A morphism $f:(X,d)\to(Y,d^{\prime})$ of matrix factorizations is a
${\mathbb{Z}}/2$-graded $\Gamma$-equivariant map $f$ such that $f\raise
1.0pt\hbox{{$\scriptstyle\circ$}}d^{\prime}=d\raise
1.0pt\hbox{{$\scriptstyle\circ$}}f$.

The resulting dg-category is denoted $\underline{Mat\\!f}_{w,\Gamma,\chi}$.
There is an important difference with the non-equivariant case: this category
is not ${\mathbb{Z}}/2$-graded. This is caused by the action of the character.
Instead, the $\Gamma$-equivariant hom-complexes are

$\displaystyle\begin{split}\mathop{\rm
Hom}\nolimits^{2k}_{\Gamma}(X,Y)&=\mathop{\rm
Hom}\nolimits(X^{0},Y^{0}(\chi^{\otimes k}))^{\Gamma}\oplus\mathop{\rm
Hom}\nolimits(X^{1},Y^{1}(\chi^{\otimes k}))^{\Gamma},\\\ \mathop{\rm
Hom}\nolimits^{2k+1}_{\Gamma}(X,Y)&=\mathop{\rm
Hom}\nolimits(X^{0},Y^{1}(\chi^{\otimes(k+1)}))^{\Gamma}\oplus\mathop{\rm
Hom}\nolimits(X^{1},Y^{0}(\chi^{\otimes k}))^{\Gamma},\end{split}$

where the differentials are defined as in the non-graded case and where
$M^{\Gamma}$ stands for the submodule of $\Gamma$-invariants of $M$, i.e.
$M^{\Gamma}=\\{x\in M\mid\gamma(x)=x\text{ for all }\gamma\in\Gamma\\}$.

######  Example 6.28.

Assume that $\Gamma$ acts on a commutative ring $R$ and let $N$ be a free
$\Gamma$-module of finite rank. Starting from ${{\bf\it f}}\in N^{\Gamma}$ and
a $\Gamma$-equivariant map $\phi:N\to R$ such that $\phi({{\bf\it f}})\in R$
is $\Gamma$-invariant, the two Koszul sequences given in Section 6.2 are
obviously $\Gamma$-equivariant. This yields a $\Gamma$-equivariant matrix
factorization for $w=\phi({{\bf\it f}})$. One can also find a
$\Gamma$-equivariant Koszul matrix factorization $\\{{{\bf\it g}},{{\bf\it
x}}\\}$ for ${\mathbb{C}}$ (with trivial $\Gamma$-action) using a
$\Gamma$-invariant IHS given by $w=\sum g_{i}x_{i}$. Below I’ll restrict
myself to the following situation:

  * •

    * –

The ring $R$ is the polynomial ring $R={\mathbb{C}}[x_{1},\dots,x_{m}]$;

    * –

$\Gamma$ is an finite extension of ${\mathbb{C}}^{*}$ acting diagonally on
${\mathbb{C}}^{m}$, i.e. there are characters

$\hskip 27.0pt\chi_{i}:\Gamma\to{\mathbb{C}}^{*},\quad i=1,\dots,m\text{ with
}\gamma(x_{i})=\chi_{i}(\gamma)x_{i}\text{ for all }\gamma\in\Gamma;$

    * –

There is a character

$\chi:\Gamma\to{\mathbb{C}}^{*}\text{ such that }G:=\mathop{\rm
Ker}\nolimits\chi\text{ is a finite group.}$

  * •

    * –

The polynomial $w\in R={\mathbb{C}}[x_{1},\dots,x_{m}]$ belongs to the
$\chi$-character subspace $R_{\chi}=\\{w\in
R\mid\gamma(w)=\chi(\gamma)w\\},\gamma\in\Gamma$ so that $w$ is invariant
under $G$,

    * –

$w=0$ has an IHS at ${\bf\it 0}$.

####  6.6.B. Hochschild cohomology

In the situation of Example 6.28, the techniques that have been used in
Section 6.5.B can be extended to the $\Gamma$-equivariant case. It is a
special case of a more general situation studied at length in [BFK14].

The calculation in loc. cit. is very technical, but the main ideas are already
present in the non-equivariant setting. There are some important special
features of the equivariant setting to keep in mind:

  1. 1.

In the equivariant setting characters of the group $\Gamma$ play a central
role which, as explain above, implies that the hom-complexes of the matrix
factorizations are not periodical, but graded by the characters and so are the
Hochschild cohomology groups;

  2. 2.

as in the non-equivariant case, Jacobian rings come up, but now involve only
part of the variables associated to a character space.

The expression for equivariant Hochschild cohomology involves an extra
variable $x_{0}$, and so one introduces

  * •

$V:={\mathbb{C}}x_{0}\oplus\cdots\oplus{\mathbb{C}}x_{m}$;

  * •

The $\Gamma$-action extends to $V$ (and hence to
${\mathbb{C}}[x_{0},\dots,x_{m}]$) through the character

(6.12)
$\chi_{0}:=\chi\otimes\prod_{i=1}^{n+1}\chi_{i}^{-1},\quad\gamma(x_{0})=\chi_{0}(\gamma)x_{0}\text{
for all }\gamma\in\Gamma;$

Moreover, for each $\gamma\in\Gamma$ set

  * •

$V^{\gamma}=\\{v\in V\mid\gamma(v)=v\\}=\bigoplus_{i\in
I^{\gamma}}{\mathbb{C}}x_{i}$, where $I^{\gamma}$ is the collection of
integers $i$ in $I:=\\{0,\dots,n+1\\}$ for which $\gamma(x_{i})=x_{i}$.

  * •

$w^{\gamma}$, the restriction of the polynomial $w$ to the polynomial
subalgebra of ${\mathbb{C}}[x_{0},\dots,x_{m}]$ spanned by the
$\gamma$-invariant variables.

  * •

$V_{\gamma}=\bigoplus_{j\in I-I^{\gamma}}{\mathbb{C}}x_{j}$, a
$\Gamma$-invariant complement of $V^{\gamma}$ in $V$.

Tensor-products and duals of $\Gamma$-modules are $\Gamma$-modules. In what
follows, the dual of ${\mathbb{C}}[x_{0},\dots,x_{m}]$ is identified with the
polynomial ring ${\mathbb{C}}[x_{0}^{*},\dots,x_{m}^{*}]$ where the
$x_{j}^{*}$ are given degree $-1$. Hence
${\mathbb{C}}[x_{0},\dots,x_{m},x_{0}^{*},\dots,x_{m}^{*}]$ is a
$\Gamma$-module as well.

In the present situation the following $\Gamma$-submodules of
${\mathbb{C}}[x_{0},\dots,x_{m},x_{0}^{*},\dots,x_{m}^{*}]$ play a role:

  * •

The Jacobian ring $\mathsf{Jac}_{w_{\gamma}}$.

  * •

The character space $\det(V_{\gamma})=\Lambda^{\dim V_{\gamma}}V_{\gamma}$
with character $\kappa_{\gamma}:=\prod_{i\in I_{\gamma}}\chi_{i}$,

  * •

For any $\Gamma$-invariant subring $S$ of
${\mathbb{C}}[x_{0},\dots,x_{m},x_{0}^{*},\dots,x_{m}^{*}]$, the character
space $S_{\chi}:=\\{x\in S\mid\gamma(x)=\chi(x)\cdot x\\}$.

Recall from Section 6.2 that the Koszul sequence associated to the regular
sequence ${{\bf\it w}}=(w_{x_{1}},\dots,w_{x_{m}})$ is denoted by $N({{\bf\it
w}})$, and similarly for the regular sequence ${{\bf\it w}}^{\gamma}$
associated to $w^{\gamma}$. Using these conventions, the result from [BFK14]
in this case reads (see [LU21, Eqn. (5.2])

$\displaystyle\mathsf{HH}^{t}(\underline{Mat\\!f}_{w,\Gamma,\chi})=\bigoplus_{\begin{subarray}{c}\gamma\in\mathop{\rm
Ker}\nolimits\chi,\ l\geq 0\\\
t-\\#I^{\gamma}=2u\end{subarray}}\left(H^{-2l}(N({{\bf\it
w}}^{\gamma})\otimes\kappa_{\gamma}\right)_{(u+\ell)\chi}\bigoplus$
$\displaystyle\displaystyle\bigoplus_{\begin{subarray}{c}\gamma\in\mathop{\rm
Ker}\nolimits\chi,\ l\geq 0\\\
t-\\#I^{\gamma}=2u+1\end{subarray}}\left(H^{-2l-1}(N({{\bf\it
w}}^{\gamma})\otimes\kappa_{\gamma}\right)_{(u+\ell+1)\chi}.$

Since $w=0$ (and hence also $w^{\gamma}=0$) is an isolated IHS at
$\mathbf{0}$, Example 6.6.1 tells that the cohomology of the Koszul sequence
${{\bf\it w}}$ (and also for ${{\bf\it w}}^{\gamma}$) is concentrated in
degree $0$ and so only the terms with $\ell=0$ contribute, i.e., the terms
involving the $H^{0}(N({{\bf\it w}}^{\gamma}))=\mathsf{Jac}_{w^{\gamma}}$.
Unraveling the above formula then yields:

######  Proposition 6.29.

Let $u\in{\mathbb{Z}}$, then

  1. (1)

The contributions to
$\mathsf{HH}^{2u+\\#I^{\gamma}}(\underline{Mat\\!f}_{w,\Gamma,\chi})$ consist
of two types of monomials in the ring
${\mathbb{C}}[x_{0},\dots,x_{m},x_{0}^{*},\dots,x_{m}^{*}]$:

    1. (a)

In case $\gamma(x_{0})\not=x_{0}$, every monomial in $S_{\chi^{\otimes u}}$,
where $S=\mathsf{Jac}_{w^{\gamma}}\otimes\kappa^{*}_{\gamma}$;

    2. (b)

If $\gamma(x_{0})=\ x_{0}$, every monomial in
$(S\otimes{\mathbb{C}}[x_{0}])_{\chi^{\otimes u}}$.

  2. (2)

The contributions to
$\mathsf{HH}^{2u+1+\\#I^{\gamma}}(\underline{Mat\\!f}_{w,\Gamma,\chi})$
consist of every monomial of the form
$x_{0}^{*}\otimes(S\otimes{\mathbb{C}}[x_{0}])_{\chi^{\otimes u}}$ provided
$\gamma(x_{0})=\ x_{0}$.

##  Chapter 7 Bigrading on symplectic cohomology as a contact-invariant

### Introduction

A Gerstenhaber algebra, introduced in Section 7.1, gives a bigrading on
Hochschild cohomology, as explained in (7.2). It turns out that a Gerstenhaber
structure exists on the symplectic cohomology of the symplectic completion of
a Liouville domain with $b_{1}=0$ and with trivializable tangent bundle, as is
the case for the Milnor fiber of an IHS. Under favorable conditions,
enumerated in Theorem 7.6, this structure is a contact invariant of the link.
It follows in particular (cf. Corollary 7.7) that for isolated cDV
singularities $(X,x)$ the resulting grading on each of the vector spaces
$\mathsf{HH}^{d}(X,x)$, $d<0$, is a contact invariant.

###  7.1. Gerstenhaber algebras

A graded complex vector space
${{\mathfrak{g}}}^{*}=\oplus_{k\in{\mathbb{Z}}}{{\mathfrak{g}}}^{k}$, is a
_Gerstenhaber algebra_ if it comes equipped with

  * •

a degree-preserving associative and graded commutative product $\cdot$, that
is, $a\cdot b=(-1)^{|a|\cdot|b|}b\cdot a$, where $|a|,|b|,\dots$, denotes the
degree of $a,b,\dots$.

  * •

a Lie-algebra bracket $[-,-]$ of degree $-1$:

    * –

$[{{\mathfrak{g}}}^{k},{{\mathfrak{g}}}^{\ell}]\subset{{\mathfrak{g}}}^{k+\ell-1}$;

    * –

$[a,b]=-(-1)^{(|a|-1)\cdot(|b|-1)}[b,a]$ (i.e. $[-,-]$ is graded anti-
symmetric);

    * –

$[a,[b,c]]=[[a,b],c]+(-1)^{(|a|-1)(|b|-1)}[b,[a,c]]$ (the Jacobi identity).

  * •

the two products are compatible: $[a,b\cdot
c]=[a,b]c+(-1)^{(|a|-1)|b|}b\cdot[a,c]$ (the Poisson identity).

The subspace ${{\mathfrak{g}}}^{1}$ is a Lie-algebra over ${\mathbb{C}}$.
Moreover, since
$[{{\mathfrak{g}}}^{1}\,,{{\mathfrak{g}}}^{k}]\subset{{\mathfrak{g}}}^{k}$,
one has a degree preserving representation of ${{\mathfrak{g}}}^{1}$ on
${{\mathfrak{g}}}$. Assume now that ${{\mathfrak{g}}}^{1}$ is a finite
dimensional Lie algebra. Then, from the usual theory of Lie-algebras (see e.g.
[Hum72, §15.3]), there is a Cartan subalgebra
${{\mathfrak{h}}}\subset{{\mathfrak{g}}}^{1}$, that is a nilpotent subalgebra
which equals its own normalizer. Cartan subalgebras are unique up to
conjugacy. In case ${{\mathfrak{g}}}^{1}$ is semisimple, a Cartan subalgebra
is the same as a maximal abelian subalgebra.

######  Example 7.1.

Consider the traceless matrices ${{\mathfrak{s}}}{{\mathfrak{l}}}(2)$ of
$2\times 2$-matrices. The $1$-dimensional vector space ${{\mathfrak{h}}}$ of
diagonal matrices is a Cartan subalgebra. The functional
$\alpha:{{\mathfrak{h}}}\to{\mathbb{C}}$ sending $H=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$ to $2$ gives an eigenvalue for the adjoint action of $H$ on
${{\mathfrak{s}}}{{\mathfrak{l}}}(2)$ with eigenvector $\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$. The functional $\alpha$ is called a root of
${{\mathfrak{s}}}{{\mathfrak{l}}}(2)$ . The only other root is $-\alpha$.

Since ${{\mathfrak{g}}}^{1}$ cannot be assumed to be semisimple, the usual
approach with root systems does not apply here. It is still true that for a
finite dimensional representation $\rho:{{\mathfrak{h}}}\to\mathop{\rm
End}\nolimits V$ the Jordan decompositions for the non-zero elements
$x\in{{\mathfrak{h}}}$ are compatible and lead to a common generalized
eigenspace decomposition

$V=\bigoplus_{\lambda}V_{\lambda},\quad V_{\lambda}=\\{v\in
V\mid(\rho(x)-\lambda(x)\text{\rm id})^{n_{\lambda}}v=0,\,\text{ for all
}x\in{{\mathfrak{h}}}\\},$

where $n_{\lambda}$ is some positive integer. Moreover, the set of eigenvalues
$\lambda(x)$ for a fixed $\lambda$ defines a linear function on
${{\mathfrak{h}}}$.

In the semisimple case the $V_{\lambda}$ are genuine eigenspaces and are
directly related to the roots. This provides $V$ with a multi-index grading as
follows. Choose a basis $\\{\alpha_{1},\dots,\alpha_{r}\\}$,
$r=\dim{{\mathfrak{h}}}$ for the roots. Then the eigenvalues occurring in a
${{\mathfrak{h}}}$-representation space $V$ are integral linear combinations
$\lambda=\sum\lambda_{i}\alpha_{i}$ to which one associates the multi-index
$\vec{\lambda}=(\lambda_{1},\dots,\lambda_{r})\in{\mathbb{Z}}^{r}$. This gives
the aimed for grading $V=\oplus V_{\vec{\lambda}}$. In this situation, the
adjoint representation of ${{\mathfrak{h}}}$ on ${{\mathfrak{g}}}^{1}$ gives
the so-called Cartan decomposition
${{\mathfrak{g}}}^{1}={{\mathfrak{h}}}\oplus_{\alpha\not=0}{{\mathfrak{g}}}^{1,\alpha},\quad\dim({{\mathfrak{g}}}^{1,\alpha})=1,\alpha\not=0$,
where now any non-zero $\alpha$ is called a root.

In the general case the $(\lambda=0)$-eigenspace for the adjoint action of
${{\mathfrak{h}}}$ on ${{\mathfrak{g}}}^{1}$ still equals the Cartan
subalgebra. However, instead of a root basis, one can only use a
${\mathbb{C}}$-basis and the multi-index becomes a complex multi-index. In the
applications below, $\dim{{\mathfrak{h}}}=1$ and there is an honest finite
complex grading on each of the ${{\mathfrak{g}}}^{k}$ resulting in the
bigrading

(7.1)
${{\mathfrak{g}}}=\bigoplus_{k\in{\mathbb{Z}},\ell\in{\mathbb{C}}}{{\mathfrak{g}}}^{k,\ell}.$

This bigrading depends on the basis $\alpha$ of ${{\mathfrak{h}}}$. Changing
it by a complex multiple $\mu\cdot\alpha$ replaces the second grading $\ell$
by $\mu\ell$. One calls it a _rescaled grading_.

Recall from § 6.5.A that the Hochschild cohomology $\mathsf{HH}^{*}(A)$ of an
associative algebra $A$ is constructed from a complex $C^{\bullet}(A)$ and if
$Z^{k}(A)$ is the sub-algebra of the $k$-cocyles, there is a a Lie-algebra
structure which comes from a Lie bracket

$Z^{k}(A)\times Z^{\ell}(A)\to Z^{k+\ell-1}(A)$

generalizing the Lie bracket on $Z^{1}(A)$, the so-called _Gerstenhaber
bracket_. See [Ger63] for the (involved) definition. This bracket together
with the cup product structure gives Hochschild cohomology
$\mathsf{HH}^{*}(A)$ the structure of a Gerstenhaber algebra.

######  Example 7.2.

Since Remark 6.26 states that the Hochschild cohomology of the category
$[\underline{Mat\\!f}_{R,w}]$ of matrix factorizations for $w$ can be computed
from a Hochschild complex of $R$-algebras, it has a natural structure of a
Gerstenhaber algebra.

Observe that since $R={\mathbb{C}}[x_{1},\dots,x_{m}]$, the $R$-algebras that
occur here are graded and so $\mathsf{HH}^{*}(\underline{Mat\\!f}_{R,w})$ as
well as $\mathsf{HH}^{*}(\underline{Mat\\!f}_{w,\Gamma,\chi})$ receive an
extra grading. This grading is an integral grading.

The above example motivates to consider the abstract situation of a _graded_
associative algebra, $A$, that is $A=\oplus_{\ell\in{\mathbb{Z}}}A^{\ell}$. As
in the example, the spaces $C^{k}(A)$ then receive an extra grading. The one
on $C^{1}(A)=\mathop{\rm Hom}\nolimits(A,A)$ is to be interpreted as an
operator $E:A\to A$ having integral eigenvalues $\ell$ on the eigenspace
$A^{\ell}$. Such an operator can be seen to be a derivation on $A$ and hence
defines a class $[E]\in\mathsf{HH}^{1}(A)$. The operator $E$ acts on
$Z^{k}(A)$ through the Gerstenhaber-bracket and preserves $B^{k}(A)$. Hence
one gets a second grading on Hochschild cohomology. Usually one shifts the
grading on $A$ to achieve that the bigrading adds up to the degree of the
Hochschild cohomology by setting

(7.2) $Z^{k,\ell}(A):=\\{c\in Z^{k}(A)\mid[E,c]=\ell
c\\}[-\ell],\quad\mathsf{HH}^{k,\ell}(A)=Z^{k,\ell}(A)/B^{k,\ell}(A),$

where $B^{k,\ell}(A)=Z^{k,\ell}(A)\cap B^{k}(A)$. It follows from kernel of
$[E,-]$ on $\mathsf{HH}^{1}(A)$ is exactly $\mathsf{HH}^{1,0}(A)$. If this
space happens to be $1$-dimensional it obviously is a Cartan subalgebra of
$\mathsf{HH}^{1}(A)$ and by unicity of such a subalgebra, one deduces:

######  Proposition 7.3.

Suppose $\dim{\mathsf{HH}^{1,0}(A)}=1$. The adjoint action of the Cartan
subalgebra $\mathsf{HH}^{1,0}(A)\subset\mathsf{HH}^{1}(A)$ on Hochschild
cohomology yields an integral bigrading (7.2) which satisfies

$\mathsf{HH}^{m}(A)=\bigoplus_{k+\ell=m}\mathsf{HH}^{k,\ell}(A).$

######  Example 7.4.

$\mathsf{HH}^{*}(\underline{Mat\\!f}_{R,w})$ as well as
$\mathsf{HH}^{*}(\underline{Mat\\!f}_{w,\Gamma,\chi})$ come from Hochschild
complexes of $R$-algebras with $R={\mathbb{C}}[x_{1},\dots,x_{m}]$.

###  7.2. How Gerstenhaber algebras lead to contact invariants

Recall that by Theorem 5.8 the symplectic cohomology of a Liouville domain $W$
with $c_{1}(W)=0$ has a graded product. More is true:

######  Theorem 7.5.

Let $(W,\omega)$ be a Liouville domain for which $TW$ is trivializable. and
such that $b_{1}(\widehat{W})=0$, where $\widehat{W}$ is the symplectic
completion of $W$. Then $\mathsf{SH}^{*}(W)$ admits a Lie-algebra bracket of
degree $-1$ and, together with the graded product $\mathsf{SH}^{*}(W)$, forms
a Gerstenhaber algebra.

There is an excellent overview of the construction of the Lie bracket in
[EL21, §4.3]. The Gerstenhaber structure then provides a Cartan subalgebra
${{\mathfrak{h}}}\subset\mathsf{SH}^{1}(W)$ which is the generalized
$0$-eigenspace $\mathsf{SH}^{1,0}(W)$ of the adjoint action of
${{\mathfrak{h}}}$ on $\mathsf{SH}^{1}(W)$. Assuming that

(7.3) $\dim{\mathsf{SH}^{1,0}(W)}=1,$

the Gerstenhaber structure gives $\mathsf{SH}^{*}(W)$ a bigraded structure as
in (7.1):

$\mathsf{SH}^{*}(W)=\bigoplus_{k\in{\mathbb{Z}},\ell\in{\mathbb{C}}}\mathsf{SH}^{k,\ell}(W).$

It is worthwhile to observe that here one has
$\mathsf{SH}^{k,\ell}(W)\subset\mathsf{SH}^{k}(W)$, contrary to the bigrading
in Hochschild cohomology discussed in the previous subsection where
$\mathsf{HH}^{k,\ell}(A)\subset\mathsf{HH}^{k+\ell}(A)$ (see Proposition 7.3).

One applies this to contact manifolds $S$ which are symplectically fillable,
say $S=\partial W$, $W$ a Liouville domain and $\widehat{W}$ its symplectic
completion. In § 5.3.C it has been explained that the groups
$\mathsf{SH}^{k}(W)$ for $k<0$ are contact invariants under the assumption
that $(S,\xi)$ is index-positive. Lemma 4.3 in [EL21] states something more
precise, namely, if in addition, every closed Reed orbit $\gamma$ has
Conley–Zehnder index $\geq\max(5-n,n-1)$, then for a suitable almost complex
structure on $W$ these orbits stay away from the cylindrical end of $W$, that
is, close to $S=\partial W$. Together with some supplementary conditions, this
implies that then the Gerstenhaber algebra structure on $\mathsf{SH}^{<0}(W)$
is a contact invariant for $S$:

######  Theorem 7.6 ([EL21, Cor. 4.5]).

Let $S$ be as above and let $W,W^{\prime}$ be Liouville domains of (real)
dimension $2n$ with $\partial W=\partial{W^{\prime}}=S$. Assume

  1. (1)

The Conley–Zehnder index of every closed Reeb orbit $\gamma$ satisfies
$\operatorname{\mu_{\mathsf{C}Z}}({\gamma})\geq\max(5-n,n-1)$;

  2. (2)

$c_{1}(W)=c_{1}(W^{\prime})=0$;

  3. (3)

$W$ and $W^{\prime}$ admit Morse functions all of whose critical points have
index $\not=1$.

Then

  1. (a)

there is an isomorphism of Lie algebras
$f^{1}:\mathsf{SH}^{1}(W)\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\sim\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\mathsf{SH}^{1}(W^{\prime})$,

  2. (b)

for each $d<0$ there is an isomorphism
$f^{d}:\mathsf{SH}^{d}(W)\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\sim\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}\mathsf{SH}^{d}(W^{\prime})$
which intertwines the induced representations given by the adjoint
representation given by the bracket operation of $\mathsf{SH}^{1}(W)$ on
$\mathsf{SH}^{d}(W)$, respectively of $\mathsf{SH}^{1}(W^{\prime})$ on
$\mathsf{SH}^{d}(W^{\prime})$. That is, for each $d<0$, one has a commutative
diagram:

$\textstyle{\mathsf{SH}^{1}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ad$\scriptstyle{f^{1}}$$\textstyle{{{\mathfrak{g}}}{{\mathfrak{l}}}(\mathsf{SH}^{d}(W))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{d}}$$\textstyle{\mathsf{SH}^{1}(W^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ad$\textstyle{{{\mathfrak{g}}}{{\mathfrak{l}}}(\mathsf{SH}^{d}(W^{\prime})).}$

This can be applied to links. Note that in case two Milnor fibers (for
different singularities) are symplectomorphic, its is not clear that the
induced contact structures on the boundary are contactomorphic. The above
result gives conditions which make it possible to read this off from Reeb
orbits near the cylindrical ends. In dimension $3$ one deduces:

######  Corollary 7.7.

Let $\\{f=0\\}\subset{\mathbb{C}}^{4}$ have a normal terminal IHS at the
origin. Then the bigraded symplectic cohomology in negative degrees of its
Milnor fiber $\mathsf{F}_{f}$ is a contact invariant of the link.

###### Proof.

By Proposition 5.16 the minimal discrepancy equals $1$. By Mclean’s theorem
5.19 the Conley–Zehnder index for every closed Reeb orbit is at least
$2=\max(5-3,3-1)$. Since the Milnor fiber is a parallellizable complex
manifold, one has $c_{1}(\mathsf{F}_{f})=0$. Also, since $\mathsf{F}_{f}$ is
diffeomorphic to a handlebody obtained from the $6$-disc by attaching $\mu$
handles of index $3$, by Corollary 1.6 there is Morse function
$\mathsf{F}_{f}\to{\mathbb{R}}$ which has only index $0$ and $3$. So all
conditions are satisfied to apply the preceding theorem. ∎

##  Chapter 8 Symplectic cohomology for invertible matrix singularities

### Introduction

In Section 8.2 it will be shown that symplectic cohomology for the Milnor
fiber of large classes of invertible matrix singularities is the same as
Hochschild cohomology for the category of equivariant matrix factorizations.
But first, in § 8.1, I shall specialize the prescription given in Section
6.6.B to the special case of invertible matrix singularities.

By Corollary 7.7 the bigraded symplectic cohomology in negative degrees of the
Milnor fiber of a $3$-dimensional normal terminal IHS is a contact invariant
of the link. In Section 8.2 it is shown that for cDV-singularities
$\\{w_{A}=0\\}$ of invertible matrix type the Gerstenhaber structure on
symplectic cohomology can be transported to $\mathsf{HH}^{*}(A,\Gamma_{A})$ in
such a way that it preserves the property of being a contact invariant of the
link. This makes this bigrading often computable in these cases.

In Section 8.3 and 8.4, following [EL21, §2.4, §3.1], I shall explain how to
calculate the Hochschild cohomology with its Gerstenhaber structure for
diagonal matrix-singularities in dimension $3$. As just explained, this also
gives contact invariants for the links. These calculations give an indication
of how to proceed in the other cases treated in [EL21].

###  8.1. General prescription

Recall that an invertible matrix
$A=(a_{ij})\in\operatorname{\mathsf{G}L}_{n+1}({{\mathbb{C}}})$ defines the
polynomial $w_{A}({\bf\it x}):=\sum_{k}x_{1}^{a_{k,1}}x_{2}^{a_{k,2}}\cdots
x_{n+1}^{a_{k,n+1}}$ which is an invertible polynomial IHS if the hypersurface
$w_{A}=0$ of ${\mathbb{C}}^{n+1}$ has an isolated singularity at ${\bf\it 0}$.
The entries of $A$ define the group

$\Gamma_{A}:=\\{(t_{0},\dots,t_{n+1})\in({\mathbb{C}}^{*})^{n+2}\mid
t_{1}^{a_{k},1}\cdots t_{n+1}^{a_{k},n+1}=t_{0}\cdots
t_{n+1},k=1,\dots,n+1\\}.$

Since $A$ is invertible, this is in fact a finite group extension of
${\mathbb{C}}^{*}$ which admits the canonical character

$\chi_{A}:\quad\Gamma_{A}\mathop{\vbox{\halign{
#\cr${\scriptstyle\hfil\;\;\quad\;\;\hfil}$\crcr\kern
2.0pt\nointerlineskip\cr\rightarrowfill\crcr}}\;}{\mathbb{C}}^{*},\quad\quad{{\bf\it
t}}:=(t_{0},\dots,t_{n+1})\mapsto t_{0}\cdots t_{n+1}$

whose kernel is the finite group

$\displaystyle G_{A}=\\{{{\bf\it t}}\in({\mathbb{C}}^{\times})^{n+2}\mid
t_{1}^{a_{k,1}}t_{2}^{a_{k,2}}\cdots
t_{n+1}^{a_{k.n+1}}=1,\,k=1,\dots,n+1,\,t_{0}=(t_{1}\cdots t_{n+1})^{-1}\\}.$

Now one can begin to specialize the description of
$\mathsf{HH}^{*}(\underline{Mat\\!f}_{w,\gamma\chi})$ given in § 6.6.B in case
$w=w_{A},\Gamma=\Gamma_{A},\chi=\chi_{A}$. First note that
$({\mathbb{C}}^{\ast})^{n+1}$ acts naturally on the polynomial ring
${\mathbb{C}}[x_{0},x_{1},\dots,x_{n+1}]$ and on

$\widetilde{R}:={\mathbb{C}}[x_{0},\dots,x_{n+1},x_{0}^{*},\dots,x_{n+1}^{*}]$

by coordinate-wise multiplication:

$(t_{0},\dots,t_{n+1})\cdot x_{j}=t_{j}x_{j},\quad(t_{0},\dots,t_{n+1})\cdot
x^{*}_{j}=t_{j}^{-1}x_{j}^{*}.$

This induces an action of $\Gamma_{A}$ on $\widetilde{R}$ through further
characters $\chi_{j}$, $j=0,\dots,n+1$, which on $(t_{0},\dots,t_{n+1})$ take
the value $t_{j}$, that is

$\gamma(x_{j})=\chi_{j}(\gamma)\cdot
x_{j},\quad\gamma(x^{*}_{j})=\chi^{-1}_{j}(\gamma)\cdot x^{*}_{j}.$

One clearly has:

######  Lemma.

$w_{A}$ is a semi-invariant for the $\Gamma_{A}$-action with character
$\chi_{A}$ and $w_{A}$ is invariant under the action of $G_{A}$.

The individual variables $x_{j}$ may or may not be invariant under the action
of $\gamma\in G_{A}$, and one accordingly divides the indexing set
$I=\\{1,\dots,n+1\\}$ in two disjoint subsets $I^{\gamma}$ and $I_{\gamma}$,

$i\in I^{\gamma}\iff x_{i}$ is fixed under the action of $\gamma$,

$i\in I_{\gamma}\iff x_{i}$ is not fixed under the action of $\gamma$.

The polynomial $w_{A}^{\gamma}$ is the trace of $w_{A}$ in the
$\gamma$-invariant polynomial ring. In other words, $w_{A}^{\gamma}$ is
obtained from $w_{A}$ upon setting all $x_{j}$, $j\in I_{\gamma}$, to zero:

$w_{A}^{\gamma}=w_{A}|_{\\{x_{j}=0,\text{ for all }j\in I_{\gamma}\\}}.$

Note that $w_{A}^{\gamma}$ only involves the $\gamma$-invariant variables.
Since $t_{0}=t_{0}\cdots t_{n+1}\cdot(t_{1}\cdots t_{n+1})^{-1}$, the
characters $\chi_{A}$ and $\chi_{j}$, $j=0,\dots,n+1$ satisfy the relation

$\chi_{0}=\chi_{A}\otimes\prod_{i=1}^{n+1}\chi_{i}^{-1},$

one finds oneself exactly in the situation of Section 6.6.B. In the present
situation one has

$\kappa_{\gamma}=\prod_{j\in I_{\gamma}}x_{j}^{*}.$

Since the contributions in the Hochschild cohomology come from monomials
${{\bf\it m}}\in\widetilde{R}$, it is convenient to use the corresponding
monomial characters $\chi_{{\bf\it m}}$ of the full torus
$({\mathbb{C}}^{*})^{n+2}$ given by

$\chi_{{\bf\it m}}:({\mathbb{C}}^{*})^{n+2}\to{\mathbb{C}}^{*},\quad{{\bf\it
t}}\mapsto t_{0}^{b_{0}}\cdots t_{n+1}^{b_{n+1}},\text{ where
}b_{j}=\deg_{x_{j}}({{\bf\it m}})-\deg_{x_{j}^{-1}}({{\bf\it m}}).$

Proposition 6.29 motivates the concept of a $\gamma$-monomial:

######  Definition 8.1.

To $\gamma\in G_{A}$ and $w_{A}$ one associates a set
$M_{\gamma}:=A_{\gamma}\cup B_{\gamma}\cup C_{\gamma}$ of monomials in
$\widetilde{R}$, the _$\gamma$ -monomials_, where

(=case 1(b) of Prop. 6.29)):

The set $A_{\gamma}$ is empty if $\gamma(x_{0})\not=x_{0}$ and if
$\gamma(x_{0})=x_{0}$ one has

$A_{\gamma}=\\{x_{0}^{b_{0}}\cdot P\cdot\prod_{i\in I_{\gamma}}x^{*}_{i}\mid
b_{0}\geq 0\text{ and }P\text{ monomial in
}\in\mathsf{Jac}_{w_{A}^{\gamma}}\\},$

a collection of monomials of involving $x_{0}$, $x_{j}$, $j\in I^{\gamma}$,
$x_{i}^{*}$, $i\in I_{\gamma}$.

(=case 2 of Prop. 6.29):

The set $B_{\gamma}$ is empty if $\gamma(x_{0})\not=x_{0}$ and if
$\gamma(x_{0})=x_{0}$ one has

$B_{\gamma}=\\{x_{0}^{b_{0}}\cdot P\cdot x^{*}_{0}\cdot\prod_{i\in
I_{\gamma}}x^{*}_{i}\mid b_{0}\geq 0\text{ and }P\text{ monomial in
}\in\mathsf{Jac}_{w_{A}^{\gamma}}\\},$

a collection of monomials involving $x_{0}$, $x_{0}^{*}$, $x_{j}$, $j\in
I^{\gamma}$, $x_{i}^{*}$, $i\in I_{\gamma}$.

(=case 1a of Prop. 6.29):

The set $C_{\gamma}$ is empty if $\gamma(x_{0})=x_{0}$ and if
$\gamma(x_{0})\not=x_{0}$ one has

$C_{\gamma}=\\{P\cdot x^{*}_{0}\cdot\prod_{i\in I_{\gamma}}x^{*}_{i}\mid\text{
and }P\text{ monomial in }\in\mathsf{Jac}_{w_{A}^{\gamma}}\\},$

a collection of monomials involving $x_{0}^{*}$, $x_{j}$, $j\in I^{\gamma}$,
$x_{i}^{*}$, $i\in I_{\gamma}$.

The condition that a $\gamma$-monomial ${{\bf\it m}}$ belongs to the
$\chi^{\otimes u}_{A}$-character space translates as $\chi_{{{\bf\it
m}}}=\chi_{A}^{\otimes u}$, suggesting the following

######  Definition 8.2.

A pair $(\gamma,{{\bf\it m}})$, consisting of $\gamma\in\text{ker}(\chi_{A})$
and a $\gamma$-monomial ${{\bf\it m}}$, is said to be a _compatible pair of
weight $u\in{\mathbb{Z}}$_ if $\chi_{{{\bf\it m}}}=\chi_{A}^{\otimes u}$.

To determine the degree of such monomials in the Hochschild cohomology, let me
provisionally introduce the _negativity_

of a $\gamma$-monomial ${{\bf\it m}}$ as the number of variables
$x_{0}^{*},\dots,x_{n+1}^{*}$ appearing in ${{\bf\it m}}$.

So, if $x_{0}$ is not fixed by $\gamma$, the only $\gamma$-monomials in
$C_{\gamma}$ are those involving $x_{0}^{*}$ and no powers of $x_{0}$. In case
$\gamma(x_{0})=x_{0}$, there are two types: The $A$-types which possibly
involve a power of $x_{0}$ but not of $x_{0}^{*}$ while the $B$-types involve
$x_{0}^{*}$ and possibly a power of $x_{0}$. The negativity of a
$\gamma$-monomial of type $A_{\gamma}$ equals $\\#I_{\gamma}$, the total
number of $x_{j}$, $j\in[1,\dots,n+1]$ that are not invariant under $\gamma$.
The other types have negativity $\\#I_{\gamma}+1$. From Proposition 6.29 it
then follows that a compatible pair $(\gamma,{{\bf\it m}})$ of weight $u$ and
negativity $h$ contributes to $\mathsf{HH}^{2u+h}(A,\Gamma_{A})$.

######  Example 8.3.

The group $G_{A}$ is finite. If $|G_{A}|=k$ and if the subgroup $H$ that fixes
each of the variables $x_{1},\dots,x_{n+1}$ has order $\ell$, then
$\\#I_{\gamma}=n+1$ for $\gamma\in G_{A}-H$. Thus $(x_{0}\cdots x_{n+1})^{*}$
is a $\gamma$-monomial of type $B$ $\iff\gamma(x_{0})=x_{0}$, and otherwise is
of type $C$. Hence one always has $(k-\ell)$ $\gamma$-monomials of negativity
$n+2$.

######  Remark 8.4.

$\gamma$-monomials can involve $x_{j}$, $j\geq 1$ but the exponent of such
$x_{j}$ is bounded by $\dim\mathsf{Jac}_{w_{A}}$. This implies that
$\gamma$-monomials of type $C$ have bounded total degree. Since $x_{0}$ can
have arbitrary high exponent for types $A$ and types $B$, the total degree of
such $\gamma$-polynomials can be arbitrarily high.

The group $\Gamma_{A}$ and its canonical character $\chi_{A}$ being defined by
$A$, as in Section 1.5 of the introductory chapter, I shall use simplified
notation:

(8.1)
$\mathsf{HH}^{*}(A,\Gamma_{A}):=\mathsf{HH}^{*}(\underline{Mat\\!f}_{w_{A},\Gamma_{A},\chi_{A}}).$

Summing up one has:

######  Theorem 8.5 ([EL21, Thm. 2.14]).

$\mathsf{HH}^{*}(A,\Gamma_{A})$ is the (possibly infinite dimensional)
${\mathbb{C}}$-vector space with basis the compatible $(\gamma,{{\bf\it
m}})$-pairs. A compatible pair $(\gamma,{{\bf\it m}})$ of weight $u$ and
negativity $h$ contributes to $\mathsf{HH}^{2u+h}(A,\Gamma_{A})$.

###  8.2. Relating symplectic cohomology to Hochschild cohomology

As mentioned in the introduction, calculating symplectic cohomology is in
general difficult. In [EL21] it is explained how one might reduce the
calculation for $n$-dimensional isolated singularities $w_{A}=0$ associated to
invertible matrices $A$, to a purely algebraic one. This explanation is highly
technical; for the benefit of a general readership I shall give here a
simplified account skipping all technical details.

There are two related categories that play the central role here. They are
associated to a perturbation of $w_{A}$ which only has ordinary double point
singularities, a so-called _Morsification_ of $w_{A}$. Such perturbations
arise e.g. in the semi-universal unfolding, which has been briefly discussed
in Section 3.3.C. To the resulting family of hypersurfaces acquiring at most
ordinary double points one can apply the technique of Lefschetz: each double
point gives a vanishing cycle and a vanishing thimble in the total space of
the family. It turns out that this association leads to two corresponding so-
called $A_{\infty}$-algebras ${\mathcal{A}}$ (for the vanishing cycles) and
${\mathcal{B}}$ (for the vanishing thimbles) which only depend on the function
$w_{A}$.111 Here one does not need to know what $A_{\infty}$-algebras or
$A_{\infty}$-categories are. The reader can learn about these structures for
instance in S. Ganatra’s thesis [Gan13]. As outlined in Section 6.5.B, these
algebras can be considered as categories as well, say
$\underline{{\mathcal{A}}}$ and $\underline{{\mathcal{B}}}$. Below a ”duality”
between invertible matrix-singularities plays a role where $A$ is replaced
with its transpose ${A}^{{\mathsf{T}}}$.

In order to understand the formulation of these conjectures, observe that
$\Gamma_{A}$ also fixes the polynomial $x_{A^{\prime}:}=x_{A}+(x_{0}\cdots
x_{n+1})$, one can thus consider the category
$\underline{Mat\\!f}_{w_{A^{\prime},\Gamma,\chi}}$. The following conjectures
play role in this story:

######  Conjecture 8.6.

  1. (A)

The homotopy category $[\underline{Mat\\!f}_{w_{A},\Gamma_{A},\chi_{A}}]$ is
equivalent to the homotopy category $[\underline{{\mathcal{A}}}]$ (associated
to the vanishing thimbles of a morsification of $w_{{A}^{{\mathsf{T}}}}$).
Moreover, the algebra ${\mathcal{A}}$ is _formal_ in the sense that its
cohomology algebra $H^{*}({\mathcal{A}})$ is quasi-isomorphic to
${\mathcal{A}}$.

  2. (B)

The homotopy category
$[\underline{Mat\\!f}_{w_{A^{\prime},\Gamma_{A},\chi_{A}}}]$ is equivalent to
the homotopy category $[\underline{{\mathcal{B}}}]$ (associated to the
vanishing thimbles of a morsification of $w_{{A}^{{\mathsf{T}}}}$).

Recalling the simplified notation (8.1), the announced relation is as follows:

######  Theorem 8.7.

Assume that $w_{A}$ has non-zero amplitude (i.e., $w_{A}$ is not of log-Calabi
type, see Example 1.2.5). If moreover

  1. (1)

$\mathsf{HH}^{2}(A,\Gamma_{A})=0$;

  2. (2)

either Conjecture 8.6.(A) or 8.6(B) holds,

then
$\mathsf{HH}^{*}(A,\Gamma_{A})\simeq\mathsf{SH}^{*}(\mathsf{F}_{w_{{A}^{{\mathsf{T}}}}})$
as Gerstenhaber algebras.

###### Sketch of the proof:.

Assumption (1) implies two crucial results used in the proof:

  * •

linking Hochschild cohomology of the category $\underline{{\mathcal{B}}}$ to
symplectic cohomology of the Milnor fiber of $w_{{A}^{{\mathsf{T}}}}$:

(8.2)
$\mathsf{HH}^{*}(\underline{{\mathcal{B}}})\simeq\mathsf{SH}^{*}(\mathsf{F}_{w_{{A}^{{\mathsf{T}}}}})\quad\text{
(as Gerstenhaber algebras)}.$

This is a consequence of [LU21, Thm. 6.4] together with [Gan13, Thm. 1.1].

  * •

linking $\underline{{\mathcal{A}}}$ and $\underline{{\mathcal{B}}}$:

${\mathsf{A}}\simeq{\mathsf{B}},\quad{\mathsf{B}}=H^{*}(\underline{{\mathcal{B}}}).$

This is a consequence of [LU21, Eq.(1.0) and Section 2]. Consequently
$\mathsf{HH}^{*}(A,\Gamma_{A})=\mathsf{HH}^{*}({\mathsf{B}})$. Assumption (1)
then implies that also ${\mathcal{B}}$ is formal so that

(8.3)
$\mathsf{HH}^{*}({\mathsf{A}})\simeq\mathsf{HH}^{*}({\mathsf{B}})=\mathsf{HH}^{*}(\underline{{\mathcal{B}}}).$

Let me first assume that Conjecture 8.6(A) holds. Hence

$\mathsf{HH}^{*}(A,\Gamma_{A})\simeq\mathsf{HH}^{*}({\mathsf{A}}),\quad{\mathsf{A}}=H^{*}(\underline{{\mathcal{A}}}).$

By (8.3) it thus follows that

(8.4)
$\mathsf{HH}^{*}(A,\Gamma_{A})\simeq\mathsf{HH}^{*}(\underline{{\mathcal{B}}}).$

Moreover, the isomorphisms preserve the Gerstenhaber-algebra structure.
Equations (8.2) and (8.4) yield
$\mathsf{HH}^{*}(A,\Gamma_{A})=\mathsf{SH}^{*}(\mathsf{F}_{w_{{A}^{{\mathsf{T}}}}})$
as Gerstenhaber algebras which proves the theorem.

Next, assume that Conjecture 8.6(B) holds. By [EL21, Theorem 2.15] assumption
(1) implies that the two homotopy categories
$[\underline{Mat\\!f}_{w_{A},\Gamma_{A},\chi_{A}}]$ and
$[\underline{Mat\\!f}_{w_{A^{\prime}},\Gamma_{A},\chi_{A}}]$, are equivalent.
Hence Conjecture 8.6.(B) together with (8.2) then imply the theorem. ∎

The present status of the conjectures is as follows:

######  Proposition 8.8.

Conjecture 8.6(A) holds if

  1. (1)

$A$ is diagonal (cf. [FU11]);

  2. (2)

$A$ is block diagonal and its blocks are either $1$-by-$1$ or $2$-by-$2$ equal
to $\begin{pmatrix}2&1\\\ 0&k\end{pmatrix}$ (cf. [FU13]);

  3. (3)

$A=f(x_{1},x_{2})+\sum_{j=3}^{n+1}x_{j}^{2}$ (cf. [HS20]).

Conjecture 8.6(B) holds if $A$ is associated to an $A$-$D$-$E$-singularity
(any dimension). (cf. [Gam20, LU21, LU22])

###  8.3. The diagonal case

#### On the $\gamma$-monomials

Assuming that $A=\text{diag}(a_{1},\dots,a_{n+1})$, I shall explain how to
find compatible pairs of $\gamma$-monomials. The first task consists of
comparing $\chi_{A}$ and the restriction of the character $\chi_{{\bf\it m}}$
to $\Gamma_{A}$. Recall that

$G_{A}=\mathop{\rm Ker}\nolimits(\chi_{A})=\\{{{\bf\it
t}}\in({\mathbb{C}}^{\times})^{n+2}\mid t_{k}^{a_{k}}=1,\,k=1,\dots,n+1,\text{
and }t_{0}=(t_{1}\cdots t_{n+1})^{-1}\\},$

which is isomorphic to the product
${\bf\it\mu}_{a_{1}}\times\cdots{\bf\it\mu}_{a_{n+1}}$ of $(n+1)$ cyclic
groups, each generated a primitive root of unity. The comparison will be done
using the group

$\widetilde{G}_{A}:=\\{(t_{0},\dots,t_{n+1})\in({\mathbb{C}}^{\times})^{n+2}\mid
t_{j}^{a_{j}}=1,\quad t=1,\dots,n+1\\}$

and the homomorphism

(8.5)
$T:\widetilde{G}_{A}\to\Gamma_{A},\quad(t_{0},\dots,t_{n+1})\mapsto(t_{0}^{m}\cdot(t_{1}\cdots
t_{n+1})^{-1},t_{0}^{\ell/a_{1}}t_{1},\dots,t_{0}^{\ell/a_{n+1}}t_{n+1}),$

where $\ell:=\text{lcm}(a_{1},\dots,a_{n+1})$ and
$m:=\ell-\sum_{i=1}^{n+1}\ell/a_{i}$. Notice that

$\chi_{A}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}T(t_{0},\dots,t_{n+1})=(t_{0}\cdots
t_{n+1})^{\ell}$

which gives a commutative diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\chi_{A}\quad}$$\textstyle{{\mathbb{C}}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bf\it\mu}_{a_{1}}\times\cdots\times{\bf\it\mu}_{a_{n+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{G}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{\quad\chi_{A}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}T\quad}$$\textstyle{{\mathbb{C}}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t\mapsto
t^{\ell}}$$\textstyle{1.}$

To check if a $\gamma$-monomial ${{\bf\it m}}$ is compatible and determine its
weight, one has a simple procedure. This uses the reduced exponents of the
monomial ${{\bf\it m}}$ where $x_{j}^{p_{j}}(x^{*}_{j})^{q_{j}}$ has reduced
exponent $p_{j}-q_{j}$. In other words, write $x_{j}^{-1}$ instead of
$x_{j}^{*}$ and allow negative exponents. (Note that for a $\gamma$-monomial
one has $q_{j}=0$ or $q_{j}=1$.) For simplicity I make the substitution
$x_{j}^{*}=x_{j}^{-1}$ and use the and use the reduced form
$x_{0}^{b_{0}}\cdots x_{n+1}^{b_{n+1}}$. This creates an ambiguitiy only for
$x_{0}$ since for $j\geq 1$, the variables $x_{j}$ and $x_{j}^{*}$ never occur
simultaneously in $\gamma$-monomials.

######  Lemma 8.9.

Let ${{\bf\it m}}=x_{0}^{b_{0}}\cdots x_{n+1}^{b_{n+1}}$ be a
$\gamma$-monomial in reduced form. Then the character $\chi_{{\bf\it m}}$ is a
power of $\chi_{A}$ if and only if integers $m_{1},\dots,m_{n+1}$ exist such
that $b_{i}=b_{0}-m_{i}a_{i}$ for $i=1,\dots,n+1$ and then $\chi_{{\bf\it
m}}=\chi_{A}^{\otimes u}$ with $u=b_{0}-\sum m_{i}$.

###### Proof.

Observe that $\chi_{{\bf\it m}}({{\bf\it t}})=t_{0}^{b_{0}}\cdots
t_{n+1}^{b_{n+1}}$ and so, using Eqn. (8.5) one finds

(8.6) $\chi_{{\bf\it m}}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}T({{\bf\it
t}})=t_{0}^{c_{0}}\cdots t_{n+1}^{c_{n+1}},\quad
c_{0}=mb_{0}+\sum_{i=1}^{n+1}b_{i}\frac{\ell}{a_{i}},\quad
c_{i}=b_{i}-b_{0},\,i=1,\dots,n+1.$

I claim that this is a power of $\chi_{A}$ if $c_{i}\equiv 0\bmod a_{i}$ for
$i=1,\dots,n+1$. In fact, writing

$b_{i}=b_{0}-m_{i}a_{i},\quad i=1,\dots,n+1,$

one finds $c_{0}=\ell(b_{0}-\sum m_{i})$ and so

$\chi_{{\bf\it m}}\raise 1.0pt\hbox{{$\scriptstyle\circ$}}T({{\bf\it
t}})=((t_{0}\cdots t_{n+1})^{\ell})^{b_{0}-\sum m_{i}}=(\chi_{A}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}T({{\bf\it t}}))^{b_{0}-\sum m_{i}}\text{ for
all }{{\bf\it t}}\in\widetilde{G}_{A}.$

Conversely, if $\chi_{{\bf\it m}}\raise
1.0pt\hbox{{$\scriptstyle\circ$}}T({{\bf\it t}})=\chi_{A}(T({{\bf\it
t}}))^{u}=((t_{0}\cdots t_{n+1})^{\ell})^{u}$, then (8.6) implies that
$b_{i}-b_{0}=\ell u$, but $\ell$ is a multiple of $a_{i}$ for all
$i=1,\dots,n+1$ and so $b_{i}-b_{0}\equiv 0\bmod a_{i}$. ∎

As a consequence, if $(\gamma,{{\bf\it m}}=x_{0}^{b_{0}}\cdots x_{n+1}^{b+1})$
is a compatible pair, then its weight equals $u=b_{0}-\sum m_{i}$. Hence the
sole exponent $b_{0}$ determines the other exponents $b_{j}$ as well as the
weight by solving the congruences

(8.7) $b_{i}\equiv b_{0}\bmod a_{i},\quad i=1,\dots,n+1.$

Now by Remark 8.4, the appearance of $x_{j}^{b_{j}}$, $j\geq 1$ is governed by
the Jacobian ring of the polynomial $w_{A}$. In the present situation all
$w_{A}^{\gamma}=\sum_{j\in I^{\gamma}}x_{j}^{a_{j}}$ are diagonal and

$\mathsf{Jac}_{w_{A}^{\gamma}}={\mathbb{C}}\cdot
1\oplus\bigoplus_{\overrightarrow{k_{j}}=\overrightarrow{1}}^{\overrightarrow{a_{j}-1}}{\mathbb{C}}\cdot\prod_{j\in
I^{\gamma}}x_{j}^{k_{j}}.$

######  Examples 8.10.

1\. I claim that
$\dim\mathsf{HH}^{n}(\Gamma_{A})\geq\prod_{i=1}^{n+1}(a_{i}-1)$. This can be
seen by the above procedure, setting $b_{0}=-1$. Then the congruences (8.7)
have a solution $b_{i}=-1$, $m_{i}=0$, $i=1,\dots,n+1$. So $x_{0}^{-1}\cdots
x_{n+1}^{-1}$ is a $\gamma$-polynomial for all $\gamma\in G_{A}$ that fix no
variable $x_{j}$, $j\geq 1$, i.e,
$\gamma\in({\bf\it\mu}_{a_{1}}-\\{1\\})\times\cdots\times({\bf\it\mu}_{a_{n+1}}-\\{1\\})$.
Hence there are $\prod_{i=1}^{n+1}(a_{i}-1)$ such compatible pairs
$(\gamma,x_{0}^{-1}\cdots x_{n+1}^{-1})$ of weight $u=-1$ and negativity
$(n+2)$ which all contribute to
$\mathsf{HH}^{n}(\Gamma_{A})=\mathsf{HH}^{-2+n+2}(\Gamma_{A})$. So its
dimension is at least $\prod_{i=1}^{n+1}(a_{i}-1)$.
2\. Likewise, setting $b_{0}=0$, one finds that $b_{i}=0$, $m_{i}=0$,
$i=1,\dots,n+1$. Hence the reduced form of ${{\bf\it m}}$ equals the constant
polynomial $1$ and so ${{\bf\it m}}=1$, an $A_{\gamma}$-type polynomial, or
${{\bf\it m}}=x_{0}x_{0}^{*}$, a $B_{\gamma}$-polynomial of negativity $1$ and
weight $0$. Hence $\mathsf{HH}^{1}(\Gamma_{A})$ is at least one-dimensional.
If $n=3$, the calculations in Section 8.4 show that this is the only class in
$\mathsf{HH}^{1}(\Gamma_{A})$. By Theorem 8.12 below, since the non-zero class
is represented by $x_{0}x_{0}^{-1}$, it belongs to
$\mathsf{HH}^{1,0}(\Gamma_{A})$ precisely because
$\dim\mathsf{HH}^{1,0}(\Gamma_{A})=1$. This is a central observation which
makes it possible to calculate the contact invariants of the examples of the
links discussed in the last section.
3\. In a similar way, for a fixed $b_{0}>0$, solving the congruences (8.7),
one checks whether ${{\bf\it m}}(b_{0}):=x_{0}^{b_{0}}\cdots
x_{n+1}^{b_{n+1}}$ is an $A_{\gamma}$-monomial for some $\gamma\in G_{A}$. By
construction it yields the compatible pairs and their weights and hence to
which degree in Hochschild cohomology the monomial contributes.

Elaborating the last example, let me outline a practical way to find a
$\gamma$-monomial ${{\bf\it m}}_{A}(b_{0})$ of $A$-type whose $x_{0}$-exponent
is a given positive integer $b_{0}>0$. Taking a look at the $A$-monomials, one
sees that for each $i\in I_{\gamma}$ the variable $x_{i}^{-1}$ appears. So
then $b_{i}=-1$ and this leads to the congruence $b_{0}+1\equiv 0\bmod a_{i}$.
Pick out all $i$ for which this is possible. Then these form $I_{\gamma}$. So
now the complementary set $I^{\gamma}$ is known and one determines the
monomials which span $\mathsf{Jac}_{w_{A}^{\gamma}}$. Now $d_{0}\bmod a_{i}$
gives a unique remainder $b_{i}<a_{i}$ and the required $\gamma$-monomial
becomes

${{\bf\it m}}_{A}(b_{0}):=x_{0}^{b_{0}}\cdot\prod_{j\in
I^{\gamma}}x_{j}^{b_{j}}\cdot\prod_{i\in I_{\gamma}}x_{i}^{-1},$

where one sets $x_{j}^{a_{j}-1}=1$. The associated $B$-monomial then is
${{\bf\it m}}_{B}(b_{0})={{\bf\it m}}_{A}(b_{0})x_{0}x_{0}^{-1}$. Notice that
if indeed $\gamma\in\Gamma_{A}$ exists with $\gamma(x_{0})=x_{0}$ and
$\gamma(x_{i})=x_{i}\iff i\in I^{\gamma}$, the pairs $(\gamma,{{\bf\it
m}}_{A}(b_{0}))$ and $(\gamma,{{\bf\it m}}_{B}(b_{0}))$ yield compatible
pairs.

######  Example 8.11.

Consider $w:=x_{1}^{2}+x_{2}^{3}+x_{3}^{5}+x_{4}^{7}$ and $b_{0}=38$. Then
$b_{0}+1=39\equiv 0\bmod 3$ but $39\not\equiv 0\bmod 2,5,7$ and so
$I_{\gamma}=\\{2\\}$. Hence $\mathsf{Jac}_{w_{\gamma}}$ is spanned by
$x_{3}^{a}x_{4}^{b}$ with $a=0,\dots,3$, $b=1,\dots,5$. Since
$b_{0}-b_{3}\equiv 3-b_{3}\equiv 0\bmod 5$, one has $b_{3}=3$ and likewise
$38-b_{4}\equiv 0\bmod 7$ gives $b_{4}=3$. Hence ${{\bf\it
m}}_{A}(38)=x_{0}^{38}x_{3}^{3}x_{4}^{3}x_{2}^{-1}$.

#### Computing the second grading

Let me now describe (without proof) the bigrading on
$\mathsf{HH}^{*}(A,\Gamma_{A})$ under the identification

$\mathsf{HH}^{*}(\underline{Mat\\!f}_{w_{A}})=\mathsf{HH}^{*}(A,\Gamma_{A}).$

This requires tracing through all of the identifications from Section 8.2.
This is quite involved. The final result is as follows:

######  Theorem 8.12 ([EL21, Lemma 4.6]).

Let $w_{A}=0$ be an $n$-dimensional isolated singularity associated to an
invertible matrix $A$. Suppose that assumption (7.3) holds. Then the bigrading
on the side of symplectic cohomology on the Milnor fibre $\mathsf{F}_{w_{A}}$
given by the representation

$\mathsf{SH}^{1}(w_{A})\rightarrow\bigoplus_{d}\mathfrak{gl}(\mathsf{SH}^{d}(\mathsf{F}_{w_{A}}))$

is scale equivalent to the bigrading by the total exponent of $x_{0}$ on
Hochschild cohomology:

$\mathsf{HH}^{d}(A,\Gamma_{A})=\bigoplus_{q}\mathsf{HH}^{d-q,q}(A,\Gamma_{A}),$

where a $\gamma$-monomial $m$ whose total exponent of $x_{0}$ in $m$ is
$b_{0}$ contributes to the bigraded piece
$\mathsf{HH}^{d-nb_{0},nb_{0}}(A,\Gamma_{A})$.

The bigrading on $\bigoplus_{d<0}\mathsf{SH}^{d}(w_{A})$ (which by Theorem
7.6) is a contact invariant of the link) can therefore be computed in terms of
the $x_{0}$-powers of the contributing $\gamma$-monomials. This information
can be used to distinguish non-isomorphic contact structures on the link of
the Milnor fibre as will be explained below in Section 8.4.B for the example
of diagonal CDV-singularities.

###  8.4. A diagonal cDV-example

I now give the full details for calculations which leads to the results stated
in Example 1.16. Here $(a_{1},a_{2},a_{3},a_{4})=(2,2,2,2k)$.

####  8.4.A. The symplectic cohomology of the Milnor fiber in terms of the
Hochschild cohomology

Recall the statement:

######  Theorem.

$\dim(\mathsf{SH}^{d}(A_{1}(2k)))=\begin{cases}0&d\geq 4\\\ 2k-1&d=3\\\
0&d=2\\\ 1&d\leq 1\end{cases}$

To show this, first note that
$G_{A}={\bf\it\mu}_{1}\times{\bf\it\mu}_{1}\times{\bf\it\mu}_{1}\times{\bf\it\mu}_{2k}$
with action given by222Here $\rho_{k}$ denotes a primitive $k$-th root of
unity.

$\displaystyle((-1)^{b_{1}},(-1)^{b_{2}},(-1)^{b_{3}},(\rho_{2k})^{b_{4}})$
$\displaystyle\cdot(x_{0},x_{1},x_{2},x_{3},x_{4})$
$\displaystyle=((-1)^{b_{1}+b_{2}+b_{3}}\rho_{2k}^{b_{4}}x_{0},(-1)^{b_{1}}x_{1},(-1)^{b_{2}}x_{2},(-1)^{b_{3}}x_{3},(\rho_{2k})^{b_{4}}x_{4}).$

Let me make this more explicit in the following tables. There I use the
convention that $\alpha_{i}$ is the generator of the $i$-th cyclic factor of
$G_{A})$ and $\alpha_{ij}=\alpha_{i}\times\alpha_{j}$. Hence
$\alpha_{1},\alpha_{2},\alpha_{3}$ have order $2$ while
$\alpha_{4}=\rho_{2k}$, and in Table 8.1 only exponents $1,\dots,2k-1$ are
allowed to occur. Observe that $\alpha_{4}^{k}=\rho_{2k}^{k}=-1$ so that
$(-1,-1,-1,\alpha_{4}^{k})$ as well as $1_{4}$, $(-1,-1,1,1),(-1,1,-1,1)$ and
$(1,-1,-1,1)$ fix $x_{0}$. This leads to Table 8.1 and Table 8.2

Table 8.1. $\gamma$ with $\gamma(x_{0})=x_{0}$ $\gamma$ | $1_{3}\times 1$ | $\alpha_{i}\times 1$ | $\alpha_{ij}\times\alpha_{4}^{k}$ | $(-1)_{3}\times\alpha_{4}^{k}$
---|---|---|---|---
other $x_{k}$ fixed | $k=1,2,3,4$ | $k\not=i,k=4$ | $k\not=i,j$ | none
$|I^{\gamma}|$ | $4$ | $3$ | $1$ | $0$
$\dim\mathsf{Jac}_{w_{A}^{\gamma}}$ | $2k-2$ | $2k-2$ | $0$ | $0$
$A_{\gamma}$-polyn. | $x_{0}^{b_{0}}x_{4}^{\ell}$ | $x_{0}^{b_{0}}x_{4}^{\ell}x_{i}^{-1}$ | $x_{0}^{b_{0}}x_{i}^{-1}x_{j}^{-1}x_{4}^{-1}$ | $x_{0}^{b_{0}}\prod_{i=1}^{4}x_{i}^{-1}$
Table 8.2. $\gamma$ with $\gamma(x_{0})\not=x_{0}$ $\gamma$ | $1_{3}\times\alpha_{4}^{\ell}$ | $\alpha_{i}\times\alpha_{4}^{\ell}$ | $\alpha_{ij}\times\alpha_{4}^{\ell}$ | $(-1)_{3}\times\alpha_{4}^{\ell}$ | $\alpha_{ij}\times 1$ | $(-1)_{3}\times 1$
---|---|---|---|---|---|---
$\gamma(x_{k})=x_{k}$ | $k=1,2,3$ | $k\not=i$ | $k\not=i,j$ | none | $k\not=i$ | $k\not=i,j$
$|I^{\gamma}|$ | $3$ | $2$ | $1$ | $0$ | $2$ | $1$
$\dim\mathsf{Jac}_{w_{A}^{\gamma}}$ | 0 | 0 | 0 | 0 | $2k-2$ | $2k-2$
$C_{\gamma}$-polyn. | $x_{4}^{-1}$ | $x_{i}^{-1}x_{4}^{-1}$ | $x_{i}^{-1}x_{j}^{-1}x_{4}^{-1}$ | $\prod_{i=1}^{4}x_{i}^{-1}$ | $x_{i}^{-1}x_{j}^{-1}x_{4}^{\ell}$ | $\prod_{i=1}^{3}x_{i}^{-1}x_{4}^{\ell}$

The next task is to find compatible pairs. This necessitates solving the
equations $b_{0}\equiv b_{i}\bmod 2$ for $i=1,2,3$ and $b_{0}\equiv b_{4}\bmod
2k$ for any given integer $b_{0}>0$. Now, by the euclidean algorithm, one may
write

$b_{0}=2pk+2q+r,\quad p\geq 0,\quad\quad 0\leq q\leq k-1,\quad r=0,1.$

One has to play this off against the $x_{i}$, $1,2,3$, having exponents
$b_{i}=0$ or $b_{i}=-1$.

Let me first find the $A_{\gamma}$-polynomials.

Case 1:

$b_{0}$ is even. Then $r=0$ and $b_{1}=b_{2}=b_{3}=0$ and $b_{4}=2q$, $0\leq
q\leq k-1$. Recalling Lemma 8.9, one finds $m_{1}=m_{2}=m_{3}=pk+q$, $m_{4}=p$
and so the weight is $u=b_{0}-\sum m_{i}=-((k+1)p+q)$. Such polynomials
$x_{0}^{b_{0}}x_{4}^{2q}$ are $A_{\gamma}$-polynomials for $\gamma=1$. These
contribute each to
$\mathsf{HH}^{2u}(A_{1}(2k))=\mathsf{HH}^{-2((k+1)p+q)}(A_{1}(2k))$.

Case 2:

$b_{0}$ is odd (so $r=1$). One sees that $b_{1}=b_{2}=b_{3}=-1$, $b_{4}=2q+1$,
$0\leq q\leq k-1$. Also, $m_{1}=m_{2}=m_{3}=pk+q+1$, $m_{4}=p$ if $q\leq k-2$
and $m_{4}=-1$ if $q=k-1$. Only the last possibility gives an
$A_{\gamma}$-polynomial, namely
$x_{0}^{b_{0}}x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}x_{4}^{-1}$ for
$\gamma=(-1)_{3}\times\alpha_{4}^{k}$ whose weight is $u=-((k+1)p+k+2)$. It
contributes to
$\mathsf{HH}^{2u+4}(A_{1}(2k))=\mathsf{HH}^{-2((k+1)p+k)}(A_{1}(2k))$.

This shows that $\dim\mathsf{HH}^{m}(A_{1}(2k))=1$ for each non-positive even
degree $m$, since $2((k+1)p+q)$, $q=0,\dots,k$ runs over all possible even
numbers (or $0$) because $p$ can be any non-negative integer.

Next, multiplying the above $A_{\gamma}$-polynomials with $x_{0}^{-1}$ gives
$B_{\gamma}$-polynomials which gives the odd negative degrees. For $b_{0}=0$
one only finds $1\in\mathsf{HH}^{0}(A_{1}(2k))$. The only other
$B_{\gamma}$-polynomial (or $C_{\gamma}$-polynomial) is $(x_{0}\cdots
x_{4})^{-1}$ which contributes one dimension to $\mathsf{HH}^{2}(A_{1}(2k))$
for each group element of the form $\gamma=(-1_{3}\times\alpha_{4}^{\ell})$,
$\ell=1,\dots,2k-1$. See Example 8.10.1. The other $C_{\gamma}$-polynomials do
not give contributions.

####  8.4.B. The bigrading and contact invariants

Recall that the link of $A_{1}(2k)$ is diffeomorphic to $S^{2}\times S^{3}$
and its contact structure is denoted $\alpha_{1,k}$. The goal is to show that
the second grading distinguishes these contact structures. The crucial tool is
Theorem 8.12 which asserts that a $\gamma$-monomial
$x^{b_{0}}_{0}x_{1}^{b_{1}}x_{2}^{b_{2}}x_{3}^{b_{3}}x_{4}^{b_{4}}$
contributes to $\mathsf{HH}^{d-3b_{0},3b_{0}}(A,\Gamma_{A})$. So in this case
the unique contribution to degree $-2$ is:

$\begin{cases}x_{0}x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}x_{4}^{-1}\text{ bidegree
}(-5,3)&\text{ for }\alpha_{1,1},\\\ x_{0}^{2}x_{4}^{2}\text{ bidegree
}(-8,6)&\text{ for }\alpha_{1,k},\ (k\geq 2).\end{cases}$

This already distinguishes $\alpha_{1,1}$ from everything else. To make
comparison of the various contact structures easier, one can rescale the
second degree for degrees $d<-2$ coming from the contribution of
$x_{0}^{b_{0}}\cdots x_{4}^{b_{4}}$ to be $4b_{0}$ for $\alpha_{1,1}$ and
$2b_{0}$ for $\alpha_{1,k}$, $k\geq 2$. Then the unique contribution in degree
$-4$ is:

$\begin{cases}x_{0}^{4}\text{ in bidegree }(-20,16)&\text{ for
}\alpha_{1,1},\\\ x_{0}^{3}x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}x_{4}^{-1}\text{ in
bidegree }(-10,6)&\text{ for }\alpha_{1,2},\\\ x_{0}^{4}x_{4}^{4}\text{ in
bidegree }(-12,8)&\text{ for }\alpha_{1,k},(k\geq 3).\end{cases}$

This distinguishes $\alpha_{1,2}$ from the other $\alpha_{1,k}$, $k\neq 2$. To
distinguish $\alpha_{1,k}$ from $\alpha_{1,K}$ with $2\leq k<K$, observe that
the unique contribution to degree $-2k$ is
$x_{0}^{2k-1}x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}x_{4}^{-1}$ in bidegree
$(-6k+2,4k-2)$ for $\alpha_{1,k}$, respectively $x_{0}^{2k}x_{4}^{2k}$ of
bidegree $(-6k,4k)$ for $\alpha_{1,K}$. The result is summarized in the table
below which shows that $\alpha_{1,k}$, and $\alpha_{1,j}$ are not
contactomorphic if $k\not=j$.

To interpret the table, recall that $\dim\mathsf{SH}^{d}(A_{2,k})=1$ for $d<0$
so that in all cases there is one generator in each bidegree so that indeed
the second degree (which is printed in red here) distinguishes the contact
structures $\alpha_{1,k}$ among each other.

Table 8.3. Bidegrees $(d-p,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}p})$ contributing to $\mathsf{SH}^{d}(A_{2,k})$, $d<0$, $d$ even. $k$ | $(d,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}p})$
---|---
$1$ | $(-2,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3})$
$2$ | $(-2,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}6})$
$1$ | $(-4,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}16})$
$2$ | $(-4,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}6})$
$k\geq 3$ | $(-4,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}8})$
$3\leq k\leq K-1$ | $(-2k,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}4k-2})$
$K$ | $(-2k,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}4k})$

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## Index

  * amplitude Examples 1.2
  * Arnold’s unimodal hyperbolic singularity Definition 1.14
  * blow up Example 2.2
  * Brieskorn-Pham singularity item 2, Examples 1.2, §2.5
  * $cA_{n}$-type singularity §2.5, Theorem 3.28
  * canonical
    * divisor §3.1
    * sheaf §3.1
    * singularity Definition 3.1
  * Cartan subalgebra §7.1
  * category
    * compact generator §6.4.C
    * derived — §6.5.A
    * dg- — §6.3.B
    * thick subcategory §6.4.C
    * triangulated — §6.4.C
  * Cohen–Macaulay module §6.1.C
  * compact generator §6.4.C
  * Conley–Zehnder index Definition 5.3
  * contact
    * field §4.2.A
    * form Definition 1.9
    * invariants §1.5.B
    * manifold Definition 1.9
    * structure Definition 1.9
  * contact structure
    * on $\\#_{\ell}S^{2}\times S^{3}$ §3.5
    * on link of Brieskorn singularities item 2
    * on $S^{2}\times S^{3}$ Example 1.16, §8.4.B
    * on $S^{4m+1}$ item 1
    * on $S^{5}$ item 3, §3.5
  * contactomorphism Definition 1.9
  * cotangent bundle item 2, Examples 1.10, Example 4.3
  * critical point
    * non-degenerate — of index $r$ Definition 1.5
  * diagonal bimodule Examples 6.24
  * discrepancy item 2
    * minimal — Remark 3.5, Definition 5.14
  * divisor
    * Cartier — §3.1
    * Weil — §3.1
  * du Val singularity Examples 1.2
  * dynamically convex Definition 5.11
  * equivalence of IHS Definition 1.14
  * exceptional locus Definition 2.1
  * filling
    * Milnor — §1.3
    * strong Milnor — §4.3
    * symplectic — Definition 1.12, §4.2.D
  * Floer
    * cohomology §5.3.B
    * homology §5.3.A
  * Fourier–Mukai transform Examples 6.24
  * Gerstenhaber
    * algebra §7.1
    * bracket §7.1
  * Gray stability §4.2.A
  * Hamiltonian
    * admissible — Definition 5.6
    * flow §4.1.A
    * vector field §4.1.A
  * handlebody §1.2
  * Hironaka
    * resolution theorem of — §2.1
  * Hochschild
    * cohomology §1.5.A, §6.5.A
    * cohomology (equivariant matrix factorization) §6.6.B
    * complex §6.5.B
  * homology sphere Example 2.12
  * hypersurface of contact type §4.2.D
  * identity
    * bimodule Examples 6.24
    * functor Examples 6.24
  * index
    * (highest) minimal – Definition 5.17
  * index positive Definition 5.11
  * Kähler
    * manifold item 2
    * metric item 2
  * knot
    * generalized trefoil – Example 2.15
    * trefoil – Example 1.7
  * Koszul
    * matrix factorization item 2
    * sequence §6.2
    * sign convention §6.3.A
  * linearized return map §4.2.D
  * link item 3
    * of 3-dimensional singularity §2.5
  * link of singularity §1.2
  * Liouville
    * domain Definition 4.15
    * field §4.1.B
  * local class group Definition 3.17
  * log-Calabi–Yau Examples 1.2
  * log-Fano Examples 1.2
  * log-general Examples 1.2
  * manifolds of dimension 5 (Smale’s theorem) Theorem 2.16
  * Maslov
    * cycle §5.1
    * index §5.1
  * matrix factorization Definition 6.1
    * equivariant — §6.6.A
  * McLean’s theorem §5.4
  * Milnor
    * fiber (Lê style) §1.2
    * fiber (of singularity) §1.2
    * filling §1.3
    * number §1.2
    * strong — filling §4.3
  * monodromy §2.2
  * Morse
    * (co)homology §5.3.A
    * function Definition 1.5
    * perfect — function §1.2
  * Morse–Smale datum §5.3.A
  * Morsification (of singularity) §8.2
  * Mumford’s theorem
    * topologically smooth=smooth for surfaces §1.5.B, §2.4
  * negativity (of $\gamma$-polynomial Definition 8.1
  * normal point Definition 2.1
  * Poincaré conjecture §2.4
  * positive symplectic
    * cohomology §5.3.C
  * positive symplectic chains §5.3.C
  * Reeb
    * Conley-Zehnder index of periodic — flow Examples 5.4
    * vector field §4.2.B
  * resolution Definition 2.1
    * crepant — §3.2
    * small — Definition 2.1, §3.3.B
    * small — and symplectic cohomology §3.5
  * Shokurov’s conjecture Conjecture 5.15
  * singularity
    * Brieskorn-Pham — item 2, Examples 1.2, §2.5
    * canonical — Definition 3.1
    * compound du Val (cDV) — Examples 1.2, Definition 3.6
    * du Val — Examples 1.2
    * Gorenstein — Definition 3.1
    * index of — Definition 3.1
    * invertible polynomial — Examples 1.2
    * isolated hypersurface — §1.1
    * locally (${\mathbb{Q}}$)-factorial — item 2
    * Morsification §8.2
    * normal— Definition 2.1
    * of $cA_{n}$-type §2.5, Theorem 3.28
    * of Fermat type §3.5
    * plane curve — §2.3
    * rational — Definition 2.1, §3.4
    * semi-universal unfolding of — §3.3.C
    * surface — §2.4
    * terminal — Definition 3.1
    * topologically smooth — §1.5.B
    * type $A$-$D$-$E$ Examples 1.2
    * unimodal — Example 1.15
  * small resolution §1.5.C, Definition 2.1
  * sphere
    * homology — Example 2.12
    * topological — §2.5
  * stabilization §6.4.A
  * Stiefel manifold Example 1.7
  * strong Milnor filling §4.3
  * surgery
    * elementary – §1.2, §1.2
  * symplectic
    * cohomology §1.3, §5.3.B
    * completion §4.2.D
    * filling Definition 1.12, §4.2.D
    * manifold Definition 1.9, §4.1.A
    * positive — chains §5.3.C
    * positive — cohomology §5.3.C
    * structure Definition 1.9
  * symplectically convex boundary Definition 4.15
  * symplectization §4.2.C
  * symplectomorphism Definition 1.9
  * terminal
    * singularity Definition 3.1
  * topologically smooth singularity §1.5.B
  * unit sphere item 1, Example 4.17, Example 5.21
  * Wang sequence §2.2
  * weighted homogeneous polynomial Examples 1.2, Example 6.12
  * $\gamma$-monomials §8.1