ be a negative line bundle over $B$ with some conditions we refer to
[117]. Define the minimal Chern number $c_{min}$ of a manifold $M$ to be the
number satisfying $c_{1}(TM)(\pi_{2}(M))=c_{min}\mathbb{Z}$. If the minimal
Chern number of $M$ is greater or equal to the rank of $H^{*}(B)$, then
$SH^{*}(M)=0$ [Corollary 7, [117]].

  4. (4)

$SH^{*}(M)=0$ if and only if $\pi^{*}_{M}c_{1}(L)$ is nilpoitent in the
quantum cohomology $QH^{*}(M)$ of $M$. In particular, the quantum cup product
reduces to the ordinary cup product, then $SH^{*}(M)=0$ [Corollary 2, [117]].

  5. (5)

Let $E\to B$ be any complex vector bundle, $L\to B$ a negative line bundle and
$M_{k}$ be the total space of $E\otimes L^{\otimes k}\to B$. Then for $k\gg
0$, $SH^{*}(M_{k})=0$ [Theorem 10, [117]].

######  Example 2.37 (Example p.7, [117]).

Let $M$ be the total space of the canonical bundle over Fano variety, then
$SH^{*}(M)=0$.

######  Proposition 2.38 (Theorem p.3, [115]).

Let $M$ be an almost locally Euclidean space, i.e. a minimal resolution of ADE
singularities on $\mathbb{C}^{2}$, which is the quotient
$\mathbb{C}^{2}/\Gamma$ of $\mathbb{C}^{2}$ by a finite subgroup $\Gamma\in
SU(2)$ or plumbings of cotangent bundles $T^{*}\mathbb{P}^{1}$ according to
ADE Dynkin diagrams(It is known to be a non-compact simply-connected
hyperkähler $4$-manifold). Then for generic $\beta\in H^{2}(M)$,
$SH^{*}(M,d\theta;\underline{\Lambda}_{\tau\beta})=0$.

####  2.6.3. Invariance from Convex Symplectic Lefschetz fibration

Understanding the convex symplectic Lefschetz fibration structure on Kaliman
modification leads the fact that symplectic cohomology is invariant under
Kaliman modification. The theorem of invariance will be used to prove theorem
4.11 below.

######  Proposition 2.39 (Theorem 2.31, [83]).

Let $M$ be an smooth affine variety of dim${}_{\mathbb{C}}M\geq 3$ with
compactification $X$ with ample divisor $D$. Let $Z$ be an irreducible divisor
in $X$ and $p\in Z\cap M$ a smooth point. Assume
$c_{1}\big{(}\mathcal{B}l_{p}M\setminus\widetilde{(Z\cap
M)}\big{)}=c_{1}(M\setminus Z)=0$. Then
$SH^{*}\big{(}\mathcal{B}l_{p}M\setminus\widetilde{(Z\cap M)}\big{)}\cong
SH^{*}(M\setminus Z)$.

####  2.6.4. Vanishing Symplectic Cohomology from Stably-Displaceability

The same idea of vanishing result on relative symplectic cohomology for
stably-displaceable compact subsets of a closed symplectic manifold in
[Theorem 5.12, [89], [134]] works for compact subsets of a completed Liouville
domain. A subset $K$ of a symplectic manifold $(M,\omega)$ is called to be
Hamiltonian displaceable if there is a Hamiltonain symplectomorphism $\phi$
satisfying $\phi(K)\cap K=\emptyset$. A subset $K\subset M$ is called to be
stably displaceable if $K\times S^{1}\subset(M\times
T^{*}S^{1},\omega+d\sigma\wedge d\tau)$ is Hamiltonian displaceable, where
$(\sigma,\tau)\in\mathbb{R}\times S^{1}\cong T^{*}S^{1}$.

######  Proposition 2.40 ([89]).

If a compact subset $\mathbb{Y}\subset M$ is stably-displaceable by
Hamiltonians on $M\times T^{*}S^{1}$, then $SH^{*}(\mathbb{Y}\subset M)=0$.

###### Proof of Proposition 2.40.

Let $\mathcal{K}_{t},c$ be a Hamiltonian on $M\times T^{*}S^{1}$ which
displaces a small neighborhood of $V$ with height $c$. Let $\mathcal{H}_{t},c$
be an admissible time-dependent Hamiltonian with height $c$. Denote
$\phi_{\mathcal{K}_{t}}$ (resp. $\phi_{\mathcal{H}_{t}}$) the one-parameter
family of the Hamiltonian vector field $X_{\mathcal{K}_{t}}$ (resp.
$X_{\mathcal{H}_{t}}$) of $\mathcal{K}_{t}$ (resp. $\mathcal{H}_{t}$). We need
the following lemmas that we provide proofs below.

######  Lemma 2.41 (Proposition 3.3.2, [134]).

There exist an isomorphism of relative symplectic cohomologies after tensoring
with the Novikov field,

$H^{*}\Big{(}\varinjlim_{\begin{subarray}{c}a^{\prime}\searrow-\infty\end{subarray}}\circ\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow\infty\end{subarray}}\circ\text{tel}_{c}[\mathcal{F}_{a^{\prime},b^{\prime}}CF^{\bullet}(\mathcal{K}_{c}+\mathcal{H}_{t}\circ\phi_{\mathcal{K}_{t},c})]\Big{)}\cong
H^{*}\Big{(}\varinjlim_{\begin{subarray}{c}a^{\prime}\searrow-\infty\end{subarray}}\circ\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow\infty\end{subarray}}\circ\text{tel}_{c}[\mathcal{F}_{a^{\prime},b^{\prime}}CF^{\bullet}(\mathcal{H}_{t}\circ\phi_{\mathcal{K}_{t},c})])\Big{)}.$

######  Lemma 2.42 (Proposition 3.3.3, [134]).

For any time-dependent Hamiltonian diffeomorphism $\phi_{\mathcal{K}_{t}}$ on
$M$,

$H^{*}\Big{(}\varinjlim_{\begin{subarray}{c}a^{\prime}\searrow-\infty\end{subarray}}\circ\varprojlim_{\begin{subarray}{c}b^{\prime}\nearrow\infty\end{subarray}}\circ\text{tel}_{c}[\mathcal{F}_{a^{\prime},b^{\prime}}CF^{\bullet}(\mathcal{H}_{t}\circ\phi_{\mathcal{K}_{t},c})]\Big{)}\cong