Surface quotient singularities and bigness of the cotangent bundle: Part II

Yohannes D. Asega, Bruno De Oliveira, Michael Weiss

University of Miami
---
1365 Memorial Dr
Miami, FL 33134
e-mail<EMAIL_ADDRESS>
e-mail<EMAIL_ADDRESS>
e-mail<EMAIL_ADDRESS>

###### Abstract

In two parts, we present a bigness criterion for the cotangent bundle of
resolutions of orbifold surfaces of general type. As a corollary, we obtain
the canonical model singularities (CMS) criterion that can be applied to
determine when a birational class of surfaces has smooth representatives with
big cotangent bundle and compare it with other known criteria. We then apply
this criterion to the problem of finding the minimal degrees $d$ for which the
deformation equivalence class of a smooth hypersurface of degree d in
$\mathbb{P}^{3}$ has a representative with big cotangent bundle; applications
to the minimal resolutions of cyclic covers of $\mathbb{P}^{2}$ branched along
line arrangements in general position are also obtained.

The CMS criterion involves invariants of canonical singularities whose values
were unknown. In this part of the work, we describe a method of finding these
invariants and obtain formulas for $A_{n}$-singularities. We also use our
approach to derive several extension results for symmetric differentials on
the complement of the exceptional locus $E$ of the minimal resolution of an
$A_{n}$-singularity; namely, we characterize the precise extent to which the
poles along $E$ of symmetric differentials on the complement are milder than
logarithmic poles.

keywords: $A_{n}$ singularities; symmetric differentials; big cotangent
bundle; extension results; surfaces of general type.

##  0 Introduction

In Part I of this work [AdOW23], we present the (quotient singularities) QS-
bigness criterion and its special case, the (canonical model singularities)
CMS-criterion, concerning bigness of the cotangent bundle of surfaces of
general type $X$. The CMS-criterion for the bigness of the cotangent bundle of
$X$ has two terms: one comes from the topology of the minimal model
$X_{\text{min}}$ of $X$ and the other comes from the contribution of the
singularities in the canonical model $X_{\text{can}}$ of $X$ to the
$m$-asymptotic growth of
$h^{1}(X_{\text{min}},S^{m}\Omega^{1}_{X_{\text{min}}})$. The main purpose of
this paper is to determine this contribution when the singularities are of
type $A_{n}$. Our approach also leads to several extension results for
symmetric differentials on the complement of the exceptional locus of the
minimal resolution of an $A_{n}$ singularity.

In Section 1, we give the background and define the strategy to determine the
$1$st cohomological $\Omega$-asymptotics, ${h}_{\Omega}^{1}(y)$, of a log
terminal (quotient) surface singularity $y$. These are the surface invariants
involved in the QS-bigness criterion. Our approach uses the theory developed
by Wahl [Wah93], Blache [Bla96] and Langer [Lan00] for the Chern classes
(local and global) and the asymptotic Riemann-Roch formulas for orbifold
vector bundles.

The $1$st cohomological $\Omega$-asymptotics, ${h}_{\Omega}^{1}(y)$, of a
surface singularity $y$ is defined by

${h}_{\Omega}^{1}(y)=\liminf_{m\to\infty}\frac{h^{0}(U_{y},R^{1}\sigma_{*}S^{m}\Omega^{1}_{U_{y}})}{m^{3}},$

where $U_{y}$ is a neighborhood germ of the surface singularity $y$ and
$\sigma:\tilde{U}_{y}\to U_{y}$ its minimal resolution. It was shown in the
proof of the QS-criterion (Theorem 2 of Part I) that if $X$ is the minimal
resolution of a surface of general type $Y$ with quotient singularities, then

$\sum_{y\in\operatorname{Sing}(Y)}{h}_{\Omega}^{1}(y)\leq
h^{1}_{\Omega}(X):=\lim_{m\to\infty}\frac{h^{1}(X,S^{m}\Omega^{1}_{X})}{m^{3}}$

The right side $h^{1}_{\Omega}(X)$ is the measurement of the $m$-asymptotic
growth of $h^{1}(X,S^{m}\Omega^{1}_{X})$. The sum on the left side is called
the localized component of $h^{1}_{\Omega}(X)$ and we denote it by
$Lh^{1}_{\Omega}(X)$. It is a lower bound for $h^{1}_{\Omega}(X)$ determined
by the singularities of $Y$.

Our approach to find ${h}_{\Omega}^{1}(y)$ uses the relations between the
local invariants appearing in the comparison between the Euler characteristics
an orbifold vector bundle $V$ on an orbifold surface $Y$ and of a vector
bundle $\tilde{V}$ on its minimal resolution $X$, $\sigma:X\to Y$, with
$V=\sigma_{*}\tilde{V}$. The key relation for us, requires $m$-asymptotic
results for $V=(S^{m}\Omega^{1}_{Y})^{\vee\vee}$ (see section 3.1), is:

$\lim_{m\to\infty}\frac{\hslash^{0}(y,m)+h^{1}(y,m)}{m^{3}}=\frac{1}{3!}\left((c_{2}(y,T_{{X}})-c_{1}^{2}(y,T_{{X}})\right)$

where $h^{1}(y,m):=h^{0}(U_{y},R^{1}\sigma_{*}S^{m}\Omega^{1}_{U_{y}})$,
$c_{2}(y,T_{{X}})$, $c_{1}^{2}(y,T_{{X}})\in\mathbb{Q}$ are the local Chern
numbers of $y$ and

$\hslash^{0}(y,m)=\dim[H^{0}(\tilde{U_{y}}\setminus
E,S^{m}\Omega_{\tilde{U_{y}}}^{1})/H^{0}(\tilde{U_{y}},S^{m}\Omega_{\tilde{U_{y}}}^{1})],$

where $(\tilde{U}_{y},E)$ is the minimal resolution of the neighborhood germ
$(U_{y},y)$.

In Section 2, we present a method to find the invariants $\hslash^{0}(y,m)$
and their asymptotics when $y$ is an $A_{n}$ singularity. We show that for
fixed $n$, $\hslash^{0}(A_{n},m)$ is a polynomial of degree 3 in $m$ up to the
linear term. In fact, it is a quasi-polynomial in $m$, i.e. coefficients are
periodic functions of $m$, with the cubic and quadratic coefficients constant.
The main interest in this work, with respect to the discussion above, lies in
finding the cubic term of such a polynomial. In Theorem 1, we give a closed
formula for the cubic and quadratic coefficients of $\hslash^{0}(A_{n},m)$. We
note that the special case of $A_{1}$ was considered in [BDO06], see [Tho13]
for the cubic coefficient and [BTVA22] for the quasi-polynomial.

######  Theorem 1.

Let $(\tilde{X},E)$ be the minimal resolution of the germ $(X,x)$ of an
$A_{n}$ singularity. Then the following holds for
$\hslash^{0}(A_{n},m)=\dim[H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})/H^{0}(\tilde{X},S^{m}\Omega^{1}_{\tilde{X}})]$:

  * (a)

$\hslash^{0}(A_{n},m)$ is given by a weighted lattice sum over a polygon
$\mathcal{P}_{n}(m)$:

$\hslash^{0}(A_{n},m)=\sum_{\begin{subarray}{c}{\bf
x}=(x_{1},x_{2})\in\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}\\\
x_{1}+(n+1)x_{2}\equiv m\bmod{2}\end{subarray}}h_{n,m}({\bf x})$

where

    * i)

the polygon $\mathcal{P}_{n}(m)$ is symmetric about the $x_{2}$-axis and
$\mathcal{P}_{n}(m)\cap\\{x_{2}\geq 0\\}$ is given by the inequalities
$-x_{1}\leq 0$, $-x_{2}\leq 0$, $x_{1}-(n-1)x_{2}\leq m$ and
$-x_{1}+(n+1)x_{2}\leq m+2$.

    * ii)

the weight function $h_{n,m}({\bf
x})=\min\left\\{\displaystyle\sum_{r=0}^{n-1}\alpha_{n,m,r}({\bf
x}),\beta_{n,m}({\bf x})\right\\}$, with

$\displaystyle\alpha_{n,m,r}({\bf x})$
$\displaystyle=\max\left\\{0,\frac{m-x_{1}+(2r-n+1)x_{2}}{2}\right\\}$
$\displaystyle\beta_{n,m}({\bf x})$
$\displaystyle=\max\left\\{0,m+1-\alpha_{n,m,-1}({\bf x})-\alpha_{n,m,n}({\bf
x})\right\\}.$

  * (b)

$\hslash^{0}(A_{n},m)=\hslash^{0}_{\Omega}(A_{n})m^{3}+3\hslash^{0}_{\Omega}(A_{n})m^{2}+O(m)$,
with

$\displaystyle\hslash^{0}_{\Omega}(A_{n}):=\lim_{m\to\infty}\frac{\hslash^{0}(A_{n},m)}{m^{3}}=\frac{4}{3}\sum^{n}_{j=1}\frac{1}{j^{2}}-\frac{12n^{4}+65n^{3}+117n^{2}+72n}{6(n+1)^{2}(n+2)^{2}}.$

To understand the weight function, some setup needs to be introduced. Let
$\tilde{X}$ be minimal resolution of the $A_{n}$ affine model
$X=\\{xz+y^{n+1}=0\\}\subset\mathbb{C}^{3}$ and $E$ the exceptional locus.
Using the smoothing $\pi:\mathbb{C}^{2}\to X$ we define a 3-gradation on the
algebra $S(\tilde{X}\setminus
E):=\bigoplus_{m=0}^{\infty}H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})$. A differential $w$ has 3-degree
$(\hat{k},i,m)$ if $w\in H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})$ and when $w$ is viewed in
$H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})^{\mathbb{Z}_{n+1}}$ it
vanishes to order $i$ at $0$, for the degree $\hat{k}$ it comes from a block
partition on the $\mathbb{Z}_{n+1}-$ invariant differential monomials, see
Section 2.2.1.

The weight function $h_{n,m}({\bf x})$ gives the dimension of the obstruction
space for differentials in the $(x_{2},x_{1},m)$-graded piece of
$S(\tilde{X}\setminus E)$ to extend regularly along $E$. The polygon
$\mathcal{P}_{n}(m)$ lives on the $(x_{1},x_{2})$-plane, $x_{1}=i$ and
$x_{2}=\hat{k}$. The weight function naturally induces a decomposition of the
polygon $\mathcal{P}_{n}(m)=\cup_{\ell=1}^{4n}\mathcal{P}_{n}^{\ell}(m)$ by
convex polygons where the weight functions $h_{n,m,\ell}({\bf
x}):=h_{n,m}({\bf x})|_{\mathcal{P}_{n}^{\ell}(m)}$ are defined by a single
polynomial of degree 1 in $x_{1}$, $x_{2}$ (and $m$).

The function $\hslash^{0}(A_{n},m)$ being a quasi-polynomial of degree 3 in
$m$ follows from the theory of polynomial weighted lattice sums over convex
polytopes $\mathcal{P}({\bf b})$, where the parameter ${\bf
b}=(b_{1},...,b_{k})$ defines $\mathcal{P}({\bf b})$ via the inequalities,
$\mu_{l}({\bf x})\leq b_{l}$, where the linear forms $\mu_{l}({\bf x})$ are
fixed, but ${\bf b}$ varies (see [Ehr62], [BV97], [BV12]). Each polygon
$\mathcal{P}_{n}^{\ell}(m)$ is of the form $\mathcal{P}({\bf b}(m))$ with
${\bf
b}(m)=(\alpha_{\ell,1}m+\beta_{\ell,1},...,\alpha_{\ell,k}m+\beta_{\ell,k}$
with $k=3,4$. Relevant to the task of finding the period of the quasi-
polynomial $\hslash^{0}(A_{n},m)$ (i.e. the lcm of the periods of the
coefficients of the quasi-polynomial) is that $\alpha_{\ell,i},\beta_{\ell,i}$
and the coefficients of the fixed linear forms $\mu_{l}({\bf x})$ are in
$\mathbb{Q}$ (and we know the denominators).

On future work we will describe non-asymptotic features of the quasi-
polynomial $\hslash^{0}(A_{n},m)$. One such feature is a divisibility
condition for the period. This result allows us to determine the quasi-
polynomials $\hslash^{0}(A_{n},m)$ for low $n$ and use them to investigate the
presence of symmetric differentials of low degrees on the resolution of
certain classes surfaces (e.g. hypersurfaces in $\mathbb{P}^{3}$) with $A_{n}$
type singularities.

In Section 3, we derive applications of our knowledge of
$\hslash^{0}(A_{n},m)$ and the theory developed in Section 2 describing the
extension properties of differentials on $\tilde{X}\setminus E$. The first
application is the main purpose of Part II of this work, that is, finding the
closed formula for the $1$st cohomological $\Omega$-asymptotics
${h}_{\Omega}^{1}(x)$ for $A_{n}$ singularities.

######  Theorem 2.

The 1st cohomological $\Omega$-asymptotics of an $A_{n}$ singularity is given
by:

$\displaystyle
h^{1}_{\Omega}(A_{n})=\frac{n^{5}+19n^{4}+83n^{3}+137n^{2}+80n}{6(n+1)^{2}(n+2)^{2}}-\frac{4}{3}\sum^{n}_{k=1}\frac{1}{k^{2}},$

In Part I [AdOW23] Section 3, Theorem 2 is used in combination with the CMS-
bigness criterion to obtain the strongest known results concerning:

i) the minimum $d_{\min}$ of the degrees $d$ for which the deformation
equivalence class of a smooth hypersurfaces of $\mathbb{P}^{3}$ of degree $d$
has representatives with big cotangent bundle. Theorem 4 of Part I shows
$d_{\min}\leq 8$, moreover in theory with Theorem 2 and considering
resolutions of hypersurfaces with only $A_{n}$ singularities one could achieve
$d_{\min}\leq 6$, but never $d_{\min}=5$;

ii) bigness of the cotangent bundle of the resolutions of cyclic coverings of
$\mathbb{P}^{2}$ branched along line arrangements, see Theorem 5 of Part I.

The second application is concerned with extension results and characterizes:
1) the precise extent to which the poles along $E$ of symmetric differentials
on the complement of the exceptional locus $E$ of the minimal resolution of an
$A_{n}$ singularity are milder than logarithmic poles ([Miy84] 4.14, see also
[Wah93] 4.7 showed that the poles are at most logarithmic); 2) a sufficient
condition for a symmetric differential on the complement of the exceptional
locus $E$ of the minimal resolution of the germ of an $A_{n}$ singularity to
extend holomorphically through $E$.

######  Theorem 3.

Let $\sigma:(\tilde{X},E)\to(X,x)$ and $\pi:(\mathbb{C}^{2},0)\to(X,x)$ be the
minimal resolution and the smoothing of the $A_{n}$ singularity germ $(X,x)$.
Then:

  * (a)

The maximal divisor $D$ such that:

$\displaystyle H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})=H^{0}(\tilde{X},S^{m}\Omega^{1}_{\tilde{X}}(\log
E)\otimes\mathcal{O}_{\tilde{X}}(-D))$

is given by

$\displaystyle
D=\sum_{r=1}^{n}\left(\sum_{j=0}^{\min(r-1,n-r)}\lceil\frac{m-2j}{n+1}\rceil\right)E_{r}$

with $E=U_{r=1}^{n}E_{r}$ the exceptional locus (ordering such that $E_{r}\cap
E_{r+1}\neq\emptyset$).

  * (b)

Let $w\in H^{0}(X\setminus{x},S^{m}\Omega^{1}_{X})$ and $\bar{w}\in
H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})$ with
$\bar{w}|_{\mathbb{C}^{2}\setminus 0}=\pi^{*}w$. Set
$ord(w):=\max_{i}\\{\bar{w}\in\mathfrak{m}^{i}S^{m}\Omega^{1}_{\mathbb{C}^{2}}\\}$,
$\mathfrak{m}$ the maximal ideal at $0$.

Then $\sigma^{*}w$ extends holomorphically to $\tilde{X}$, if ord$(w)\geq nm$.

The second author would like to thank Fedor Bogomolov and Margaridade Mendes
Lopes for comments on this work and the faculty and staff of the Center for
Mathematical Analysis, Geometry and Dynamical Systems of the University of
Lisbon for the support on his stay at Center.

##  1 Background

Let $Y$ be an orbifold surface (i.e. with only quotient singularities) of
general type and $\sigma:X\to Y$ its minimal resolution. In this section we
succinctly describe the invariants of orbifold surface singularities involved
in determining the presence of symmetric differentials on $X$ with an emphasis
on the asymptotics
$\lim_{m\to\infty}\frac{h^{0}(X,S^{m}\Omega^{1}_{X})}{m^{3}}$ (for more
details see Section 1 of Part I of this work [AdOW23]).

Riemann-Roch gives:

$\displaystyle
h^{0}(X,S^{m}\Omega^{1}_{X})=\int_{X}\text{ch}({S^{m}\Omega^{1}_{X}})\text{td}(X)+h^{1}(X,S^{m}\Omega^{1}_{X})-h^{2}(X,S^{m}\Omega^{1}_{X})$
(1.1)

The first term $\int_{X}\text{ch}({S^{m}\Omega^{1}_{X}})\text{td}(X)$ is a
polynomial of degree 3 in $m$ with the coefficients involving the Chern
classes of $X$. The term $h^{2}(X,S^{m}\Omega^{1}_{X})$ vanishes for $m\geq 3$
due to Bogomolov’s vanishing, [Bog78]. The term $h^{1}(X,S^{m}\Omega^{1}_{X})$
has the following decomposition (for further details see proof of Theorem 2 in
Part I [AdOW23]):

$\displaystyle
h^{1}(X,S^{m}\Omega^{1}_{X})=Lh^{1}(X,S^{m}\Omega^{1}_{X})+NLh^{1}(X,S^{m}\Omega^{1}_{X})$
(1.2)

where the localized component (at the singularities):

$\displaystyle
Lh^{1}(X,S^{m}\Omega^{1}_{X}):=\sum_{y\in\text{Sing}(Y)}h^{1}(y,S^{m}\Omega^{1}_{X})$
(1.3)

with
$h^{1}(y,S^{m}\Omega^{1}_{X}):=h^{0}(U_{y},R^{1}\sigma_{*}S^{m}\Omega^{1}_{\tilde{X}})$,
where $U_{y}$ is an affine neighborhood with $U_{y}\cap\text{Sing}(Y)=\\{y\\}$
and the non-localized component
$NLh^{1}(X,S^{m}\Omega^{1}_{X}):=h^{1}(Y,\sigma_{*}S^{m}\Omega^{1}_{X})$.

The QS-bigness criterion (Theorem 2 of Part I) gives the criterion for the
bigness of the cotangent bundle of $X$
($\lim_{m\to\infty}\frac{h^{0}(X,S^{m}\Omega^{1}_{X})}{m^{3}}\neq 0$):

$\displaystyle\sum_{y\in\operatorname{Sing}(Y)}\liminf_{m\to\infty}\frac{h^{1}(y,S^{m}\Omega^{1}_{X})}{m^{3}}+\frac{s_{2}(X)}{3!}>0\implies\Omega^{1}_{X}\text{
big}$ (1.4)

where $\frac{s_{2}(X)}{3!}=\frac{c^{2}_{1}(X)-c_{2}(X)}{6}$ is the cubic
coefficient and the leading term in $m$ of
$\int_{X}\text{ch}({S^{m}\Omega^{1}_{X}})\text{td}(X)$.

The main goal of this work is to provide a method to find
$h^{1}(y,S^{m}\Omega^{1}_{X})$ and determine its asymptotics,
$\lim_{m\to\infty}\frac{h^{1}(y,S^{m}\Omega^{1}_{X})}{m^{3}}$, when $y$ is an
$A_{n}$ singularity (we will see that this limit exists). We use an indirect
approach coming from the theory developed by Wahl [Wah93], Blache [Bla96] and
Langer [Lan00] concerning orbifold vector bundles, their Chern classes, and
Riemann-Roch formulas (with their asymptotics).

Notation. Hereon, $h^{1}(y,m):=h^{1}(y,S^{m}\Omega^{1}_{X})$ and
$h^{1}(A_{n},m):=h^{1}(y,m)$ where $y$ is the singularity $A_{n}$.

For further background on the general theory concerning what follows, see
Section 1 of Part I [AdOW23] (or see [Bla96], [Lan00], [Wah93]).

The difference between the Euler charateristics of the vector bundle
$S^{m}\Omega^{1}_{X}$ on $X$ and of the orbifold vector bundle
$\hat{S}^{m}\Omega^{1}_{Y}:=(\sigma_{*}\Omega^{1}_{X})^{\vee\vee}$ on $Y$ is
measured by the sum over all the singular points $y$ of $Y$ of the local Euler
characteristic of $\hat{S}^{m}\Omega^{1}_{Y}$ at $y$:

$\displaystyle\chi(y,m):=\chi(y,S^{m}\Omega^{1}_{X}):=\hslash^{0}(y,m)+h^{1}(y,m)$
(1.5)

$\displaystyle\hslash^{0}(y,m)$
$\displaystyle:=\dim[H^{0}(\tilde{U_{y}}\setminus
E,S^{m}\Omega_{X}^{1})/H^{0}(\tilde{U_{y}},S^{m}\Omega_{{X}}^{1})]$ (1.6)

The relations between the Euler characteristic of $\hat{S}^{m}\Omega^{1}_{Y}$
on $Y$, the orbifold Euler characteristic of $\hat{S}^{m}\Omega^{1}_{Y}$ on
$Y$, and the orbifold Euler characteristic of $S^{m}\Omega^{1}_{X}$ on $X$,
where the orbifold Euler characteristic of an orbifold vector bundle
${\mathcal{F}}$ on a orbifold $Z$ is
$\chi_{\operatorname{orb}}(Z,{\mathcal{F}}):=\int_{Z}\text{ch}_{\operatorname{orb}}({\mathcal{F}})\text{td}_{\operatorname{orb}}(Z)$
(involving orbifold Chern classes) give:

$h^{1}(y,m)=\mu(y,m)-\chi_{\operatorname{orb}}(y,m)-\hslash^{0}(y,m)$ (1.7)

with:

$\displaystyle\mu(y,m):=\frac{1}{|G_{y}|}\sum_{g\in
G_{y}\setminus\\{\text{Id}\\}}\frac{\text{Tr}(\rho_{\hat{S}^{m}\Omega_{{Y}}^{1}}(g))}{\det(\text{Id}-g)}$
(1.8)

$\chi_{\text{orb}}(y,m):=\frac{s_{2}(y)}{3!}m^{3}-\frac{1}{2}c_{2}(y)m^{2}-\frac{c_{1}^{2}(y)+3c_{2}(y)}{12}m+\frac{c_{1}^{2}(y)+c_{2}(y)}{12}$
(1.9)

where:

i) $\mu(y,m)$ is the contribution that the singularity $y$ gives to the
discrepancy between the Euler characteristic of $\hat{S}^{m}\Omega^{1}_{Y}$
and the orbifold Euler characteristic of $\hat{S}^{m}\Omega^{1}_{Y}$ on $Y$.
$G_{y}\subset GL(2,\mathbb{C})$ is the local fundamental group and
$\rho_{\hat{S}^{m}\Omega_{{Y}}^{1}}$ the representation of $G_{y}$ associated
to the orbifold vector bundle $\hat{S}^{m}\Omega_{{Y}}^{1}$ (see [Bla96] 2.6
for the bijective association of isomorphism classes of representations of
$G_{y}$ to isomorphism classes of germs of orbifold vector bundles at the
quotient singularity with local fundamental group $G_{y}$).

ii) $\chi_{\text{orb}}(y,m)$ is the contribution that the singularity $y$
gives to the discrepancy between the orbifold Euler characteristics of
$S^{m}\Omega^{1}_{X}$ on $X$ and of $\hat{S}^{m}\Omega^{1}_{Y}$ on $Y$.

$c^{2}_{1}(y):=c_{1}^{2}(y,T_{{X}})=c_{1}^{2}(y,\Omega_{{X}}^{1})\in\mathbb{Q},$
$c_{2}(y):=c_{2}(y,T_{{X}})=c_{2}(y,\Omega_{{X}}^{1})\in\mathbb{Q},$
$s_{2}(y)=c_{1}^{2}(y)-c_{2}(y)$

are local Chern numbers at the singularity $y$. They are obtained from the
local Chern classes $c_{i}(y,T_{X})\in
H^{2i}_{\text{dRc}}((\tilde{U}_{y},E),\mathbb{Q})$,
$\tilde{U}_{y}=\sigma^{-1}(U_{y})\subset X$ and dRc stands for de Rham
cohomology with compact support ([Bla96, §3]).

##  2 The invariants $\hslash^{0}(y,m)$ and their asymptotics for $A_{n}$
singularities

###  2.1  $A_{n}$ model

The affine model for the $A_{n}$ singularity that will be used is
$X=\\{xz-y^{n+1}=0\\}\subset\mathbb{C}^{3}$. The affine surface $X$ is the
quotient space obtained from $\mathbb{C}^{2}$ via the diagonal action of
$\mathbb{Z}_{n+1}$ coming from the representation
$\rho:\mathbb{Z}_{n+1}=\langle\tau\rangle\to SL(2,\mathbb{C})$ with:

$\displaystyle\rho(\tau)=\left[\begin{array}[]{cl}\epsilon&0\\\
0&\epsilon^{n}\end{array}\right]$ (2.1)

where $\epsilon$ is a $n+1$-primitive root of unity.

The standard smoothing of ${X}$, $\pi:\mathbb{C}^{2}\to X$, is given by
$\pi(z_{1},z_{2})=(z_{1}^{n+1},z_{1}z_{2},z_{2}^{n+1})$. Let
$\sigma:\tilde{X}\to X$ be the minimal good resolution of $X$, which can be
obtained via successive blow ups of the ambient space $\mathbb{C}^{3}$ at
points infinitesimally near the origin ($n$ blow ups for $n$ odd and $n-1$ for
$n$ even). Let $\varphi:\mathbb{C}^{2}\dashrightarrow\tilde{X}$ be the $n+1$
to 1 rational map $\sigma^{-1}\circ\pi$, whose indeterminacy locus is
$\text{Ind}(\varphi)=\\{(0,0)\\}$.

The resolution $\tilde{X}$ has a covering consisting of $n+1$ open sets
$U_{r}$, $r=0,...,n$, isomorphic to the affine plane. The isomorphisms
$\phi_{r}:\mathbb{C}^{2}\to U_{r}$, with $u_{1}$ and $u_{2}$ coordinaters for
$\mathbb{C}^{2}$, can be chosen such that the following diagram holds:

$\mathbb{C}^{2}$$\mathbb{C}^{2}$$U_{r}\subset$$\tilde{X}$$X=\\{xz-y^{n+1}=0\\}\subset\mathbb{C}^{3}$$\phi_{r}$
$\sigma$ $\pi$,$(z_{1}^{n+1},z_{1}z_{2},z_{2}^{n+1})$ $\varphi_{r}$ $\varphi$

and

$\displaystyle\varphi_{r}^{*}u_{1}=z_{1}^{n+1-r}z_{2}^{-r}\hskip
43.36243pt\varphi_{r}^{*}u_{2}=z_{1}^{r-n}z_{2}^{r+1}$ (2.2)

The exceptional locus of $\sigma$, $E=E_{1}+...+E_{n}$, is a sum of $n$
$(-2)$-curves that intersect transversally with intersection properties given
by the $A_{n}$-Dykin diagram. The relation between the open covering
$\\{U_{r}\\}$ with coordinates $u_{i,r}=u_{i}\circ\phi_{r}^{-1}$, $i=1,2$, and
the exceptional set is given by:

$\hskip 28.90755ptE_{j}\subset U_{j-1}\cup U_{j}\hskip 72.26999ptE_{j}\cap
U_{r}=\emptyset\hskip 7.22743pt\text{if}\hskip 7.22743ptr\not=j-1,j$

$\displaystyle E_{j}\cap U_{j-1}=\\{u_{1,j-1}=0\\}\hskip 36.135ptE_{j}\cap
U_{j}=\\{u_{2,j}=0\\}$ (2.3)

Also relevant is the extension of the exceptional set $\hat{E}=E\cup E_{0}\cup
E_{n+1}$, where $E_{0}=\\{u_{2,0}=0\\}\subset U_{0}$ and
$E_{n+1}=\\{u_{1,n}=0\\}\subset U_{n}$. The following holds for all
$r=0,...,n$:

$\displaystyle\tilde{X}\setminus\hat{E}=U_{r}\setminus\\{u_{1,r}u_{2,r}=0\\}$
(2.4)

Note: To obtain uniformity in the formulas ahead we also consider $r=-1$ and
$r=n+1$, where we define $\varphi_{r}:\mathbb{C}^{2}\to\mathbb{C}^{2}$ for
$r=-1,n+1$ using the same formulas as in (2.2).

The following diagram coming, via restrictions, from diagram (2.2) is also
relevant:

$\mathbb{C}^{*}\times\mathbb{C}^{*}$$\mathbb{C}^{*}\times\mathbb{C}^{*}$$X^{*}:=(\mathbb{C}^{*}\times\mathbb{C}^{*})/\mathbb{Z}_{n+1}$(2.5)$\sigma\circ\phi_{r}$
$\pi$ $\varphi_{r}$

###  2.2 The 3-gradation of the algebra of symmetric differentials
$S(\mathbb{C}^{*}\times\mathbb{C}^{*})$

####  2.2.1 Block partition

The algebra of regular symmetric differentials on
$\mathbb{C}^{*}\times\mathbb{C}^{*}\subset\mathbb{C}^{2}$:

$S(\mathbb{C}^{*}\times\mathbb{C}^{*})=\bigoplus_{m=0}^{\infty}H^{0}(\mathbb{C}^{*}\times\mathbb{C}^{*},S^{m}\Omega_{\mathbb{C}^{2}}^{1})=\mathbb{C}[z_{1},z_{1}^{-1},z_{2},z_{2}^{-1},dz_{1},dz_{2}]$

has the natural bi-gradation with bi-graded pieces:

$S(\mathbb{C}^{*}\times\mathbb{C}^{*})_{(i,m)}=H^{0}(\mathbb{C}^{*}\times\mathbb{C}^{*},S^{m}\Omega^{1}_{\mathbb{C}^{2}})_{(i)}:=\mathbb{C}[z_{1},z_{1}^{-1},z_{2},z_{2}^{-1}]_{(i)}\otimes\mathbb{C}[dz_{1},dz_{2}]_{(m)}$

with $i\in\mathbb{Z}$ and $m\in\mathbb{Z}_{\geq 0}$. Also relevant to us is
the subalgebra $S(\mathbb{C}^{2})=\mathbb{C}[z_{1},z_{2},dz_{1},dz_{2}]$ of
$S(\mathbb{C}^{*}\times\mathbb{C}^{*})$ with induced bi-gradation with bi-
graded pieces
$H^{0}(\mathbb{C}^{2},S^{m}\Omega_{\mathbb{C}^{2}}^{1})_{(i,m)}$,
$i,m\in\mathbb{Z}_{\geq 0}$.

A symmetric differential $w\in S(\mathbb{C}^{*}\times\mathbb{C}^{*})_{(i,m)}$
is said to be bi-homogeneous of $order$ $i$ and $degree$ $m$. The symmetric
differentials $z_{1}^{i_{1}}z_{2}^{i_{2}}(dz_{1})^{m_{1}}(dz_{2})^{m_{2}}$
will be called monomials. We have:

$\displaystyle
z_{1}^{i_{1}}z_{2}^{i_{2}}(dz_{1})^{m_{1}}(dz_{2})^{m_{2}}\text{ is
}\mathbb{Z}_{n+1}\text{-invariant}\iff
i_{1}+m_{1}+n(i_{2}+m_{2})\stackrel{{\scriptstyle\mathclap{\mbox{\tiny{n+1}}}}}{{\equiv}}0$
(2.6)

Denote by $S(\mathbb{C}^{*}\times\mathbb{C}^{*})^{\mathbb{Z}_{n+1}}$ the
subalgebra of $\mathbb{Z}_{n+1}$-invariant differentials.

From diagram (2.5) one obtains for each $r=0,...,n$:

$S(\mathbb{C}^{*}\times\mathbb{C}^{*})^{\mathbb{Z}_{n+1}}$$S(\mathbb{C}^{*}\times\mathbb{C}^{*})$$S(X^{*})$(2.7)$(\sigma\circ\phi_{r})^{*}$
$\cong$ $\pi^{*}$ $\cong$ $\varphi_{r}^{*}$ $\cong$

The goal is to introduce a 3-gradation of
$S(\mathbb{C}^{*}\times\mathbb{C}^{*})$, with graded pieces spanned by
symmetric differential monomials, that is respected by the isomorphisms
$\varphi^{*}_{r}$. The structure of the pullback of symmetric differential
monomials under the isomorphisms
$\varphi_{r}^{*}:S(\mathbb{C}^{*}\times\mathbb{C}^{*})\to
S(\mathbb{C}^{*}\times\mathbb{C}^{*})^{\mathbb{Z}_{n+1}}$ is the motivation
for our choice of 3-gradations appearing below.

The pullback by $\varphi_{r}$ of a monomial of degree $m$, using (2.2), is of
the form:

$\displaystyle\varphi_{r}^{*}(u_{1}^{i_{1}}u_{2}^{i_{2}}(du_{1})^{m-q}(du_{2})^{q})=\sum_{l=0}^{m}c_{ql}(r)z_{1}^{j_{1}(i_{1},i_{2},m,q,r)+l}z_{2}^{j_{2}(i_{1},i_{2},m,q,r)-l}(dz_{1})^{m-l}(dz_{2})^{l}$
(2.8)

where

$\displaystyle j_{1}(i_{1},i_{2},m,q,r)$
$\displaystyle=(n+1-r)i_{1}+(r-n)i_{2}+(n-r)m+(2r-2n-1)q$ $\displaystyle
j_{2}(i_{1},i_{2},m,q,r)$ $\displaystyle=(-r)i_{1}+(r+1)i_{2}+(-r)m+(2r+1)q,$
(2.9)

and the coefficients $c_{ql}(r)$ are determined by the equality:

$\displaystyle\left[(n+1-r)X-rY\right]^{m-q}\left[(r-n)X+(r+1)Y\right]^{q}=\sum_{l=0}^{m}c_{ql}(r)X^{m-l}Y^{l}.$
(2.10)

(all monomials on the right side of (2.8) are $\mathbb{Z}_{n+1}$-invariant).

The expression (2.8) jointly with (2.9) motivates the partition of the set of
all monomials of degree $m$, say in the coordinates $(z_{1},z_{2})$, into the
following $blocks$ of $m+1$ monomials:

$\displaystyle
B_{k,i,m}:=\left\\{z_{1}^{i-k+l}z_{2}^{k-l}(dz_{1})^{m-l}(dz_{2})^{l})\right\\}_{l=0,\dots,m},$
(2.11)

where $i,k\in\mathbb{Z}$. The monomials in a single block are either all
$\mathbb{Z}_{n+1}$-invariant or all not:

$\displaystyle B_{k,i,m}\cap
S(\mathbb{C}^{*}\times\mathbb{C}^{*})^{\mathbb{Z}_{n+1}}=\begin{cases}B_{k,i,m},&\hskip
3.61371pt\text{if }2k{\equiv}i+m\bmod{n+1}\\\ \varnothing,&\hskip
3.61371pt\text{otherwise.}\end{cases}$

When dealing with blocks of $\mathbb{Z}_{n+1}$-invariant monomials a distinct
indexing will be useful. The $\mathbb{Z}_{n+1}$-invariance condition
corresponds to $\hat{k}=\frac{2k-i-m}{n+1}\in\mathbb{Z}$. Set:

$\displaystyle{\hat{B}}_{{\hat{k}},i,m}:=B_{k({\hat{k}}),i,m}\hskip
28.90755ptk(\hat{k})=\frac{i+m}{2}+\frac{n+1}{2}\hat{k}$ (2.12)

The permissible indices $(\hat{k},i,m)$ satisfy:

$\displaystyle(n+1)\hat{k}\equiv i+m\bmod{2}$ (2.13)

Condition (2.13) has a dichotomy:

$\displaystyle\noindent(n+1)\hat{k}\equiv
i+m\bmod{2}\iff\begin{cases}i+m\equiv 0\bmod{2}&\hskip 3.61371ptn\text{
odd}\\\ \hat{k}\equiv i+m\bmod{2}&\hskip 3.61371ptn\text{ even}\end{cases}$

####  2.2.2 3-gradations for $S(\mathbb{C}^{*}\times\mathbb{C}^{*})$ and
$S(\mathbb{C}^{2})^{\mathbb{Z}_{n+1}}$

Set the 3-gradation of $S(\mathbb{C}^{*}\times\mathbb{C}^{*})$ to be:

$\displaystyle
S(\mathbb{C}^{*}\times\mathbb{C}^{*})=\bigoplus_{\begin{subarray}{c}m\in\mathbb{Z}_{\geq
0}\\\ i,k\in\mathbb{Z}\end{subarray}}V_{k,i,m}$ (2.14)

where the graded pieces are:

$V_{k,i,m}:=\text{Span}(B_{k,i,m})$

($V_{k,i,m}V_{k^{\prime},i^{\prime},m^{\prime}}\subset
V_{k+k^{\prime},i+i^{\prime},m+m^{\prime}}$).

We have two variants of a 3-gradation of
$S(\mathbb{C}^{*}\times\mathbb{C}^{*})^{\mathbb{Z}_{n+1}}$:

$\displaystyle
S(\mathbb{C}^{*}\times\mathbb{C}^{*})^{\mathbb{Z}_{n+1}}=\bigoplus_{\begin{subarray}{c}m\in\mathbb{Z}_{\geq
0}\\\ i,k\in\mathbb{Z}\\\ 2k-i-m\equiv 0\bmod{n+1}\end{subarray}}V_{k,i,m}$
(2.15)

and the other using
$\hat{V}_{\hat{k},i,m}:=\text{Span}(\hat{B}_{\hat{k},i,m})$ is:

$\displaystyle
S(\mathbb{C}^{*}\times\mathbb{C}^{*})^{\mathbb{Z}_{n+1}}=\bigoplus_{\begin{subarray}{c}m\in\mathbb{Z}_{\geq
0}\\\ i,\hat{k}\in\mathbb{Z}\\\ (n+1)\hat{k}\equiv
i+m\bmod{2}\end{subarray}}\hat{V}_{\hat{k},i,m}$ (2.16)

For each $r=0,...,n$ the pullback morphisms $\varphi_{r}^{*}$ give the
isomorphisms:

$\displaystyle
S(\mathbb{C}^{*}\times\mathbb{C}^{*})\xrightarrow[]{\varphi_{r}^{*}}S(\mathbb{C}^{*}\times\mathbb{C}^{*})^{\mathbb{Z}_{n+1}}$

that by (2.8) respects the 3-gradations. The relations (2.9) describe which
graded pieces are sent to which graded pieces:

$\displaystyle\varphi_{r}^{*}V_{k-r\left(\frac{2k-m-i}{n+1}\right),i+(n-2r)\left(\tiny{\frac{2k-m-i}{n+1}}\right),m}=V_{k,i,m}$

We will be interested in the following reformulation of the above:

$\displaystyle\varphi_{r}^{*}V_{\hat{k},i,m,r}=\hat{V}_{\hat{k},i,m}$
$\displaystyle
V_{\hat{k},i,m,r}:=V_{\frac{i+m}{2}+(\frac{n+1}{2}-r)\hat{k},i+(n-2r)\hat{k},m}$
(2.17)

The above gradation of $S(\mathbb{C}^{*}\times\mathbb{C}^{*})$ induces a
3-gradation on $S(\mathbb{C}^{2})$ :

$\displaystyle
S(\mathbb{C}^{2})=\bigoplus_{\begin{subarray}{c}i,m\in\mathbb{Z}_{\geq 0}\\\
0\leq k\leq m+i\end{subarray}}V_{k,i,m}^{\text{reg}}$ (2.18)

The graded pieces are:

$\displaystyle V_{k,i,m}^{\text{reg}}:=V_{k,i,m}\cap
H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})=\text{Span}(B_{k,i,m}^{\text{reg}})$

where
$B_{k,i,m}^{\text{reg}}:=\left\\{z_{1}^{i_{1}}z_{2}^{i_{2}}dz_{1}^{m_{1}}dz_{2}^{m_{2}}\in
B_{k,i,m}|i_{1},i_{2}\geq 0\right\\}.$

Remark: The condition $0\leq k\leq m+i$ is equivalent to
$B_{k,i,m}^{\text{reg}}\not=\emptyset$.

We will also use with $j=1,2$:

$B_{k,i,m}^{\text{reg},j}:=\left\\{z_{1}^{i_{1}}z_{2}^{i_{2}}dz_{1}^{m_{1}}dz_{2}^{m_{2}}\in
B_{k,i,m}\mid i_{j}\geq 0\right\\}$

For the algebra $S(\mathbb{C}^{2})^{\mathbb{Z}_{n+1}}$ we will consider the
3-gradation:

$\displaystyle
S(\mathbb{C}^{2})^{\mathbb{Z}_{n+1}}=\bigoplus_{\begin{subarray}{c}i,m\in\mathbb{Z}_{\geq
0}\\\ |\hat{k}|\leq\frac{i+m}{n+1}\\\ (n+1)\hat{k}\equiv
i+m\bmod{2}\end{subarray}}\hat{V}^{\text{reg}}_{\hat{k},i,m}$ (2.19)

The condition $|\hat{k}|\leq\frac{i+m}{n+1}$ is equivalent to
$\hat{B}_{\hat{k},i,m}^{\text{reg}}\not=\emptyset$.

###  2.3 Finding $\hslash^{0}(A_{n},m)$ and its asymptotics

In this section we derive, for any fixed $n$, a formula for the function
(notation of section 2.1):

$\displaystyle\hslash^{0}(A_{n},m)=\dim\left(H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})\middle/H^{0}(\tilde{X},S^{m}\Omega^{1}_{\tilde{X}})\right)$

giving the dimension of the space of obstructions for $m$-differentials on
$\tilde{X}\setminus E$ to extend holomorphically through $E$. We will observe
that $\hslash^{0}(A_{n},m)$ is a quasi-polynomial of degree 3 in $m$. The main
purpose is to determine the leading asymptotics:

$\displaystyle\hslash^{0}_{\Omega}(A_{n})=\lim_{m\to\infty}\frac{\hslash^{0}(A_{n},m)}{m^{3}}$

####  2.3.1 From $\tilde{X}$ to $\mathbb{C}^{2}$

In this section, $\tilde{X}$ is the minimal resolution of the affine model $X$
of the $A_{n}$ singularity as described in section 2.1. We have the
commutative diagram involving the resolution $\sigma$ and the smoothing $\pi$
of $X$:

${(\mathbb{C}^{2},0)}$${(\tilde{X},E)}$${(X,x)}$$\scriptstyle{\varphi}$$\scriptstyle{\pi}$$\scriptstyle{\sigma}$

The map $\varphi$ induces the isomorphisms between the algebras of symmetric
differentials:

$\displaystyle S\left(\tilde{X}\setminus
E\right)\xrightarrow[\cong]{\varphi^{*}}S\left(\mathbb{C}^{2}\right)^{\mathbb{Z}_{n+1}}\text{
, }\text{
}S\left(\tilde{X}\setminus\hat{E}\right)\xrightarrow[\cong]{\varphi^{*}}S\left(\mathbb{C}^{*}\times\mathbb{C}^{*}\right)^{\mathbb{Z}_{n+1}}$

(recall: $\hat{E}=E+E_{0}+E_{n+1}$ as in 2.1). The first isomorphism follows
from

$H^{0}\left(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}}\right)\xleftarrow[\cong]{\sigma^{*}}H^{0}\left({X}\setminus\\{x\\},S^{m}\Omega^{1}_{{X}}\right)\xrightarrow[\cong]{\pi^{*}}H^{0}\left(\mathbb{C}^{2}\setminus\\{0\\},S^{m}\Omega_{\mathbb{C}^{2}}^{1}\right)^{\mathbb{Z}_{n+1}}$

plus the equality

$H^{0}\left(\mathbb{C}^{2},S^{m}\Omega_{\mathbb{C}^{2}}^{1}\right)^{\mathbb{Z}_{n+1}}=H^{0}\left(\mathbb{C}^{2}\setminus\\{0\\},S^{m}\Omega_{\mathbb{C}^{2}}^{1}\right)^{\mathbb{Z}_{n+1}}$

due to the reflexivity of the sheaf $S^{m}\Omega_{\mathbb{C}^{2}}^{1}$. The
second isomorphism follows from similar arguments.

We make use of the following diagram:

${H^{0}(\tilde{X}\setminus\hat{E},S^{m}\Omega^{1}_{\tilde{X}})}$${H^{0}(\mathbb{C}^{*}\times\mathbb{C}^{*},S^{m}\Omega^{1}_{\mathbb{C}^{2}})^{{}^{\mathbb{Z}_{n+1}}}}$${H^{0}(\mathbb{C}^{*}\times\mathbb{C}^{*},S^{m}\Omega^{1}_{\mathbb{C}^{2}})}$${H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})}$${H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})^{{}^{\mathbb{Z}_{n+1}}}}$${H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})}$$\scriptstyle{\varphi^{*}}$$\scriptstyle{\cong}$$\scriptstyle{\varphi_{r}^{*}}$$\scriptstyle{\cong}$$\scriptstyle{\varphi_{r}^{*}}$$\scriptstyle{\varphi^{*}}$$\scriptstyle{\cong}$

to turn the task of finding $\hslash^{0}(A_{n},m)$ from the spaces $\tilde{X}$
and $\tilde{X}\setminus E$ to $\mathbb{C}^{2}$, via:

$\displaystyle\hslash^{0}(A_{n},m)=\dim\left(H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})^{{}^{\mathbb{Z}_{n+1}}}\middle/\bigcap_{r=0}^{n}\varphi_{r}^{*}H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})\right)$
(2.20)

The above equality holds since a differential $w\in H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})$ has no poles along $E_{r}$ and $E_{r+1}$ if
and only if
$\varphi^{*}w\in\varphi_{r}^{*}H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})$.

Notice that whilst for each $r$,
$\varphi_{r}^{*}H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})\not\subset
H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})$ (differentials will be
$\mathbb{Z}_{n+1}-invariant$ but not necessarily holomorphic along
$\\{z_{1}z_{2}=0\\}$), one has

$\bigcap_{r=0}^{n}\varphi_{r}^{*}H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})\subset
H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})$

since
$w\in\varphi_{0}^{*}H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})\cap\varphi_{n}^{*}H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})$
is $w=\varphi^{*}\tilde{w}$ with $\tilde{w}\in H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})$.

####  2.3.2 3-gradation breakdown of $\hslash^{0}(A_{n},m)$

The 3-gradation breakdown of $\hslash^{0}(A_{n},m)$ follows from the the
pullback mappings, $\varphi_{r}^{*}$, sending graded piece to graded piece of
the 3-gradation of
$H^{0}(\mathbb{C}^{*}\times\mathbb{C}^{*},S^{m}\Omega^{1}_{\mathbb{C}^{2}})$
given in (2.14).

Let $w\in
H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})^{{}^{\mathbb{Z}_{n+1}}}$.
Since no monomial belongs to two blocks that generate two distinct graded
pieces, the following are equivalent:

  1. i)

$(\varphi^{*})^{-1}w$ extends regularly along $E$

  2. ii)

$(\varphi^{*})^{-1}w_{\hat{k},i,m}$ extends regularly along $E$, $\forall$
$w_{\hat{k},i,m}$ in the $(\hat{k},i,m)$-decomposition of $w$.

Also relevant to this equivalence and what will follow is the decomposition
for each $r=0,...,n$:

$\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }(\varphi_{r}^{*})^{-1}w_{\hat{k},i,m}=\sum
w_{\hat{k},i,m,r}\text{ },\text{ }\text{ }\text{ }\text{ }\text{
}w_{\hat{k},i,m,r}\in V_{\hat{k},i,m,r}$

with this decomposition, the following statements are equivalent to
$(\varphi^{*})^{-1}w_{\hat{k},i,m}$ extending regularly along $E$:

  1. a)

$w_{\hat{k},i,m,r}\in V^{\text{reg}}_{\hat{k},i,m,r}$, for $r=0,...,n$

  2. b)

$w_{\hat{k},i,m,r}\in V^{\text{reg,1}}_{\hat{k},i,m,r}$, for $r=-1,...,n$

  3. c)

$w_{\hat{k},i,m}\in\bigcap_{r=-1}^{n}\varphi_{r}^{*}V_{\hat{k},i,m,r}^{\operatorname{reg},1}$

See the note below (2.4) for the meaning of the case $r=-1$. The artificial
case of $r=-1$ in (b) and (c) will be used to streamline formulas. We observe
that
$\varphi_{r-1}^{*}V^{\text{reg,1}}_{\hat{k},i,m,r-1}=\varphi^{*}_{r}V^{\text{reg,2}}_{\hat{k},i,m,r}$,
for $r=0,...,n+1$.

######  Proposition 1.

Let $(\tilde{X},E)$ be the germ of the minimal resolution of $(X,x)$ the germ
of an $A_{n}$ singularity. Then:

$\displaystyle\hslash^{0}(A_{n},m)=\sum_{\begin{subarray}{c}0\leq i\leq
mn-1\\\ |\hat{k}|\leq\frac{i+m}{n+1}\\\ (n+1)\hat{k}\equiv
i+m\bmod{2}\end{subarray}}\hslash^{0}(A_{n},\hat{k},i,m)$ (2.21)

where
$\hslash^{0}(A_{n},\hat{k},i,m):=\dim\left(\frac{\hat{V}_{\hat{k},i,m}^{\text{reg}}}{\bigcap_{r=-1}^{n}\varphi_{r}^{*}V_{\hat{k},i,m,r}^{\operatorname{reg},1}}\right)$.

Note. The inclusion
$\bigcap_{r=-1}^{n}\varphi_{r}^{*}V_{\hat{k},i,m,r}^{\operatorname{reg},1}\subset\hat{V}_{\hat{k},i,m}^{\text{reg}}$
holds since
$\hat{V}_{\hat{k},i,m}^{\text{reg}}=\varphi_{-1}^{*}V_{\hat{k},i,m,-1}^{\operatorname{reg},1}\cap\varphi_{n}^{*}V_{\hat{k},i,m,n}^{\operatorname{reg},1}$.

###### Proof.

The result follows from (2.20), the decomposition:

$H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})^{{}^{\mathbb{Z}_{n+1}}}=\bigoplus_{\begin{subarray}{c}i\in\mathbb{Z}_{\geq
0}\\\ |\hat{k}|\leq\frac{i+m}{n+1}\\\ (n+1)\hat{k}\equiv
i+m\bmod{2}\end{subarray}}\hat{V}^{\text{reg}}_{\hat{k},i,m}$

and from the equivalences i) $\iff$ ii) and a) $\iff$ b) $\iff$ c) that imply:

$\bigcap_{r=0}^{n}\varphi_{r}^{*}H^{0}(\mathbb{C}^{2},S^{m}\Omega^{1}_{\mathbb{C}^{2}})=\bigoplus_{\begin{subarray}{c}i\in\mathbb{Z}_{\geq
0}\\\ |\hat{k}|\leq\frac{i+m}{n+1}\\\ (n+1)\hat{k}\equiv
i+m\bmod{2}\end{subarray}}\bigcap_{r=-1}^{n}\varphi_{r}^{*}V_{\hat{k},i,m,r}^{\operatorname{reg},1}$

For the upper bound $i\leq mn-1$, see Theorem 3(b). ∎

####  2.3.3 Independence of the conditions to have no poles along the $E_{r}$
and the formula for $h^{0}(A_{n},\hat{k},i,m)$

It follows from Proposition 1 that we need to get a handle on the subspaces
$\varphi_{r}^{*}V_{\hat{k},i,m,r}^{\text{reg},1}\subset\hat{V}_{\hat{k},i,m}$,
$r=-1,...,n$ and determine how they intersect each other.

We will show that the subspaces
$\varphi_{r}^{*}V_{\hat{k},i,m,r}^{\text{reg},1}$ of $\hat{V}_{\hat{k},i,m}$
are in general position, which implies that the dimension of their
intersection is the expected dimension. It also means that the conditions for
a given differential not to have poles along the components $E_{r}$ are
independent of each other.

######  Lemma 1 (Independence of the conditions to have no poles along the
$E_{r}$).

The subspaces
$\varphi_{r}^{*}V_{\hat{k},i,m,r}^{\operatorname{reg,1}}\subset\hat{V}_{\hat{k},i,m}$,
$r=-1,...,n$, of $\hat{V}_{\hat{k},i,m}$ are in general position.

###### Proof.

Set
$\mu_{q}:=z_{1}^{i_{1}(\hat{k},i,m)+q}z_{2}^{i_{2}(\hat{k},i,m)-q}(dz_{1})^{m-q}(dz_{2})^{q}$,
$q=0,...,m$, the monomials spanning $\hat{V}_{\hat{k},i,m}$ and
$\mu_{q,r}:=u_{1}^{i_{1}(\hat{k},i,m,r)+q}u_{2}^{i_{2}(\hat{k},i,m,r)-q}(du_{1})^{m-q}(du_{2})^{q}$,
$q=0,...,m$, the monomials spanning ${V}_{\hat{k},i,m,r}$.

Consider the commutative diagram of isomorphisms:

${V_{\hat{k},i,m,r}}$${\hat{V}_{\hat{k},i,m}}$${\mathbb{C}[X,Y]_{(m)}}$$\scriptstyle{\varphi_{r}^{*}}$$\scriptstyle{\Psi_{r}}$$\scriptstyle{\Psi}$

where the $\mathbb{C}$-linear map $\Psi$ is defined by
$\Psi(\mu_{q})=X^{m-q}Y^{q}$.

From (2.8) and (2.10) of section 2.2, it follows that:

$\Psi_{r}(\mu_{r,q})=\left[(n+1-r)X-rY\right]^{m-q}\left[(r-n)X+(r+1)Y\right]^{q}$

and hence the zero locus $Z\left(\Psi_{r}(\mu_{r,q})\right)=(m-q)\text{
p}_{r,1}+q\text{ p}_{r,2}\subset\mathbb{P}^{1}$, where
$\text{p}_{r,1}=[r:n+1-r]$ and $\text{p}_{r,2}=[r+1:r-n]$.

The monomials $\mu_{r,q}$ with no poles along $E_{r+1}\cap
U_{r}=\\{u_{1}=0\\}$, generating $V^{reg,1}_{\hat{k},i,m,r}$, have
$q\geq\max\\{0,-i_{1}(\hat{k},i,m,r)\\}$, hence:

$\displaystyle\Psi_{r}(V^{reg,1}_{\hat{k},i,m,r})=\big{\\{}P\in\mathbb{C}[X,Y]\text{
}\big{|}\text{ }Z(P)\geq\max\\{0,-i_{1}(\hat{k},i,m,r)\\}\text{
p}_{r,2}\big{\\}}$

Since the points $\text{p}_{r,2}$, $r=-1,...,n$, are all distinct, the
membership conditions for $P\in\Psi_{r}(V^{reg,1}_{\hat{k},i,m,r})$,
$r=-1,...,n$, are independent. Therefore the subspaces
$\Psi_{r}(V^{reg,1}_{\hat{k},i,m,r})\subset\mathbb{C}[X,Y]_{(m)}$ are in
general position and the result follows.

∎

######  Corollary 1.

For each triple $(\hat{k},i,m)$, the contribution
$\hslash^{0}(A_{n},\hat{k},i,m)$ to $\hslash^{0}(A_{n},m)$ is:

$\displaystyle\hslash^{0}(A_{n},\hat{k},i,m)=\min\left\\{\sum_{r=0}^{n-1}\operatorname{codim}(V_{\hat{k},i,m,r}^{{}_{\operatorname{reg},1}},V_{\hat{k},i,m,r})\text{
},\text{ }\dim\hat{V}_{\hat{k},i,m}^{{}_{\operatorname{reg}}}\right\\}$ (2.22)

where:

  * i)

$\operatorname{codim}(V_{\hat{k},i,m,r}^{{}_{\operatorname{reg},1}},V_{\hat{k},i,m,r})=\max\left\\{0,\frac{m-i}{2}+\frac{2r-n+1}{2}\hat{k}\right\\}\leq
m$ $\operatorname{(2.23)}$

  * ii)

$\dim\hat{V}_{\hat{k},i,m}^{{}_{\text{reg}}}=\max\left\\{0,m+1-\sum_{r=-1,n}\operatorname{codim}(V_{\hat{k},i,m,r}^{{}_{\operatorname{reg},1}},V_{\hat{k},i,m,r})\right\\}$

###### Proof.

The formula (2.22) follows directly from the subspaces
$\varphi_{r}^{*}V_{\hat{k},i,m,r}^{{}_{\operatorname{reg},1}}$ of
$\hat{V}_{\hat{k},i,m}$ being in general position (lemma 1) plus the fact that
$\hslash^{0}(A_{n},\hat{k},i,m)=\operatorname{codim}(\bigcap_{r=-1}^{n}\varphi_{r}^{*}V_{\hat{k},i,m,r}^{\operatorname{reg},1}\text{
},\text{ }\hat{V}_{\hat{k},i,m}^{{}_{\operatorname{reg}}})$.

Towards the expression (2.30), we have that the monomials spanning
$V_{\hat{k},i,m,r}$ are $u_{1}^{i_{1}+l}u_{2}^{i_{2}-l}dz_{1}^{m-l}dz_{2}^{l}$
where $l=0,...,m$ and $i_{1}=\frac{i-m}{2}+\frac{-2r+n-1}{2}\hat{k}$ (a
consequence of (2.17). Hence there are $\max\\{0,-i_{1}\\}$ monomials which
are not regular along $\\{u_{1}=0\\}$. Note that in the range
$|\hat{k}|\leq\lfloor\frac{i+m}{n+1}\rfloor$ we have $\max\\{0,-i_{1}\\}\leq
m$.

Identity ii) follows from the note after proposition 1 and lemma 1. ∎

####  2.3.4 The geometry of $h^{0}(A_{n},\hat{k},i,m)$

In the previous section, the function $\hslash^{0}(A_{n},\hat{k},i,m)$ was
fully determined via (2.22) and (2.23) of corollary 1. To understand
$\hslash^{0}(A_{n},m)$ the geometric properties of the function
$\hslash^{0}(A_{n},\hat{k},i,m)$, when $n$ and $m$ are fixed, play a important
role and this section uncovers them.

The function $\hslash^{0}(A_{n},\hat{k},i,m)$ is an even function relative to
$\hat{k}$, a consequence of
$\operatorname{codim}(V_{\hat{k},i,m,r}^{{}_{\operatorname{reg},1}},V_{\hat{k},i,m,r})=\operatorname{codim}(V_{-\hat{k},i,m,n-1-r}^{{}_{\operatorname{reg},1}},V_{-\hat{k},i,m,n-1-r})$.

Fix $n$, using corollary 1 we associate to each $m\geq 0$ a polygonal region
$\mathcal{P}_{n}(m)$ in the $(i,\hat{k})-plane$. The polygon
$\mathcal{P}_{n}(m)$ is the closure of the region where
$\hslash^{0}(A_{n},\hat{k},i,m)$ is nonvanishing (with $i$ and $\hat{k}$ being
considered as continuous parameters). The polygon $\mathcal{P}_{n}(m)$ has the
polygonal decomposition

$\mathcal{P}_{n}(m)=\cup_{j=1}^{2n+2}\mathcal{P}_{n}^{j}(m)$

where the $\mathcal{P}_{n}^{j}(m)$, $j=1,...,2n+2,$ are the regions where
$\hslash^{0}(A_{n},\hat{k},i,m)$ is given by a single affine function on $i$
and $\hat{k}$, making this decomposition unique, see below for more details.

Remark: All the polygons $\mathcal{P}_{n}^{j}(m)$ are convex with the
exception of two. One can do a convex subdivision of
$\mathcal{P}_{n}(m)=\cup_{j=1}^{2n+2}\mathcal{P}_{n}^{j}(m)$ in a natural way,

$\mathcal{P}_{n}(m)=\cup_{r=1}^{4n}\mathcal{P^{\prime}}_{n}^{r}(m)$

and more importantly the convex polygons $\mathcal{P}_{n}^{{}^{\prime}r}(m)$
change with $m$ via
$\mathcal{P}_{n}^{{}^{\prime}r}(m)=\\{x\in\mathbb{R}^{2}\text{ }\big{|}\text{
}\mu_{r,l}(x)\leq
a_{r,l}m+b_{r,l},l=1,...,k_{r},a_{r,l},b_{r,l}\in\mathbb{Q}\\}$ with
$\mu_{r,l}$ linear forms with $\mathbb{Z}$-coefficients. This type of
decomposition will allow the use of the theory of polynomial weighted lattice
sums over convex polytopes (see [Ehr62], [BV97], [BV12]) to derive properties
of the function $\hslash^{0}(A_{n},m)$, such as being a quasi-polynomial of
degree 3 in $m$.

The fact, that there are $2n+2$ polygons $\mathcal{P}_{n}^{j}(m)$ follows from
the discussion below and the properties of the function
$\hslash^{0}(A_{n},\hat{k},i,m)$ derived from corollary 1. The polygons
$\mathcal{P}_{n}^{j}(m)$ have some of the properties described in the remark:
the coordinates of the vertices are affine functions of $m$ and the slopes of
the edges are independent of $m$.

To determine the polygonal decomposition defined above, we need to consider:

1) The $n+2$ lines coming from the components $E_{r}$, $r=0,...,n+1$, of
$\hat{E}$. These lines are given by
$\frac{m-i}{2}+\frac{2(r-1)-n+1}{2}\hat{k}=0$ and separate the half planes
where the
$\operatorname{codim}(V_{\hat{k},i,m,r-1}^{{}_{\operatorname{reg},1}},V_{\hat{k},i,m,r-1})$
equals either $\frac{m-i}{2}+\frac{2(r-1)-n+1}{2}\hat{k}$ or $0$. All the
lines pass through the point $(m,0)$ in the $(i,\hat{k})$-plane. The relevant
half plane $\\{i\geq 0\\}$ is therefore separated in $2(n+2)$ radial sectors
with center the point $(m,0)$.

2) Two extra lines are required to determine the regions where
$\dim\hat{V}_{\hat{k},i,m}^{{}_{\text{reg}}}$ either $=0$ or
$=m+1-\sum_{r=-1,n}\operatorname{codim}(V_{\hat{k},i,m,r}^{{}_{\operatorname{reg},1}},V_{\hat{k},i,m,r})$).
The two lines of 1) associated with $E_{0}$ and $E_{n+1}$ are also required to
determine these regions. We have the following:

${\hat{k}=\frac{i-m}{n+1}}$${\hat{k}=-\frac{m+i+2}{n+1}}$${\hat{k}=\frac{m-i}{n+1}}$${\hat{k}=\frac{m+i+2}{n+1}}$i+1${}_{\frac{m+i}{2}-\frac{n+1}{2}(\hat{k}+1)}$${}_{\frac{m+i}{2}+\frac{n+1}{2}(\hat{k}+1)}$m+1$0$$0$$0\,$
Figure 1: Graph of $\dim\hat{V}_{\hat{k},i,m}^{{}_{\text{reg}}}$ on the
$(\hat{k},i)$ plane

3) The $2(n+2)$ sectors defined in 1), except for three between the lines
associated to $E_{n}$,$E_{n+1}$,$E_{0}$ and $E_{1}$ and where
$\sum_{r=0}^{n-1}\operatorname{codim}(V_{\hat{k},i,m,r}^{{}_{\operatorname{reg},1}},V_{\hat{k},i,m,r})=0$,
each have a line segment separating the regions where
$\hslash^{0}(A_{n},\hat{k},i,m)=\sum_{r=0}^{n-1}\operatorname{codim}(V_{\hat{k},i,m,r}^{{}_{\operatorname{reg},1}},V_{\hat{k},i,m,r})$
or
$\hslash^{0}(A_{n},\hat{k},i,m)=\dim\hat{V}_{\hat{k},i,m}^{{}_{\operatorname{reg}}}$.

4) Altogether there are $2n+2$ polygons $\mathcal{P}_{n}^{j}(m)$ where
$\hslash^{0}(A_{n},\hat{k},i,m)$ as a function in $i$ and $\hat{k}$ is defined
by a single nontrivial affine expression (the case $n=3$ is illustrated in the
next figure). This follows from 1), 2) and 3) plus the fact that
$\dim\hat{V}_{\hat{k},i,m}^{{}_{\text{reg}}}$ has the same expression in the
sectors bonded by the lines $E_{n}$, $E_{n+1}$, $E_{0}$ and $E_{1}$. Recall,
that there is the symmetry coming from $\hslash^{0}(A_{n},\hat{k},i,m)$ being
an even function relative to $\hat{k}$.

5) The above are statements are for fixed $n$ and independent of $m$. As for
dependence in $m$. The coordinates of the vertices of the $2n+2$ polygonal
regions are affine functions in $m$ with rational coefficients. Also important
if one wants to give a complete description of the function
$\hslash^{0}(A_{n},m)$, the slopes of the boundary lines are independent of
$m$.

6) The affine functions
$\hslash^{0}(A_{n},\hat{k},i,m)_{j}:=\hslash^{0}(A_{n},\hat{k},i,m)|_{\mathcal{P}_{n}^{j}(m)}$
have the following structure for each $j=1,...,2n+2$,

$\displaystyle\hslash^{0}(A_{n},\hat{k},i,m)_{j}=a_{j}(n)i+b_{j}(n)\hat{k}+c_{j}(n)m+d_{j}(n)$
(2.24)

$a_{j}(n),b_{j}(n),c_{j}(n),d_{j}(n)\in\mathbb{Q}$. It is relevant to note the
fact that the coefficients in $i$ and $\hat{k}$ are independent of $m$.

Below we illustrate the function $\hslash^{0}(A_{n},\hat{k},i,m)$ as a
function of $\hat{k}$ and $i$ for $n=3$. All key features of the general case
are present.

$m+1$$\displaystyle\frac{m+2}{4}$$3m+2$$-m-1$$m$$-\displaystyle\frac{m+2}{4}$$\displaystyle\frac{m+i}{2}-2\hat{k}+1$$\displaystyle
i+1$$\displaystyle\frac{m+i}{2}+2\hat{k}+1$$\displaystyle\frac{3}{2}(m-i)$$\displaystyle\frac{m-i}{2}+\hat{k}$$\displaystyle\frac{m-i}{2}-\hat{k}$$m-i-\hat{k}$$m-i+\hat{k}$$E_{4}$$E_{0}$$E_{1}$$E_{3}$$E_{2}$
Figure 2: The function $\hslash^{0}(A_{3},\hat{k},i,m)$ with $m$ fixed on the
$(i,\hat{k})$-plane. Recall that $\hslash^{0}(A_{3},\hat{k},i,m)$ is only
defined at integral points with $4\hat{k}\equiv i+m\bmod{2}$.

####  2.3.5 Closed formula for the asymptotics of $\hslash^{0}(A_{n},m)$

As mentioned in the introduction, one of the aims of this paper is to find the
contribution that each $A_{n}$ singularity gives towards the the $m$-growth
asymptotics of $h^{1}(X,S^{m}\Omega_{X})$ when $X$ is the minimal resolution
of a surface of general type $Y$ with canonical singularities. We saw in
section 1 that this contribution can be derived from
$\hslash^{0}_{\Omega}(A_{n})=\lim_{m\to\infty}\frac{\hslash^{0}(A_{n},m)}{m^{3}}$.
In this section we give a closed formula for $\hslash^{0}_{\Omega}(A_{n})$.

###### Proof.

(of Theorem 1) Part (a) follows directly from Proposition 1

$\displaystyle\hslash^{0}(A_{n},m)=\sum_{\begin{subarray}{c}0\leq i\leq
mn-1\\\ |\hat{k}|\leq\frac{i+m}{n+1}\\\ (n+1)\hat{k}\equiv
i+m\bmod{2}\end{subarray}}\hslash^{0}(A_{n},\hat{k},i,m),$ (2.27)

corollary 1 describing
$\hslash^{0}(A_{n},x_{2},x_{1},m)(=h_{n,m}(x_{1},x_{2}))$ and the polygonal
geometry associated to $\hslash^{0}(A_{n},\hat{k},i,m)$ described in 2.4.4.

Part b) asserts that:

$\displaystyle\hslash^{0}_{\Omega}(A_{n}):=\lim_{m\to\infty}\frac{\hslash^{0}(A_{n},m)}{m^{3}}=\frac{4}{3}\sum^{n}_{j=1}\frac{1}{j^{2}}-\frac{12n^{4}+65n^{3}+117n^{2}+72n}{6(n+1)^{2}(n+2)^{2}}.$
(2.25)

and

$\displaystyle\hslash^{0}(A_{n},m)=\hslash^{0}_{\Omega}(A_{n})m^{3}+3\hslash^{0}_{\Omega}(A_{n})m^{2}+O(m),$
(2.26)

Consider:

$I(A_{n},m):=\iint_{\mathcal{P}_{n}(m)}h_{n,m}(x)dx$

Claim 1:

$\sum_{x\in\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}}h_{n,m}(x)=I(A_{n},m)+O(m)$

If $x=(x_{1},x_{2})$, set
$S_{x}:=[x_{1}-\frac{1}{2},x_{1}+\frac{1}{2}]\times[x_{2}-\frac{1}{2},x_{2}+\frac{1}{2}]$.
Consider:

$C_{\mathcal{P}_{n}(m)}=\\{x\in\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}\text{ | }S_{x}\subset\mathcal{P}_{n}(m)\\}$

$PC_{\mathcal{P}_{n}(m)}=\\{x\in\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}\text{ | }S_{x}\not\subset\mathcal{P}_{n}(m)\\}$

$O_{\mathcal{P}_{n}(m)}=\\{x\in\mathbb{Z}^{2}\setminus(\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2})\text{ | }S_{x}\cap\mathcal{P}_{n}(m)\neq\emptyset\\}$

Note $\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}=C_{\mathcal{P}_{n}(m)}\cup
PC_{\mathcal{P}_{n}(m)}$ and:

$\mathcal{P}_{n}(m)\subset\cup_{x\in C_{\mathcal{P}_{n}(m)}\cup
PC_{\mathcal{P}_{n}(m)}\cup O_{\mathcal{P}_{n}(m)}}S_{x}$

We start by showing:

$\displaystyle\sum_{x\in\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}}h_{n,m}(x)=\sum_{x\in\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}}\iint_{S_{x}}h_{n,m}(y)dy+O(m)$
(2.28)

Note: $h_{n,m}(y)=0$ if $y\not\in\mathcal{P}_{n}(m)$.

We need a further decomposition of $C_{\mathcal{P}_{n}(m)}$ with:

$SC_{\mathcal{P}_{n}(m)}=\\{x\in C_{\mathcal{P}_{n}(m)}{|}\exists
j,S_{x}\subset\mathcal{P}_{n}^{j}(m)\\}$

and $MC_{\mathcal{P}_{n}(m)}=C_{\mathcal{P}_{n}(m)}\setminus
SC_{\mathcal{P}_{n}(m)}$. The points $x\in C_{\mathcal{P}_{n}(m)}$, $S_{x}$
contained in a single polygon $\mathcal{P}_{n}^{j}(m)$ and hence $h_{n,m}(x)$
given by a single affine function on $S_{x}$, satisfy:

$\displaystyle h_{n,m}(x)=\iint_{S_{x}}h_{n,m}(y),\hskip 14.45377pt\text{
}\forall x\in SC_{\mathcal{P}_{n}(m)}$ (2.29)

If $x\in MC_{\mathcal{P}_{n}(m)}$ the equality above doesn’t hold, but:

$\displaystyle\big{|}h_{n,m}(x)-\iint_{S_{x}}h_{n,m}(y)\big{|}<C(n),\hskip
14.45377pt\text{ }\forall x\in MC_{\mathcal{P}_{n}(m)}$ (2.30)

where $C(n)=\max\\{|a_{j}(n)|,|b_{j}(n)|;j=1,...,2n+2\\}$, with $a_{j}(n)$ and
$b_{j}(n)$ from (2.24), is a bound on the partial derivatives of
$\hslash^{0}(A_{n},y,m)$ (bound independent from $m$).

Let $x\in PC_{\mathcal{P}_{n}(m)}$, using the fact that $0\leq h_{n,m}(z)\leq
1$ $\forall z\in\partial\mathcal{P}_{n}(m)$ and $C(n)$, it follows that
$\forall y\in S_{x}$, $|h_{n,m}(y)|<1+C(n)$ ($h_{n,m}(xy=0$ if
$y\not\in\mathcal{P}_{n}(m)$). Hence:

$\displaystyle\big{|}h_{n,m}(x)-\iint_{S_{x}}h_{n,m}(y)\big{|}<1+C(n),\hskip
14.45377pt\text{ }\forall x\in PC_{\mathcal{P}_{n}(m)}$ (2.31)

The subclaim (2.28) follows from $\\#(MC_{\mathcal{P}_{n}(m)}\cup
PC_{\mathcal{P}_{n}(m)})=O(m)$, it grows with $m$ as the sum of the lengths of
all edges of the polygonal decomposition of $\mathcal{P}_{n}(m)$.

To prove the claim 1, it remains to show that:

$\sum_{x\in
O_{\mathcal{P}_{n}(m)}}\iint_{S_{x}}h_{n,m}(y)dy=\iint_{\mathcal{P}_{n}(m)}h_{n,m}(y)dy-\sum_{x\in\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}}\iint_{S_{x}}h_{n,m}(y)dy=O(m)$

which follows from $|h_{n,m}(y)|<1+C(n)$ $\forall y\in S_{x}$ if $x\in
O_{\mathcal{P}_{n}(m)}$ and $\\#O_{\mathcal{P}_{n}(m)}=O(m)$.

Claim 2:

$\sum_{x\in\mathcal{P}_{n}(m)\cap\mathbb{Z}^{2}}h_{n,m}(x)=2\hslash^{0}(A_{n},m)+O(m)$

Set

$\hslash^{0}(A_{n},\hat{k},m):=\sum^{n(m+1)-1}_{\begin{subarray}{c}i=0\\\
(n+1)\hat{k}\equiv i+m\bmod{2}\end{subarray}}\hslash^{0}(A_{n},\hat{k},i,m)$

then $\hslash^{0}(A_{n},m)=\sum_{|\hat{k}|\leq
m+1}\hslash^{0}(A_{n},\hat{k},m)$. It is enough to show $\exists$ $A(n)>0$
such that:

$\displaystyle\left|\sum^{n(m+1)-1}_{i=0}\hslash^{0}(A_{n},\hat{k},i,m)-2\hslash^{0}(A_{n},\hat{k},m)\right|<A(n)\hskip
21.68121pt\forall|\hat{k}|\leq m+1$ (2.32)

For fixed $\hat{k}$ (and also $n$ and $m$), the function
$\hslash^{0}(A_{n},\hat{k},i,m)$ is a piecewise affine function in $i$ with at
most $n+2$ affine pieces (and continuous if $i$ is considered a continuous
parameter).

One has
$\hslash^{0}(A_{n},\hat{k},i,m)=\frac{1}{2}[\hslash^{0}(A_{n},\hat{k},i-1,m)+\hslash^{0}(A_{n},\hat{k},i+1,m)]$
if $i-1,i,i+1$ is the same affine piece. Otherwise one has
$|\hslash^{0}(A_{n},\hat{k},i,m)-\frac{1}{2}[\hslash^{0}(A_{n},\hat{k},i-1,m)+\hslash^{0}(A_{n},\hat{k},i+1,m)]|\leq
C(n)$ with $C(n)$ as above. One also has the boundary condition that
$0\leq\hslash^{0}(A_{n},\hat{k},0,m),\hslash^{0}(A_{n},\hat{k},n(m+1)-1,m)\leq
1$. All the above implies $A(n)=2+(n+2)C(n)$ works in (2.32), with $2+C(n)$
bounding the noncancelation at end points and $(n+1)C(n)$ bounding the
noncancelation at the (at most) $n+1$ transition points between the affine
functions.

Claims 1 and 2 and the symmetry of both $\mathcal{P}_{n}(m)$ and $h_{n,m}(x)$
with respect to the $x_{1}$-axis (i.e. $i$-axis) imply that

$\hslash^{0}(A_{n},m)=\iint_{{\mathcal{P}^{+}_{n}(m)}}h_{n,m}(x)dy+O(m)$

where $\mathcal{P}^{+}_{n}(m):=\mathcal{P}_{n}(m)\cap\\{x_{2}\geq 0\\}$. The
polygon $\mathcal{P}^{+}_{n}(m)$ has a polygonal decomposition
$\mathcal{P}^{+}_{n}(m)=\cup_{\ell=0}^{n+1}\mathcal{P}^{\ell}_{n}(m)$, where
for each $\ell=0,...,n+1$,

$h_{n,m,\ell}(x):=h_{n,m}(x)|_{\mathcal{P}^{\ell}_{n}(m)}$

is given by a single affine expression in $x=(x_{1},x_{2})$ (and $m$). The
polygons $\mathcal{P}^{\ell}_{n}(m)$, $\ell=0,...,n+1$ will be described via
their vertices. To that end, for $j=1,...,n$ consider the points:

$\displaystyle
v_{j}:=\left(\frac{2j(m+1)}{j-n-3}+\frac{2(j-1)(m+1)}{-j+n+2}+m,\frac{2(m+1)}{(j-n-3)(j-n-2)}\right)$

  1. (i)

$\mathcal{P}^{0}_{n}(m)$:

$\left(0,\frac{m}{n+1}\right)\to
v_{1}\to\left(\frac{mn-2}{2+n},0\right)\to(0,0)\to\left(0,\frac{m}{n+1}\right),$

and where $h_{n,m,0}(x)=x_{1}+1.$

  2. (ii)

$\mathcal{P}^{1}_{n}(m)$:

$\displaystyle v_{1}\to
v_{2}\to\left(m,0\right)\to\left(\frac{mn-2}{2+n},0\right)\to v_{1},$

and $h_{n,m,1}(x)=\frac{n(m-x_{1})}{2}$.

  3. (iii)

$\mathcal{P}^{j}_{n}(m)$, $j=2,\dots,n$:

For $j=2,\dots,n-1$,

$\displaystyle v_{j}\to v_{j+1}\to(m,0)\to v_{j}$

and for $j=n$,

$\displaystyle v_{n}\to\left(n(m+1)-1,m+1\right)\to(m,0)\to v_{n}$

and the function $h_{n,m,j}(x)=\frac{1}{2}((j-1)-n)(x_{1}-(j-1)x_{2}-m)$.

  4. (iv)

$\mathcal{P}^{n+1}_{n}(m)$:

$\displaystyle\left(0,\frac{m+2}{n+1}\right)$
$\displaystyle\to\left(n(m+1)-1,m+1\right)\to v_{n}\to\cdots$
$\displaystyle\cdots$ $\displaystyle\to v_{2}\to
v_{1}\to\left(0,\frac{m}{n+1}\right)\to\left(0,\frac{m+2}{n+1}\right)$

and where $h_{n,m,n+1}(x)=\frac{1}{2}(x_{1}-x_{2}(n+1)+m+2)$.

The above $\mathcal{P}^{\ell}_{n}(m)$ and $h_{n,m,\ell}(x)$, $\ell=0,...,n+1$
give:

$\displaystyle\hslash^{0}(A_{n},m)=\sum_{\ell=0}^{n+1}\iint_{{\mathcal{P}^{\ell}_{n}(m)}}h_{n,m,\ell}(x)dx=\hslash^{0}_{\Omega}(A_{n})m^{3}+3\hslash^{0}_{\Omega}(A_{n})m^{2}+O(m)$

with $\hslash^{0}_{\Omega}(A_{n})$ as in (2.25). ∎

######  Corollary 2.

The invariants $\hslash^{0}_{\Omega}(A_{n})$ increase with $n$ and are
bounded. Moreover:

$\displaystyle\lim_{n\to\infty}\hslash^{0}_{\Omega}(A_{n})=\frac{2\pi^{2}}{9}-2$

Remark:

  1. i)

The invariant $\hslash^{0}_{\Omega}(A_{1})$ was known [Tho13] (appeared in
[BDO06] unfortunately with an error). The $A_{1}$ case is quite direct since
the exceptional locus has a single component. Some cases of
$\hslash^{0}_{\Omega}(A_{n})$ for low $n$ were known to the authors [DOW19]
and [Wei20].

  2. ii)

The function $\hslash^{0}(A_{n},m)$ for fixed $n$ is a quasi-polynomial in $m$
of degree 3. This follows from the theory of polynomial weighted lattice sums
over convex polytopes $\mathcal{P}({\bf b})$, ${\bf b}=(b_{1},...,b_{k})$,
defined by $k$ inequalities, $\mu_{l}(x)\leq b_{l}$, where the linear forms
$\mu_{l}(x)$ are fixed, but ${\bf b}$ varies ([Ehr62], [BV97], [BV12]). A
natural convex polygon decomposition of the polygons $\mathcal{P}_{n}^{+}(m)$
with the required properties can be found by decomposing
$\mathcal{P}^{n+1}_{n}(m)$ into $n$ polygons by introducing vertical line
segments above the points $v_{2}$,…,$v_{n}$.

  3. iii)

In Theorem 1, we showed that the cubic and quadratic coefficients of
$\hslash^{0}(A_{n},m)$ have period 1. In future work, we describe a
divisibility condition for the the least common multiple of the periods of the
coefficients of the quasi-polynomial for all $n$, which allows us to determine
$\hslash^{0}(A_{n},m)$ for low $n$. In the case of $A_{1}$ the least common
multiple of the periods is 6 [BTVA22]. Knowing the functions
$\hslash^{0}(A_{n},m)$ can be used to obtain information on the degrees of
symmetric differentials that occur on a surface which is a resolution of a
surface with $A_{n}$ singularities.

Example. For $A_{2}$ singularities the lcm of the periods of the coefficients
is also $6$ and $\hslash^{0}(A_{2},m)$ is described by the polynomials:

$\displaystyle\hslash^{0}(A_{2},m)=\begin{cases}\frac{29}{216}m^{3}+\frac{29}{72}m^{2}+\frac{1}{12}m\text{
}\text{ }&m\equiv 0\text{ }(\text{mod}\text{ }6)\\\
\frac{29}{216}m^{3}+\frac{29}{72}m^{2}+\frac{1}{8}m-\frac{143}{216}\text{
}\text{ }&m\equiv 1\text{ }(\text{mod}\text{ }6)\\\
\frac{29}{216}m^{3}+\frac{29}{72}m^{2}+\frac{7}{36}m-\frac{2}{27}\text{
}\text{ }&m\equiv 2\text{ }(\text{mod}\text{ }6)\\\
\frac{29}{216}m^{3}+\frac{29}{72}m^{2}+\frac{1}{8}m+\frac{3}{8}\text{ }\text{
}&m\equiv 3\text{ }(\text{mod}\text{ }6)\\\
\frac{29}{216}m^{3}+\frac{29}{72}m^{2}+\frac{1}{12}m-\frac{10}{27}\text{
}\text{ }&m\equiv 4\text{ }(\text{mod}\text{ }6)\\\
\frac{29}{216}m^{3}+\frac{29}{72}m^{2}+\frac{17}{72}m-\frac{7}{216}\text{
}\text{ }&m\equiv 5\text{ }(\text{mod}\text{ }6)\end{cases}$

##  3 Applications

###  3.1 The $1$-st cohomological $\Omega$-asymptotics of $A_{n}$
singularities

Let $X$ be the minimal resolution of an orbifold surface $Y$ of general type.
The asymptotics of the localized component $Lh^{1}(X,S^{m}\Omega^{1}_{X})$ of
$h^{1}(X,S^{m}\Omega^{1}_{X})$ described in (1.3) plays a role in the QS-
bigness criterion (1.4).

In this section, we establish the formula for the contribution given by each
$A_{n}$ singularity to $Lh^{1}(X,S^{m}\Omega^{1}_{X})$. This contribution
consists of:

$\displaystyle
h^{1}_{\Omega}(A_{n}):=\lim_{m\to\infty}\frac{h^{1}(A_{n},m)}{m^{3}}$

and it is called the $1-$cohomological $\Omega$-asymptotics of $A_{n}$ (the
limit exists).

###### Proof.

(of Theorem 2) Using relation (1.7) for $A_{n}$ singularities,

$h^{1}(A_{n},m)=\mu(A_{n},m)-\chi_{\operatorname{orb}}(A_{n},m)-\hslash^{0}(A_{n},m)$

we find $h^{1}_{\Omega}(A_{n})$ to be

$\displaystyle
h^{1}_{\Omega}(A_{n})=\lim_{m\to\infty}\frac{\mu(A_{n},m)}{m^{3}}-\lim_{m\to\infty}\frac{\chi_{\operatorname{orb}}(A_{n},m)}{m^{3}}-\hslash^{0}_{\Omega}(A_{n}).$

The invariant $\hslash^{0}_{\Omega}(A_{n})$ was determined in Theorem 1. The
invariants $\chi_{\text{orb}}(A_{n},m)$ are given by formula (1.9) along with
the local Chern numbers

$\displaystyle c_{1}^{2}(A_{n})=0\hskip 21.68121pt\text{and}\hskip
21.68121ptc_{2}(A_{n})=e(E)-\frac{1}{|G_{A_{n}}|}=\frac{n(n+2)}{n+1},$

where $e(E)$ is the topological Euler characteristic of the exceptional locus
of the minimal resolution and $|G_{A_{n}}|$ is the order of the local
fundamental group of the $A_{n}$ singularity ([Bla96] 3.18). Hence:

$\displaystyle\lim_{m\to\infty}\frac{\chi_{\operatorname{orb}}(A_{n},m)}{m^{3}}=\frac{s_{2}(A_{n})}{3!}=-\frac{n(n+2)}{6(n+1)}.$

Now, we turn our attention to the invariants $\mu(A_{n},m)$. The key feature
of these invariants is that

$\displaystyle\lim_{m\to\infty}\frac{\mu(A_{n},m)}{m^{3}}=0$

This is shown in [Bla96] 4.4 or [Lan00]) (it follows from general results on
reflexive sheaves on quotient singularities, see also [Wah93]). A complete
description of the invariants $\mu(A_{n},m)$ is known to the authors and will
appear in future work. For example, for fixed $n$, $\mu(A_{n},m)$ is a quasi-
polynomial of degree 1 in $m$. ∎

######  Corollary 3.

The invariants $h^{1}_{\Omega}(A_{n})$ increase with $n$ and:

$\lim_{n\to\infty}h^{1}_{\Omega}(A_{n})=\infty$

Remark: The QS-bigness criterion (1.4) has $h^{1}_{\Omega}(A_{n})$ as the
contribution of $A_{n}$ to the localized component
$Lh^{1}(X,S^{m}\Omega^{1}_{X})$ of $h^{1}(X,S^{m}\Omega^{1}_{X})$, while in
the bigness criterion in [RR14] the contribution of $A_{n}$ to
$h^{1}(X,S^{m}\Omega^{1}_{X})$ is
$\frac{1}{2}\lim_{m\to\infty}\frac{\chi(A_{n},m)}{m^{3}}$. It follows from
Theorem 1, Corollary 2, Theorem 2 and Corollary 3 that the contribution of
each $A_{n}$ in the QS-bigness criterion is always larger and approaches twice
the contribution in the criterion of [RR14] as $n\to\infty$. In Part I of this
work, we present the implications of this remark towards the wider range of
pair of Chern numbers for which the CMS-bigness criterion can hold when
compared to the criterion in [RR14].

Table 1:

$n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---
$h^{1}_{\Omega}(A_{n})$ | $\dfrac{4}{27}$ | $\dfrac{67}{216}$ | $\dfrac{1283}{2700}$ | $\dfrac{577}{900}$ | $\dfrac{106819}{132300}$ | $\dfrac{1030727}{1058400}$ | $\dfrac{5431459}{4762800}$

###  3.2  Extension of symmetric differentials for $A_{n}$ singularities

Let $(\tilde{X},E)$ be the germ of the minimal resolution of a quotient
singularity $(X,x)$. It is well known that symmetric differentials $w\in
H^{0}(\tilde{X}\setminus E,S^{m}\Omega^{1}_{\tilde{X}})$ acquire mild poles
along the components of the exceptional divisor $E$. More precisely, the poles
of $w\in H^{0}(\tilde{X}\setminus E,S^{m}\Omega^{1}_{\tilde{X}})$ along $E$
are at worst logarithmic, i.e. $H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})=H^{0}(\tilde{X},S^{m}\Omega^{1}_{\tilde{X}}(\text{log}E))$
([Miy84] 4.14, see also [Wah93] 4.7).

We show that for $A_{n}$ singularities, the poles that $w\in
H^{0}(\tilde{X}\setminus E,S^{m}\Omega^{1}_{\tilde{X}})$ can acquire at the
exceptional locus are milder (and to what extent) than logarithmic poles (see
[Tho13] for $A_{1}$). More precisely, in Theorem 3, we give the maximal
effective divisor $D$ such that

$H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})=H^{0}(\tilde{X},S^{m}\Omega^{1}_{\tilde{X}}(\text{log}E)\otimes\mathcal{O}_{\tilde{X}}(-D))$

.

####  3.2.1 Decomposition and the order of differentials in
$S(\tilde{X}\setminus E)$

In this section, $\tilde{X}$ is the minimal resolution of the affine model $X$
of the $A_{n}$ singularity as described in section 2.1. We have the
commutative diagram involving the resolution $\sigma$ and the smoothing $\pi$
of $X$:

${(\mathbb{C}^{2},0)}$${(\tilde{X},E)}$${(X,x)}$$\scriptstyle{\varphi}$$\scriptstyle{\pi}$$\scriptstyle{\sigma}$

The map $\varphi$ induces the isomorphisms between the algebras of symmetric
differentials (see section 2.3.2):

$\displaystyle S\left(\tilde{X}\setminus E\right)$
$\displaystyle\xrightarrow[\cong]{\varphi^{*}}S\left(\mathbb{C}^{2}\right)^{\mathbb{Z}_{n+1}}$
$\displaystyle S\left(\tilde{X}\setminus\hat{E}\right)$
$\displaystyle\xrightarrow[\cong]{\varphi^{*}}S\left(\mathbb{C}^{*}\times\mathbb{C}^{*}\right)^{\mathbb{Z}_{n+1}}$
(3.1)

(recall that $\hat{E}=E+E_{0}+E_{n+1}$ as in 2.1).

Using the isomorphism (3.1) and the 3-gradation of
$S\left(\mathbb{C}^{*}\times\mathbb{C}^{*}\right)^{\mathbb{Z}_{n+1}}$
described in 2.2.2, we obtain the $(\hat{k},i,m)-$decomposition of
differentials $w\in S\left(\tilde{X}\setminus\hat{E}\right)$:

$\displaystyle w=\sum_{\begin{subarray}{c}m\in\mathbb{Z}_{\geq 0}\\\
i,\hat{k}\in\mathbb{Z}\end{subarray}}w_{\hat{k},i,m}$ (3.2)

where the $w_{\hat{k},i,m}\in
H^{0}(\tilde{X}\setminus\hat{E},S^{m}\Omega^{1}_{\tilde{X}})$ are such that
$\varphi^{*}w_{\hat{k},i,m}\in\hat{V}_{\hat{k},i,m}$. If $w\in
S\left(\tilde{X}\setminus E\right)$, then we have

$\displaystyle w=\sum_{\begin{subarray}{c}m,i\in\mathbb{Z}_{\geq 0}\\\
|\hat{k}|\leq\frac{i+m}{n+1}\end{subarray}}w_{\hat{k},i,m}.$ (3.3)

In this case, we additionally have that $\varphi^{*}w\in S(\mathbb{C}^{2})$
and hence $\varphi^{*}w_{\hat{k},i,m}\in\hat{V}^{\text{reg}}_{\hat{k},i,m}$.

We say a differential $w\in S\left(\tilde{X}\setminus\hat{E}\right)$ is of
$type$ $(\hat{k},i,m)$ if $\varphi^{*}w\in\hat{V}_{\hat{k},i,m}$. The
expression (3.2) is the decomposition of $w$ relative to the
$(\hat{k},i,m)$-types.

We define the $order$ of a symmetric differential $w\in
S\left(\tilde{X}\setminus\hat{E}\right)$ to be:

$\displaystyle\text{ord}(w)=\min\left\\{i\text{ }\Big{|}\text{
}\varphi^{*}w=\sum_{\begin{subarray}{c}i,\hat{k}\in\mathbb{Z}\end{subarray}}w_{\hat{k},i,m}\text{
and }w_{\hat{k},i,m}\neq 0\right\\},$ (3.4)

i.e. $\operatorname{ord}(w)$ is the smallest order of the differential
monomials appearing in the monomial decomposition of $\varphi^{*}w$. The
ord$(w)$ can also be described as the order of vanishing of $\varphi^{*}w$ at
$0\in\mathbb{C}^{2}$, this is consistent with the definition of order in
theorem B for $w\in H^{0}(X\setminus x,S^{m}\Omega^{1}_{X})$.

####  3.2.2 Comparison with logarithmic poles

We characterize the allowed poles of a differential $w\in
H^{0}(\tilde{X}\setminus E,S^{m}\Omega^{1}_{\tilde{X}})$ along $E$ via a
comparison to the maximum poles allowed on logarithmic symmetric differentials
$\mu\in H^{0}(\tilde{X}\setminus E,S^{m}\Omega^{1}_{\tilde{X}}(\log E))$.

###### Proof.

(of Theorem 3(a)) Let $w\in H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})$. The worst pole of $w$ along a component
$E_{r}$ of $E$ will be the worst pole attained by one of its
$(\hat{k},i,m)$-type components in the decomposition (3.3).

Let $w$ be of $(\hat{k},i,m)$-type. To determine the poles of $w$ along the
components of $E$, we examine $w$ on the coordinate patches $U_{r}$,
$r=0,...,n$, (described in 2.1) covering $\tilde{X}$. Set $w_{r}:=w|_{U_{r}}$,
note $w_{r}=(\varphi_{r}^{*})^{-1}(\varphi^{*}w)$ hence in
$(\varphi_{r}^{*})^{-1}\hat{V}_{\hat{k},i,m}$. Using 2.17, it follows that
$w_{r}$ is in the span of the monomials in the block
$B_{\frac{i+m}{2}+(\frac{n+1}{2}-r)\hat{k},i+(n-2r)\hat{k},m}$.

The relevant observation towards the comparison of the poles of $w\in
H^{0}(\tilde{X}\setminus E,S^{m}\Omega^{1}_{\tilde{X}})$ to logarithmic poles
is the fact that we can rewrite the monomials in a block $B_{k,i,m}$, in the
following form:

$B_{k,i,m}=\Big{\\{}z_{1}^{i+m-k}z_{2}^{k}\frac{dz_{1}^{m-q}}{z_{1}^{m-q}}\frac{dz_{2}^{q}}{z_{2}^{q}}\Big{\\}}_{q=0,...,m}$

Hence $w_{r}$ (of type $(\hat{k},i,m)$) is given by a sum of the following
monomials:

$\displaystyle\hskip
28.90755ptu_{1}^{\frac{i+m}{2}+(\frac{n-1}{2}-r)\hat{k}}u_{2}^{\frac{i+m}{2}+(\frac{n+1}{2}-r)\hat{k}}\frac{du_{1}^{m-q}}{u_{1}^{m-q}}\frac{du_{2}^{q}}{u_{2}^{q}},\hskip
14.45377ptq=0,...,m$ (3.5)

Next we prove the result for $A_{n}$ with $n\geq 2$ (the case $n=1$ follows
from the same argument after a minor setup adjustment). For $r=1,...,n-1$:

$S^{m}\Omega^{1}_{\tilde{X}}(\log
E)|_{U_{r}}=\bigoplus_{q=0}^{m}\mathcal{O}_{U_{r}}\frac{du_{1}^{m-q}}{u_{1}^{m-q}}\frac{du_{2}^{q}}{u_{2}^{q}}$

(recall: for each $r=1,...,n-1$, $E_{r}\cap U_{r}=\\{u_{2}=0\\}$ and
$E_{r+1}\cap U_{r}=\\{u_{1}=0\\}$).

It follows from (3.5) that the order of the poles along $E_{r}$ of the
monomials involved in the sum giving $w_{r}$ deviates from the highest pole
order possible for a logarithmic symmetric differential by subtracting
$\frac{i+m}{2}+(\frac{n+1}{2}-r)\hat{k}$. Hence:

$\displaystyle w_{r}\in H^{0}\left(U_{r},S^{m}\Omega^{1}_{\tilde{X}}(\log
E)\otimes\mathcal{O}(-D)\right)$

with

$\displaystyle
D=\left[\frac{i+m}{2}+\left(\frac{n+1}{2}-r\right)\hat{k}\right]E_{r}+\left[\frac{i+m}{2}+\left(\frac{n-1}{2}-r\right)\hat{k}\right]E_{r+1}$

An easy consequence of the above is:

$H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})=H^{0}(\tilde{X},S^{m}\Omega^{1}_{\tilde{X}}(\log
E))$

This follows since the condition that $D\geq 0$ in all $r=1,...,n-1$ is
equivalent to $|\hat{k}|\leq\frac{i+m}{n-1}$, which holds since the
$(\hat{k},i,m)$-components of $w\in H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})$ satisfy $|\hat{k}|\leq\frac{i+m}{n+1}$.

The full strength of the result follows from:

$a_{r}=\min\begin{cases}\frac{i+m}{2}+(\frac{n+1}{2}-r)\hat{k}\text{
}\big{|}\text{ }i\geq 0,|\hat{k}|\leq\frac{i+m}{n+1},(n+1)\hat{k}\equiv
i+m\bmod{2}\Big{\\}}\end{cases}$

satisfies:

$a_{r}:=\begin{cases}\sum_{j=1}^{r}\lceil\frac{m+2-2j}{n+1}\rceil&1\leq
r\leq\lfloor\frac{n+1}{2}\rfloor\\\
\sum_{j=1}^{n+1-r}\lceil\frac{m+2-2j}{n+1}\rceil&\lfloor\frac{n+1}{2}\rfloor<r\leq
n\end{cases}$

Set $D=\sum_{r=1}^{n}a_{r}E_{r}$. By construction all the
$(\hat{k},i,m)$-components of $w$, and hence $w$, belong to
$H^{0}(\tilde{X},S^{m}\Omega^{1}_{\tilde{X}}(\log
E)\otimes\mathcal{O}_{\tilde{X}}(-D))$ and the maximality of $D$ is guaranteed
by the definition of $a_{r}$ and the fact.

∎

####  3.2.3  Order and holomorphic extension

Part (b) of Theorem 3 gives a criterion for the holomorphic extension to
$\tilde{X}$ of symmetric differentials on $\tilde{X}\setminus E$ involving the
order of the differentials. This result is used through out section 2, it
makes explicit that the $h^{0}(A_{n},m)$ are finite since it implies that the
polygon $\mathcal{P}_{n}(m)$ over which the lattice weighted sum giving
$h^{0}(A_{n},m)$ is bounded. We note that this result can also be derived from
corollary 1.

###### Proof.

(of Theorem 3 part (b)) If $w\in H^{0}(\tilde{X}\setminus
E,S^{m}\Omega^{1}_{\tilde{X}})$, then the $(\hat{k},i,m)$-decomposition of $w$
has $w=\sum w_{\hat{k},i,m}$, with $i\in\mathbb{Z}_{\geq 0}$ and
$|\hat{k}|\leq\frac{i+m}{n+1}$. The differential $w$ extends regularly to
$\tilde{X}$ if all the $w_{\hat{k},i,m}$ do. Note that each
$w_{\hat{k},i,m}\in H^{0}(\tilde{X}\setminus E,S^{m}\Omega^{1}_{\tilde{X}})$,
hence no need to check about poles along $E_{0}$ or $E_{n+1}$.

The conditions to guarantee no poles of $w_{\hat{k},i,m}$ along $E_{r}$,
$r=1,...,n$ are obtained using (3.5). From (3.5) it follows that all $w$ of
type $(\hat{k},i,m)$ do not acquire a pole along $E_{r}$ if and only if:

$\displaystyle\frac{i+m}{2}+\left(\frac{n+1}{2}-r\right)\hat{k}\geq m$ (3.6)

First observation is that the $r$ conditions (3.6) on the triples
$(\hat{k},i,m)$ can be reduced to the conditions for $r=1$ and $r=n$. Second
observation is that for $i<m$ the conditions can’t hold. Finally, if $i\geq
m$, the conditions become:

$\displaystyle|\hat{k}|\leq\frac{i-m}{n-1}$

The condition above holds for the possible $\hat{k}$, i.e. in the range given
by $|\hat{k}|\leq\frac{i+m}{n+1}$, if $i\geq nm$.

∎

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