# Motivic zeta function of the Hilbert schemes of points on a surface

Luigi Pagano

###### Abstract

Let $K$ be a discretely-valued field. Let $X\rightarrow\operatorname{Spec}K$
be a surface with trivial canonical bundle. In this paper we construct a weak
Néron model of the schemes $\operatorname{Hilb}^{n}(X)$ over the ring of
integers $R\subseteq K$. We exploit this construction in order to compute the
Motivic Zeta Function of $\operatorname{Hilb}^{n}(X)$ in terms of $Z_{X}$. We
determine the poles of $Z_{\operatorname{Hilb}^{n}(X)}$ and study its
monodromy property, showing that if the monodromy conjecture holds for $X$
then it holds for $\operatorname{Hilb}^{n}(X)$ too.

Excerptum

Sit $K$ corpus cum absoluto ualore discreto. Sit
$X\rightarrow\operatorname{Spec}K$ leuigata superficies cum canonico fasce
congruenti $\pazocal{O}_{X}$. In hoc scripto defecta Neroniensia paradigmata
$\operatorname{Hilb}^{n}(X)$ schematum super annulo integrorum in $K$ corpo,
$R\subset K$, constituimus. Ex hoc, Functionem Zetam Motiuicam
$Z_{\operatorname{Hilb}^{n}(X)}$, dato $Z_{X}$, computamus. Suos polos
statuimus et suam monodromicam proprietatem studemus, coniectura monodromica,
quae super $X$ ualet, ualere super $\operatorname{Hilb}^{n}(X)$ quoque
demostrando.

## Introduction

The Motivic Zeta Function, introduced by Denef and Loeser, is a powerful
invariant attached to hypersurface singularities. This invariant is a formal
series with coefficients in a Grothendieck ring of varieties which has been
proved to be rational in [14], in a sense that will become clear in §4. The
most discussed open question concerning the motivic zeta function is the
_Monodromy Conjecture_ which claims the existence of a relationship between
the poles of the function and the eigenvalues of the local monodromy operator
associated to the hypersurface singularity. In this paper we focus on a
natural counterpart, introduced in [13], of Denef’s and Loeser’s zeta
function, whose argument consists of a punctured neighbourhood of the
hypersurface singularity, together with a volume form on such neighbourhood,
rather than involving the hypersurface itself. The two invariants are closely
related and the questions about the Monodromy Conjecture make sense in both
cases.

Let $K$ be a complete discretely valued field. Let $R$ be its valuation ring
and let $k$ be its residue field, which we assume to be algebraically closed.
Let $X\rightarrow\operatorname{Spec}K$ be a smooth algebraic variety. This
datum is the arithmetic analog of the datum of a family of complex varieties
over a small punctured disk, where one usually uses the fundamental group of
the punctured disk to give a monodromy action on the cohomology. In our case
the absolute Galois group of $K$, being canonically isomorphic to the étale
fundamental group of $\operatorname{Spec}K$, replaces the usual fundamental
group and acts on $X$, inducing an action on its cohomology:
$H^{\ast}(X_{\overline{K}},\mathbb{Q}_{l})$, where $l$ is any prime number
coprime with $\operatorname{char}k$. We will assume that the wild inertia
subgroup acts trivially on $X$, so that the action of
$\operatorname{Gal}(\overline{K}|K)$ can be identified with the action of its
tame quotient, which admits a single topological generator $\sigma$, hence we
can study the monodromy action by focusing on the action of the linear
operator $\sigma^{\ast}\colon
H^{\ast}(X_{\overline{K}},\mathbb{Q}_{l})\rightarrow
H^{\ast}(X_{\overline{K}},\mathbb{Q}_{l})$. This operator is quasi-unipotent,
meaning that $\exists a,b\in\mathbb{N}$ such that
${\left((\sigma^{\ast})^{a}-\operatorname{id}\right)}^{b}=0$, hence its
eigenvalues, often called _monodromy eigenvalues_ , are roots of the unity.

Now assume $X$ is a Calabi-Yau variety, _i.e._ a smooth algebraic variety
whose canonical sheaf $\omega_{X}\coloneqq\Omega_{X/K}^{\dim_{X}}$ is a
trivial line bundle and let $\omega$ be a volume form on $X$. Using this
datum, we denote by $Z_{X,\omega}(T)\in\mathscr{M}_{k}[[T]]$, the Motivic Zeta
Function of Definition 4.2. This invariant is supposed to be closely related
to the monodromy operator described above in a sense explained in the
following:

######  Conjecture (Monodromy conjecture).

Let $q\in\mathbb{Q}$ be a pole of $Z_{X,\omega}(T)$, then $e^{2\pi iq}$ is a
monodromy eigenvalue.

The monodromy conjecture has been proven in several classes of varieties:
Halle and Nicaise proved it for Abelian varieties, [10], while Jaspers proved
it in [12] when $X$ is a K3 surfaces admitting a Crauder-Morrison model and
Overkamp proved it for Kummer K3 surfaces in [18]. Yet we do not know whether
all the K3 surfaces satisfy the Monodromy conjecture.

In this paper we study this conjecture for Hilbert schemes of points on
surfaces; we report here a simplified version of our main result, Theorem 7.1;
a more precise statement is provided in §7, after having introduced the
notions involved:

######  Theorem.

Let $X$ be a surface with trivial canonical bundle satisfying the monodromy
conjecture.

Then the conjecture holds also for $\operatorname{Hilb}^{n}(X)$, $\forall
n\in\mathbb{N}$.

###### Sketch of the proof.

We will outline here the main ideas involved in the proof.

The computation of the MZF relies on the tools of motivic integration, thus on
the construction of a weak Néron model of $\operatorname{Hilb}^{n}(X)$. After
fixing an _sncd_ model of $X$ over the ring $R$ (satisfying a technical
assumption introduced in §6.1), call $\mathfrak{X}$ such a model, then for all
integers $m$ divisible enough and coprime with $\operatorname{char}K$, it is
possible to construct semistable models
$\mathfrak{X}(m)\rightarrow\Delta(m)=\operatorname{Spec}R(m)$ of
$X\times_{K}K(m)$, where $K(m)/K$ is the unique extension of degree $m$ such
that the integral closure $R(m)=\overline{R}\subseteq K(m)$ is totally
ramified over $R$. We give an explicit construction of such models, see §3,
satisfying the following additonal properties:

  * •

The natural action of $\mu_{m}=\operatorname{Gal}(K(m)|K)$ on $X(m)$ extends
to $\mathfrak{X}(m)$;

  * •

For all points $x\in\mathfrak{X}(m)_{k,\operatorname{sm}}$, the stabilizer
$\operatorname{Stab}_{x}\subseteq\mu_{m}$ acts trivially on the whole
irreducible component $E\subseteq\mathfrak{X}(m)_{k}$ containing $x$.

The relative Hilbert functor provides a proper scheme
$\operatorname{Hilb}^{n}(\mathfrak{X}(m)/\Delta(m))\rightarrow\Delta(m)$ whose
generic fibre is $\operatorname{Hilb}^{n}(X(m))$. This model inherits from
$\mathfrak{X}(m)$ an action of $\mu_{m}$, in particular it is a proper,
equivariant model of $X(m)$ over $\Delta(m)$. If we remove the critical points
of the map $\mathfrak{X}(m)\rightarrow\Delta(m)$, then we repeat the
construction obtaining a smooth $\mu_{m}-$equivariant subscheme
$\operatorname{Hilb}^{n}(\mathfrak{X}(m)_{\operatorname{sm}}/\Delta(m))\subseteq\operatorname{Hilb}^{n}(\mathfrak{X}(m)/\Delta(m))$.
By the means of Weil restriction of scalars (see §2), we construct a smooth
model of $\operatorname{Hilb}^{n}(X)$ over $R$, namely
${\left(\operatorname{Res}_{\Delta(\tilde{n})/\Delta}\operatorname{Hilb}^{n}(\mathfrak{X}(m)_{\operatorname{sm}}/\Delta(\tilde{n}))\right)}^{\mu_{\tilde{n}}}$.

Under a suitable choice of $m$ ($m=\tilde{n}$ under the notation of §6), the
model above satisfies the weak extension property with respect to
$\operatorname{Spec}K\subseteq\operatorname{Spec}R$, hence it is a weak Néron
model of $\operatorname{Hilb}^{n}(X)$. This model provides a direct way to
compute the motivic integral $\int_{\operatorname{Hilb}^{n}(X)}\omega^{[n]}$;
choosing $m=m_{1}\tilde{n}$ allows us to compute the motivic integral of the
Hilbert schemes
$\operatorname{Hilb}^{n}(X(m_{1})/K(m_{1}))$, for $m_{1}\in\mathbb{N}_{+}$,
therefore we get the formal series defining
$Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}$, since
$\operatorname{Hilb}^{n}(X)\times_{K}K(m)=\operatorname{Hilb}^{n}(X(m))$. The
value of the Zeta function is, indeed, written implicitly in Theorem 6.8 and
explicitly in Equation (6.2), i.e.

$Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}=\sum_{\alpha\dashv
n}\prod_{j=1}^{\infty}\left(\mathbb{L}^{(j-1)\alpha_{j}}\operatorname{Sym}^{\alpha_{j}}\left(Z_{X(j),\omega(j)}\right)\right)\,\text{.
}$

Using this explicit formula and the results developed in §5.3, we find out
that every pole of $Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}$ is the sum of
$n$ poles of $Z_{X,\omega}$, possibly repeated, we do not know yet if the
converse is true, i.e. whether every possible sum of $n$ poles of
$Z_{X,\omega}$ is a pole of $Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}$.
Putting the description of the poles of the Zeta Function together with the
description of the cohomology of $\operatorname{Hilb}^{n}(X)$ in [7], one
finds out the claimed result about the monodromy conjecture. ∎

A further byproduct of the computations we carried out in §5, we also achieved
Proposition 7.2:

######  Proposition.

Let $Y,Z$ be two Calabi-Yau varieties endowed with volume forms
$\omega_{1},\omega_{2}$ satisfying the monodromy conjecture. Let $\omega$ be
volume forms on $Y\times Z$ the volume form
$\omega\coloneqq\operatorname{pr}_{Y}^{\ast}\omega_{1}\wedge\operatorname{pr}_{Z}^{\ast}\omega_{2}$.
Then also $Y\times Z$, endowed with the volume form $\omega$, satisfies the
monodromy conjecture.

We conclude the paper by discussing the Monodromy conjecture in an explicit
example, i.e. comparing the poles of the zeta functions of a quartic surface
embedded in $\mathbb{P}^{3}_{K}$ and of its Hilbert schemes of degree 2.

### Organization of the paper

In §1 we introduce the Grothendieck rings of varieties and their equivariant
versions, using them to define the main actors of the manuscript.

In §2 we recall the notion of a weak Néron model and show some techniques that
we will use in order to construct them.

In §3 we adapt the theory of potential semistable reduction of families of
surfaces in our case. We construct a specific equivariant semistable model
satisfying a good property with respect to the Galois action of the extension.

In §4 we recall the notion of motivic integration and the definition of the
Motivic Zeta Function. Then we explain the notion of rationality for power
series with coefficients in $\mathscr{M}_{k}$, in
$\mathscr{M}_{k}\left[(\mathbb{L}^{r}-1)^{-1}\colon 0<r\in\mathbb{N}\right]$
and in $\widehat{\mathscr{M}_{k}}$, giving also a definition of a pole of such
functions. We conclude by stating the Monodromy Conjecture in two forms.

In §5 we discuss some facts concerning the poles of rational functions with
coefficients in our motivic ring. These properties will be useful when applied
to the formula that we will produce for the motivic zeta function of Hilbert
schemes.

In §6 we give a construction of the weak Néron models of Hilbert schemes of
points on surfaces and use those models to compute their motivic zeta
function.

We finally discuss the monodromy property in §7.

### Acknowledgements

The author is very grateful to his PhD advisor, Prof. Lars Halle, for
introducing him to the problem and for many interesting ideas and
conversations. The author is also grateful to Prof. Johannes Nicaise for many
helpful conversations.

This project has received funding from the European Union Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 801199.

##  1 Grothendick ring of varieties

In this section we introduce the rings containing the coefficients of the
formal series we shall study later on.

###  1.1 A motivic ring

#### 1.1.1

Fix a field $k$ and consider the category of algebraic varieties
$\operatorname{Var}_{k}$. Let $K_{0}(\operatorname{Var}_{k})$ be the group
whose generators are isomorphism classes in $\operatorname{Var}_{k}$ and whose
relations, called _scissor relations_ , are generated by elements in the form

$X-Y-(X\backslash Y)\,\text{, }$

whenever $X$ is an algebraic variety and $Y\subseteq X$ is a closed subscheme.
We denote by $[X]$ the class of $X\in\operatorname{Var}_{k}$ in
$K_{0}(\operatorname{Var}_{k})$.

#### 1.1.2

There is a unique ring structure on $K_{0}(\operatorname{Var}_{k})$ such that
for all $X,Y\in\operatorname{Var}_{k}$ one has $[X]\cdot[Y]=[X\times_{k}Y]$.
With this ring structure, $K_{0}(\operatorname{Var}_{k})$ is called _the
Grothendieck ring of varieties_. It is also characterized by the following
universal property:

##### Universal property of $K_{0}(\operatorname{Var}_{k})$.

Let $R$ be a ring and let $\Psi\colon\operatorname{Var}_{k}\rightarrow R$ a
multiplicative and additive invariant, i.e. a function which associate to a
variety $X$ an element $\Psi(X)\in R$ such that
$\Psi(X\times_{k}Y)=\Psi(X)\Psi(Y)$ and if $X=Y\cup Z$, then
$\Psi(X)+\Psi(Y\cap Z)=\Psi(Y)+\Psi(Z)$. Then there is a unique ring
homomorphism $\varphi\colon K_{0}(\operatorname{Var}_{k})\rightarrow R$ such
that

$\forall X\in\operatorname{Var}_{k},\mbox{ one has that
}\;\varphi([X])=\Psi(X)\,\text{. }$

Due to this universal property, $K_{0}(\operatorname{Var}_{k})$ naturally
embeds in the ring of Chow motives.

###  1.2 Localised Grothendieck ring

#### 1.2.1

A ring that is worth some consideration is obtained as a localization of
$K_{0}(\operatorname{Var}_{k})$.

######  Definition 1.1 (Localised Grothendieck ring of varieties).

Let us denote by $\mathbb{L}$ the element $[\mathbb{A}^{1}_{k}]\in
K_{0}(\operatorname{Var}_{k})$.

The localisation of $K_{0}(\operatorname{Var}_{k})$ with respect to
$\mathbb{L}$,

$\mathscr{M}_{k}\coloneqq
K_{0}(\operatorname{Var}_{k})[\mathbb{L}^{-1}]\,\text{, }$

is called _the localised Grothendieck ring of varieties_.

#### 1.2.2

By combining the universal property of localization and of
$K_{0}(\operatorname{Var}_{k})$, one can define $\mathscr{M}_{k}$ as a
universal ring for all the invariants $\operatorname{Var}_{k}\rightarrow R$
which send $\mathbb{A}_{k}^{1}$ in $R^{\ast}$.

###  1.3 Completed Grothendieck ring

#### 1.3.1

Consider the filtration
$\pazocal{F}^{n}\mathscr{M}_{k}\coloneqq\left\langle\mathbb{L}^{r}[X]|r\in\mathbb{Z}\,\text{,
}\dim[X]+r\leq-n\right\rangle_{\mathbb{Z}}$.

######  Definition 1.2 (Completed Grothendieck ring of varieties).

The completed Grothendieck ring of varieties $\widehat{\mathscr{M}_{k}}$ is
the completion of $\mathscr{M}_{k}$ with respect to the filtration
$\pazocal{F}^{\bullet}$.

###  1.4 Equivariant setting

#### 1.4.1

All the three rings above have an equivariant version, i.e. can be constructed
in the category $\operatorname{Var}_{k}^{G}$ of algebraic varieties endowed
with the action of an algebraic group $G$.

#### 1.4.2

As before, the _equivariant Grothendieck group of varieties_
$K_{0}(\operatorname{Var}_{k}^{G})$ is the group generated on the isomorphism
classes of $G-$varieties with relation of two kinds:

Scissor relations

Let $X$ be a $G-$variety and $Y$ a $G-$invariant closed subscheme, then

$X-Y-(X\backslash Y)\,\text{, }$

is $0$ in $K_{0}(\operatorname{Var}^{G}_{k})$.

Trivializing relations

Let $S\in\operatorname{Var}_{k}^{G}$ and let $V\rightarrow S$ be a
$G-$equivariant affine bundle of rank $d$. Then

$V-(S\times\mathbb{A}_{k}^{d})\in\operatorname{Var}_{k}^{G}$

is set to $0$, where $G$ acts trivially on the second factor of
$S\times\mathbb{A}_{k}^{d}$.

There is a unique ring structure on $K_{0}(\operatorname{Var}_{k}^{G})$ such
that for every two $X,Y\in\operatorname{Var}_{k}^{G}$, we have
$[X]\cdot[Y]\coloneqq[X\times_{k}Y]$, where the action of $G$ on
$X\times_{k}Y$ is the diagonal action.

#### 1.4.3

We use again the symbol $\mathbb{L}\coloneqq[\mathbb{A}_{k}^{1}]$, where the
group $G$ acts trivially on the affine line; thus the trivializing relations
tell us nothing more than:

$[V]=\mathbb{L}^{d}[S]\,\text{, }$

whenever $V\rightarrow S$ is an equivariant affine bundle of rank $d$.

#### 1.4.4

Similarly to what we did in the previous sections, we define the localisation
$\mathscr{M}_{k}^{G}\coloneqq
K_{0}(\operatorname{Var}_{k}^{G})[\mathbb{L}^{-1}]$ and its completion
$\widehat{\mathscr{M}_{k}^{G}}$ with respect to the filtration

$\pazocal{F}^{n}\mathscr{M}_{k}\coloneqq\left\langle\mathbb{L}^{r}[X]|r\in\mathbb{Z}\,\text{,
}\dim[X]+r\leq-n\right\rangle_{\mathbb{Z}}\,\text{. }$

##  2 Weak Néron models

###  2.1 Definition and basic constructions

#### 2.1.1

In order to define the main objects involved in this manuscript we shall need
to introduce the notion of a weak Néron model and to develop some techniques
involved in the construction of such models.

#### 2.1.2

The results we are going to state in this section hold in the context of
algebraic spaces, but we shall not work in such generality, thus we state them
only in the context of schemes.

######  Definition 2.1.

Let $R$ be a DVR and $K$ its fraction field and denote by
$\Delta\coloneqq\operatorname{Spec}R$. Let $X\rightarrow\operatorname{Spec}K$
be a smooth morphism of schemes. We say that the scheme
$\mathfrak{X}\rightarrow\Delta$ is a model for $X$ over $\Delta$ if the
morphism $\mathfrak{X}\rightarrow\Delta$ is flat and there is an isomorphism
$X\cong\mathfrak{X}_{K}$ in $\operatorname{Sch}_{K}$.

Moreover we say that the model $\mathfrak{X}$ has a property $\mathbf{P}$
(e.g. smooth, proper) if the morphism $\mathfrak{X}\rightarrow\Delta$ has such
property.

#### 2.1.3

The notion we are mostly interested in is that of _Weak Néron Model_ :

######  Definition 2.2.

We say that a model $\mathfrak{X}\rightarrow\Delta$ of $X$ has the weak
extension property if for any finite étale morphism $Z\rightarrow\Delta$,
there is a bijection
$\operatorname{Hom}_{\Delta}(Z,\mathfrak{X})\tilde{\rightarrow}\operatorname{Hom}_{K}(Z_{K},X)$,
$(f\colon Z\rightarrow\mathfrak{X})\mapsto f|_{Z_{K}}$.

A smooth model $\mathfrak{X}\rightarrow\Delta$ of
$X\rightarrow\operatorname{Spec}K$ that satisfies the weak extension property
is said to be a _weak Néron model_ of $X$.

######  Example 2.3.

Let $X\rightarrow\operatorname{Spec}K$ be a smooth and proper variety and let
$\pazocal{X}\rightarrow\operatorname{Spec}R$ be a proper regular model of $X$,
then the smooth locus of $\pazocal{X}\rightarrow\operatorname{Spec}R$ is a
weak Néron model of $X$, because the sections
$\operatorname{Spec}R\rightarrow\pazocal{X}$ do not meet the singular locus of
$\pazocal{X}_{0}\coloneqq X\times_{\operatorname{Spec}R}\operatorname{Spec}k$.

Since we will mostly work with _sncd_ models, i.e. regular models whose
central fibre $\pazocal{X}_{0}$ sits in $\pazocal{X}$ as a divisor with strict
normal crossings, this construction will be often used throughout this
manuscript.

###  2.2 The weak Néron model of a Hilbert scheme

#### 2.2.1

Let $V\rightarrow S$ be a flat morphism of schemes and let
$n\in\mathbb{N}_{+}$ a positive integer. Consider the scheme
$\operatorname{Sym}^{n}(V/S)\coloneqq V^{n}/{\Sigma_{n}}$, where the symmetric
group acts on the product by permuting the factors. There is a natural
morphism

$\operatorname{Hilb}^{n}(V/S)\rightarrow\operatorname{Sym}^{n}(V/S)$

called Hilbert-Chow morphism, sending a subscheme of $V$ to a weighted sum of
finite $S-$subschemes of $V$.

#### 2.2.2

If $V\rightarrow S$ is a smooth map of relative dimension $2$, then
$\operatorname{Hilb}^{n}(V/S)$ is a smooth scheme of relative dimension $2n$,
in particular the Hilbert-Chow morphism is a resolution of singularities of
$\operatorname{Sym}^{n}(V/S)$.

#### 2.2.3

Our main motivation for studying the Hilbert schemes of points on a K3 surface
comes from the following:

######  Theorem 2.4.

If $X\rightarrow\operatorname{Spec}\mathbb{C}$ is a K3 surface, then
$\operatorname{Hilb}^{n}(X/\mathbb{C})$ is an irreducible holomorphic
symplectic variety of dimension $2n$.

#### 2.2.4

Also the case of an abelian surface $A\rightarrow\operatorname{Spec}K$ is
interesting, because in that case the Hilbert scheme is anyway a smooth
Calabi-Yau variety, i.e. a variety with trivial canonical bundle; moreover all
IHS varieties of Kummer type over $\mathbb{C}$ are obtained as a deformation
of a subscheme of $\operatorname{Hilb}^{n}(A/\mathbb{C})$.

#### 2.2.5

We shall actually construct a weak Néron model of $\operatorname{Hilb}^{n}(X)$
in §6.1 and in particular in Proposition 6.1, but we anticipate here the
strategy we are going to follow.

We start with a regular sncd model $\mathfrak{X}\rightarrow\Delta$. Its smooth
locus $\mathfrak{X}_{\operatorname{sm}}$ is a weak Néron model of $X$ and its
Hilbert schemes
$\operatorname{Hilb}^{n}(\mathfrak{X}_{\operatorname{sm}}/\Delta)$ are smooth
models of $\operatorname{Hilb}^{n}(X)$ but they do not satisfy the weak
extension property for $n>1$. This is due to the fact that a closed subscheme
$Z\subseteq X$ of length $n$ is not necessarily supported on $K-$rational
points.

On the other hand, at least for $n<\operatorname{char}K$ (if it is positive),
there is an integer $\tilde{n}$ such that the support of all the closed
subschemes of $X$ of length $\leq n$ is defined on a suitable field extension
$K(\tilde{n})/K$. It turns out that base-changing with respect to the field
extension $K(\tilde{n})/K$ induces a correspondence between closed subschemes
$Z\subseteq X$ of length $n$ and
$\operatorname{Gal}(K(\tilde{n})|K)-$invariant closed subschemes of
$X(\tilde{n})$ of length $n$. Via equivariant semistable reduction, which we
explain in §3, it will be possible to construct smooth models of
$X_{K(\tilde{n})}$ endowed with an action of
$\operatorname{Gal}(K(\tilde{n})|K)$ extending its natural action on
$X_{K(\tilde{n})}$.

Thus we will be able to construct a smooth model of
$\operatorname{Hilb}^{n}(X_{K(\tilde{n})})$ endowed with an action of
$\operatorname{Gal}(K(\tilde{n})|K)$ which has a property slightly weaker than
the weak extension property. Nevertheless, using what we are going to explain
in this section, namely the Weil restriction of scalars, we will be finally
able to construct a weak Néron model for $\operatorname{Hilb}^{n}(X)$ and use
it to compute the motivic integral.

###  2.3 Weil restriction of scalars

#### 2.3.1

In this paragraph we study some generalities about the functor of the
restriction of scalars. The main content of this paragraph is Proposition 2.7,
which shall allow us to construct weak Néron models of a finite, tamely
ramified base-change of $X$.

######  Definition 2.5.

Let $S^{\prime}\rightarrow S$ be a morphism of schemes and
$\pazocal{Y}\rightarrow S^{\prime}$ be a scheme. The functor
$\operatorname{Res}_{S^{\prime}/S}(\pazocal{Y})\colon(\operatorname{Sch}_{S})^{\operatorname{opp}}\rightarrow\operatorname{Sets}$
defined by $T\mapsto\pazocal{Y}(T\times_{S}S^{\prime})$ is called the Weil
restriction of scalars of $\pazocal{Y}$ along $S^{\prime}\rightarrow S$.

When $\operatorname{Res}_{S^{\prime}/S}(\pazocal{Y})$ is represented by a
scheme, we say that the Weil restriction of $\pazocal{Y}$ along
$S^{\prime}\rightarrow S$ exists.

#### 2.3.2

It follows from [3, Theorem 7.6] that if $S^{\prime}\rightarrow S$ is a
finite, flat and locally of finite presentation and $\pazocal{Y}\rightarrow
S^{\prime}$ is quasi-projective, then the Weil restriction of $\pazocal{Y}$
along $S^{\prime}\rightarrow S$ exists. This will always be the case,
throughout this manuscript.

######  Remark 2.6.

Consider arbitrary morphisms of schemes $S^{\prime}\rightarrow S$ and
$\pazocal{X}\rightarrow S$; then the universal property of fibered products
implies the following:

$\operatorname{Res}_{S^{\prime}/S}(\pazocal{X}\times_{S}S^{\prime})=\underline{\operatorname{Hom}}_{S}(S^{\prime},\pazocal{X})\,\text{,
}$

where $\underline{\operatorname{Hom}}_{S}(S^{\prime},\pazocal{X})$ is the
sheaf
$T\mapsto\operatorname{Hom}_{T}(S^{\prime}\times_{S}T,\pazocal{X}\times_{S}T)$.

#### 2.3.3

Let $R$ be a complete DVR and let $K$ be its fraction field and $k$ be its
residue field. We suppose $k$ is algebraically closed. Let $K\subseteq L$ be a
finite, tame, Galois extension with
$\pazocal{G}\coloneqq\operatorname{Gal}(L|K)$ and denote by $R_{L}$ the
integral closure of $R$ in $L$. For an arbitrary scheme
$T\rightarrow\operatorname{Spec}R$, the action of $\pazocal{G}$ on
$\operatorname{Spec}R_{L}$ induces an action on $T_{R_{L}}$, thus, by
precomposition, an action on the Weil restriction,
$\operatorname{Res}_{R_{L}/R}(\pazocal{X})$, of an $R_{L}-$scheme
$\pazocal{X}$: more precisely, all $g\in\pazocal{G}$, it induce a
automorphisms $g\colon T_{R_{L}}\rightarrow T_{R_{L}}$, the action of
$\pazocal{G}$ sends the point corresponding to the morphism $\psi\colon
T_{R_{L}}\rightarrow\pazocal{X}$ to the composition $g(\psi)\coloneqq\psi\circ
g^{-1}$.

######  Proposition 2.7.

Let $\mathfrak{X}^{\prime}\rightarrow R_{L}$ be a weak Néron model for
$X_{L}$, then
$\mathfrak{X}\coloneqq\left(\operatorname{Res}_{R_{L}/R}\mathfrak{X}^{\prime}\right)^{\pazocal{G}}$
is a weak Néron model for $X$.

###### Proof.

It follows from [4, Proposition A.5.2] that the operations of taking the
generic fibre and taking the restriction of scalars commute, therefore

$(\operatorname{Res}_{R_{L}/R}\mathfrak{X}^{\prime})_{K}=\operatorname{Res}_{L/K}X_{L}=\underline{\operatorname{Hom}}_{K}(\operatorname{Spec}L,X)\,\text{,
}$

where the last equality follows from Remark 2.6.

Let $T\rightarrow\operatorname{Spec}K$ be a scheme, then a morphism
$T_{L}\rightarrow X\times_{\operatorname{Spec}K}T$ is $\pazocal{G}-$invariant
if and only if it factors through $T_{L}\rightarrow T_{L}/\pazocal{G}=T$, this
gives a bijection between
$\operatorname{Hom}_{T}(T_{L},X\times_{\operatorname{Spec}K}T)$ and the set of
sections of $X\times_{\operatorname{Spec}K}T\rightarrow T$, which in turn is
$\operatorname{Hom}_{K}(T,X)=X(T)$. Therefore we have that

$\left((\operatorname{Res}_{R_{L}/R}\mathfrak{X^{\prime}}){G}\right)_{K}\cong
X\,\text{. }$

Since $\mathfrak{X}^{\prime}\rightarrow\operatorname{Spec}R_{L}$ is a weak
Néron model for $X_{L}$, it is in particular a smooth morphism, thus, by [4,
Proposition A.5.2] also
$\operatorname{Res}_{R_{L}/R}(\mathfrak{X}^{\prime})\rightarrow\operatorname{Spec}R$
is smooth. It follows by [6, Proposition 3.4] that the $\pazocal{G}-$fixed
locus $\mathfrak{X}$ is smooth as well.

In order to conclude the proof, we only need to show that all the $K-$valued
points of $X$ extend to $R-$valued points of $\mathfrak{X}$, since we assumed
$R$ to be complete.

A morphism $\operatorname{Spec}K\rightarrow X$ induces a unique morphism,
which is also $\pazocal{G}-$invariant, $\operatorname{Spec}L\rightarrow
X_{L}$. Since $\mathfrak{X}^{\prime}$ is a weak Néron model for $X_{L}$, such
map extends to a unique map
$\operatorname{Spec}R_{L}\rightarrow\mathfrak{X}^{\prime}$, which is also
$\pazocal{G}-$invariant and correspond, by the definition of the restriction
of scalars, to a map
$\operatorname{Spec}R_{L}\rightarrow\operatorname{Res}_{R_{L}/R}\mathfrak{X}^{\prime}$
and, by $\pazocal{G}-$invariance it factors through a map
$\operatorname{Spec}R_{L}\rightarrow\mathfrak{X}\subseteq\operatorname{Res}_{R_{L}/R}\mathfrak{X}^{\prime}$.
∎

#### 2.3.4

The following lemma says that this construction is well behaved with respect
to a tower of extensions:

######  Lemma 2.8.

Let $K\subseteq F\subseteq L$ be a tower of Galois extensions such that also
$K\subseteq L$ is normal. Let $G\coloneqq\operatorname{Gal}(L|K)$,
$N\coloneqq\operatorname{Gal}(L/F)$ and
$G/N=H\coloneqq\operatorname{Gal}(F|K)$. Denote by $R_{F}$ and $R_{L}$ the
integral closures of $R$ in $F$ and $L$, respectively, and $\Delta_{F}$,
$\Delta_{L}$ their spectra. Let
$\pazocal{F}\colon\operatorname{Sch}_{\Delta_{L}}^{\operatorname{opp}}\rightarrow\operatorname{Sets}$
be a functor endowed with an action of $G$ compatible with its action on
$\Delta_{L}$.

Then the following two functors are naturally isomorphic:

${\left(\operatorname{Res}_{\Delta_{L}/\Delta}\pazocal{F}\right)}^{G}\cong{\left(\operatorname{Res}_{\Delta_{F}/\Delta}{\left(\operatorname{Res}_{\Delta_{L}/\Delta_{F}}\pazocal{F}\right)}^{N}\right)}^{H}\,\text{.
}$

###### Proof.

The left hand side is equal to

${\left(\operatorname{Res}_{\Delta_{F}/\Delta}\left(\operatorname{Res}_{\Delta_{L}/\Delta_{F}}\pazocal{F}\right)\right)}^{G}={\left({\left(\operatorname{Res}_{\Delta_{F}/\Delta}\left(\operatorname{Res}_{\Delta_{L}/\Delta_{F}}\pazocal{F}\right)\right)}^{N}\right)}^{H}\,\text{,
}$

where $N$ acts as a subgroup of $G$ and $H=G/N$ inherits the action of $G$ on
the $N-$invariant locus. Thus, we only need to show that there is an
$H-$equivariant isomorphism of functors:

$\operatorname{Res}_{\Delta_{F}/\Delta}{\left(\operatorname{Res}_{\Delta_{L}/\Delta_{F}}\pazocal{F}\right)}^{N}\cong{\left(\operatorname{Res}_{\Delta_{F}/\Delta}\left(\operatorname{Res}_{\Delta_{L}/\Delta_{F}}\pazocal{F}\right)\right)}^{N}\,\text{.
}$

Given a scheme morphism $T\rightarrow\Delta$, we have that

$\displaystyle\operatorname{Res}_{\Delta_{F}/\Delta}{\left(\operatorname{Res}_{\Delta_{L}/\Delta_{F}}\pazocal{F}\right)}^{N}(T)$
$\displaystyle=$
$\displaystyle{\left(\operatorname{Res}_{\Delta_{L}/\Delta_{F}}\pazocal{F}\right)}^{N}(T\times_{\Delta}\Delta_{F})$
$\displaystyle=$
$\displaystyle{\left(\pazocal{F}(T\times_{\Delta}\Delta_{L})\right)}^{N}$
$\displaystyle=$
$\displaystyle{\left(\operatorname{Res}_{\Delta_{F}/\Delta}\left(\operatorname{Res}_{\Delta_{L}/\Delta_{F}}\pazocal{F}\right)(T)\right)}^{N}\,\text{,
}$

and we are done. ∎

#### 2.3.5

The construction above can be made more explicit: in the following paragraphs
we will describe the central fibre and the canonical divisor of $\mathfrak{X}$
in terms of $\mathfrak{X}^{\prime}$.

###  2.4 Weil restriction and the central fibre

#### 2.4.1

This subsection and the next one summarize some results contained in an
unpublished manuscript of Lars Halle and Johannes Nicaise. I am grateful to
them for letting me use these results which are crucial for the computation of
the motivic integral in §6.3. We keep the notation of the previous paragraph.

#### 2.4.2

The inclusion
$\mathfrak{X}\subseteq\operatorname{Res}_{R_{L}/R}\mathfrak{X}^{\prime}$
corresponds, according to the definition of the restriction of scalars, to a
map of $R_{L}-$schemes

$h\colon\mathfrak{X}\times_{R}R_{L}\rightarrow\mathfrak{X}^{\prime}\,\text{,
}$

which gives, over the special fibres a morphism of $k-$schemes:

$h_{k}\colon\mathfrak{X}_{k}\rightarrow\mathfrak{X}_{k}^{\prime}\,\text{. }$

#### 2.4.3

On the other hand, we can characterize $\mathfrak{X}_{k}$ in a different way,
using the _Greenberg schemes_.The following definition will cover the cases we
shall use:

Let $d=[L:K]$ and let $\mathfrak{m}$ be the maximal ideal of $R_{L}$; for
$i\in\\{0,\dots,d-1\\}$, let $R_{L,i}\coloneqq R_{L}/(\mathfrak{m}^{i+1})$.
For a separated, smooth morphism
$\pazocal{A}\rightarrow\operatorname{Spec}R_{L}$, consider the functor

$\operatorname{Gr}_{i}(\pazocal{A})\coloneqq\operatorname{Res}_{R_{L,i}/k}(\pazocal{A}\times_{R_{L}}R_{L,i})\,\text{,
}$

which is representable by a separated, smooth scheme, as it follows from the
proof of [3, Proposition 7.6]; this is also called the called _level $i$
Greenberg scheme of $\pazocal{A}$_.

Clearly, $\operatorname{Gr}_{0}(\pazocal{A})=\pazocal{A}_{k}$, while
$\operatorname{Gr}_{d-1}(\pazocal{A})=\operatorname{Res}_{R_{L}/R}(\pazocal{A})_{k}$,
since $d=[L:K]$ is also the ramification index of $R_{L}\rightarrow R$ at
their closed points.

#### 2.4.4

In our case we have that
$\mathfrak{X}_{k}=(\operatorname{Gr}_{d-1}(\mathfrak{X}^{\prime}){G}$. Indeed,
if $T$ is a $k-$scheme, we have that

$\mathfrak{X}_{k}(T)={\left\\{{f\colon
T\times_{R}R_{L}\rightarrow\mathfrak{X}^{\prime}}\right\\}}^{\pazocal{G}}=(\operatorname{Gr}_{d-1}(\mathfrak{X}^{\prime})(T))^{\pazocal{G}}\,\text{.
}$

#### 2.4.5

The natural truncation maps of Greenberg schemes define
$\pazocal{G}-$equivariant affine bundles, in particular
$\operatorname{Gr}_{d-1}(\mathfrak{X}^{\prime})\rightarrow\operatorname{Gr}_{0}(\mathfrak{X^{\prime}})=\mathfrak{X}^{\prime}_{k}$
is a composition of affine bundles. By taking the $\pazocal{G}-$invariant loci
of this map, we get a description, at least locally, of $\mathfrak{X}_{k}$ as
an affine bundle over $(\mathfrak{X}^{\prime}_{k}){G}$, in the sense that for
each connected component $C\subseteq\mathfrak{X}_{k}$, there is a connected
component $C^{\prime}\subseteq(\mathfrak{X}^{\prime}_{k}){G}$ such that $C$ is
an $\mathbb{A}_{k}^{r}-$bundle over $C^{\prime}$, where
$r=\dim(\mathfrak{X}^{\prime}_{k})-\dim C^{\prime}$. In particular the
following relation holds in $K_{0}(\operatorname{Var}_{k})$:

$[C]=\mathbb{L}^{\dim(\mathfrak{X}_{k})-\dim C^{\prime}}[C^{\prime}]\,\text{.
}$

###  2.5 Weil restriction and canonical divisor

#### 2.5.1

Let us keep the notation introduced in the previous paragraph, but we also
assume that $X$ is a Calabi-Yau variety, i.e. it has trivial canonical bundle,
and that a volume form $\omega\in\Omega_{X/K}^{\dim X}(X)$ is given. Let
$\omega_{L}\in\Omega_{X_{L}/L}^{\dim X}(X_{L})$ be the pull-back of $\omega$
under the base-change map. In this paragraph we shall study the order of
vanishing of $\omega$ on each component of $\mathfrak{X}_{k}$, which we will
define as follows, adapting the definition given in [13, §4.1].

#### 2.5.2

Let $p\in\mathfrak{X}_{k}$ be a closed point. Since $R$ is an Henselian ring
and since $\mathfrak{X}\rightarrow\operatorname{Spec}R$ is smooth, there is at
least a section $\psi\colon\operatorname{Spec}R\rightarrow\mathfrak{X}$ such
that $\psi(0)=p$. Consider the line bundle $L\coloneqq\psi^{\ast}\Omega^{\dim
X}_{\mathfrak{X}/R}$ over $\operatorname{Spec}R$. There is $a\in\mathbb{Z}$
such that $\pi^{a}\omega$ extends to a global section of
$\omega^{\prime}\in\Omega^{\dim X}_{\mathfrak{X}/R}(\mathfrak{X})$, where
$\pi\in R$ is the uniformizer. So its pull-back $\psi^{\ast}(\omega^{\prime})$
is a global section of $L$. Let $M\coloneqq
L/\psi^{\ast}\omega^{\prime}\pazocal{O}_{\operatorname{Spec}R}$ be the
quotient of $\pazocal{O}_{\operatorname{Spec}R}-$modules.

######  Definition 2.9.

The order of $\omega$ at $p$ is defined as:

$\operatorname{ord}_{p}(\omega)\coloneqq\inf\\{b\in\mathbb{N}\colon\pi^{b}M=0\\}-a\,\text{.
}$

If $C\subseteq\mathfrak{X}_{k}$ is a connected component, then
$\operatorname{ord}_{p}(\omega)$ does not depend on the coice of the closed
point $p\in C$, so we define $\operatorname{ord}_{C}(\omega)$ as the order of
$\omega$ at any of its closed point. If $\operatorname{ord}_{C}(\omega)>0$ we
say that $C$ is a _zero_ of $\omega$, if $\operatorname{ord}_{C}(\omega)<0$ we
say that it is a _pole_ of $\omega$.

#### 2.5.3

Let $Z$ be a smooth scheme defined over a field $F$ and let $V\rightarrow Z$
be a vector bundle over $Z$. Consider a cyclic group $G\cong\mu_{d}$ acting
equivariantly on $V\rightarrow Z$ and let $z\in Z$ be a fixed point. There is
a unique sequence of integers $(j_{1},j_{2},\dots,j_{\operatorname{rk}V})$
such that $0\leq j_{1}\leq j_{2}\leq\cdots\leq j_{\operatorname{rk}V}\leq d-1$
such that $V_{z}$ has a basis $v_{1},\dots,v_{\operatorname{rk}V}$ of
eigenvectors such that $\zeta\star v_{i}=\zeta^{-j_{i}}\cdot v_{i}$ (where
$\zeta$ is any generator of $\mu_{d}$); the tuple $(j_{i})_{i}$ is called the
_tuple of exponents_ of the $G-$action.

######  Definition 2.10.

We define the _conductor_ of the action of $G$ in $z$ as the sum:

$c(V,z)\coloneqq\sum_{i=1}^{\operatorname{rk}V}j_{i}\,\text{. }$

If $C\subseteq Z^{G}$ is an irreducible subscheme, then for all
$z,z^{\prime}\in C$ one has that $c(V,z)=c(V,z^{\prime})$, so we simply denote
by $c(V,C)$ either of the conductors. Moreover we denote $c(Z,C)$ the
conductor $c(T_{Z},C)$.

######  Lemma 2.11.

Let $C$ be a connected component of $\mathfrak{X}_{k}$ and let
$C^{\prime}=h(C)$, where $h$ is the map defined in §2.4.2. Then:

$\operatorname{ord}_{C}(\omega)=\frac{\operatorname{ord}_{C^{\prime}}(\omega_{L})-c(\mathfrak{X}^{\prime}_{k},C^{\prime})}{[L\colon
K]}\,\text{. }$

###### Proof.

The map $\mathfrak{X}\times_{R}R_{L}\rightarrow\mathfrak{X}^{\prime}$ induces
a monomorphism

$\alpha\colon(\psi^{\prime})^{\ast}\Omega_{\mathfrak{X}^{\prime}/R_{L}}\rightarrow\psi^{\ast}\Omega_{\mathfrak{X}/R}\otimes_{R}R_{L}$

sending $\omega_{L}$ to $\omega\otimes 1$; in particular

$\operatorname{length}\left({\alpha\left({(\psi^{\prime})^{\ast}\Omega_{\mathfrak{X}^{\prime}/R_{L}}}\right)/\langle\omega\otimes
1\rangle}\right)=[L:K]\operatorname{length}\left({(\psi^{\prime})^{\ast}\Omega_{\mathfrak{X}^{\prime}/R_{L}}/\langle\omega_{L}\rangle}\right)\,\text{,
}$

thus the statement shall follow from the fact that
$\operatorname{coker}\alpha=c(\mathfrak{X}^{\prime}_{k},C^{\prime})$.

On the other hand, under the identification

$T_{\mathfrak{X}/R}=\underline{\operatorname{Hom}}_{R}(R[\varepsilon]/(\varepsilon^{2}),\mathfrak{X})=\underline{\operatorname{Hom}}_{R_{L}}(R_{L}[\varepsilon]/(\varepsilon^{2}),\mathfrak{X}^{\prime})^{\operatorname{Gal}(L|K)}=T_{\mathfrak{X}^{\prime}/{R_{L}}}^{\operatorname{Gal}(L|K)}\,\text{,
}$

the tangent map $T_{h}\colon T_{\mathfrak{X}/R\times_{R}R_{L}}\rightarrow
T_{\mathfrak{X}^{\prime}/R_{L}}$ induces a map

$\beta\colon(\psi^{\prime})^{\ast}(T_{\mathfrak{X}^{\prime}/R_{L}})^{\operatorname{Gal}(L|K)}\otimes_{R}R_{L}\rightarrow(\psi^{\prime})^{\ast}T_{\mathfrak{X}^{\prime}/R_{L}}\,\text{.
}$

Fix a base of eigenvectors of $T_{\mathfrak{X}_{k}^{\prime}}$; the upcoming
Lemma 2.12, applied to the subspaces generated by each element of the base,
implies that
$\operatorname{coker}(\beta)=\oplus_{i=1}^{d}R_{L}/\mathfrak{m}_{L}^{j_{i}}$,
hence

$\operatorname{coker}(\alpha)=\bigwedge_{i=1}^{d}R_{L}/\mathfrak{m}_{L}^{j_{i}}=R_{L}/\mathfrak{m}_{L}^{c(\mathfrak{X},C^{\prime})}\,\text{.
}$

∎

######  Lemma 2.12.

Let $M$ a free $R_{L}-$module of rank 1. Assuming that
$\operatorname{Gal}(L|K)$ acts $R-$linearly on $M$ from the left. Let $j$ be
the exponent of the action induced on $M\otimes_{R_{L}}k$, as in Definition
2.10. Then the natural morphism
$M^{\operatorname{Gal}(L|K)}\otimes_{R}R_{L}\rightarrow M$ has cokernel
isomorphic to $R_{L}/\mathfrak{m}_{L}^{j}$.

###### Proof.

Let us fix an element $v\in M$ such that $0\neq v\otimes 1\in
M\otimes_{R_{L}}k$; by our hypothesis we have that $(\zeta\ast v)\otimes
1=\zeta^{-j}v\otimes 1$. Let $\pi_{L}$ a uniformizer for $R_{L}$ such that
$\pi_{L}^{d}\in K$; if $0\leq b\leq d-1$ the vectors
$v_{b}\coloneqq\pi_{L}^{b}v\otimes 1\in M\otimes_{R}k$ form a base of the
vector space $M\otimes_{R}k\cong M\otimes_{R_{L}}R_{L}/\mathfrak{m}_{L}^{d}$.
Moreover that one is a base of $G-$eigenvectors, for $\zeta\ast
v_{b}=\zeta^{b-j}v_{b}$.

By Henselianity we can lift the base $\\{v_{b}\\}$ to an $R-$base
$\\{w_{b}\colon 0\leq b\leq d-1\\}$ of $M$ such that $\zeta\ast
w_{b}=\zeta^{b-j}w_{b}$.

Now let $x=a_{0}w_{0}+\cdots+a_{d-1}w_{d-1}\in M$ be an arbitrary element. We
have that $x\in M^{G}$ iff $x-\zeta\ast x=0$, i.e. iff

$\sum_{b=0}^{d-1}a_{b}(1-\zeta^{b-j})w_{b}=0\,\text{, }$

therefore, the $R-$module $M^{G}$ is generated by $w_{j}$.

It follows that $M^{G}\otimes R_{L}$ is sent onto $w_{j}M\subseteq M$, which
leads to our coveted statemet. ∎

##  3 Equivariant semistable reduction

###  3.1 Preliminaries on logarithmic geometry

#### 3.1.1

As seen in Example 2.3, every proper regular model of a smooth variety
contains a weak Néron model, on the other hand it is not always possible to
construct a semistable model over a DVR of all the varieties defined on its
fraction field $K$. Given a variety $X$ over $K$ that admits a regular _sncd_
model, under suitable assumptions there exists a finite extension $F/K$ such
that the basechange $X_{F}$ admits a semistable model over the ring of
integers in $F$. In this section we will show a way to produce such models
over Galois extensions of a given field in such a way that the Galois group
acts naturally on the model.

#### 3.1.2

From now on, let $R$ be a DVR and, let $K$ be its fraction field and $k$ its
residue field, which we assume being perfect and containing all the roots of
unity. Let $\Delta\coloneqq\operatorname{Spec}R$, $0\in\Delta$ be its closed
point and $\Delta^{\ast}\coloneqq\Delta\backslash 0=\operatorname{Spec}K$. Due
to our assumptions, given a positive integer $m$, not divisibe by the
characheristic of $k$ (if positive), there is a unique finite morphism of
degree $m$, $\Delta(m)\rightarrow\Delta$, which is tame and totally ramified
over $0$ and such that $\Delta^{\ast}(m)\rightarrow\Delta^{\ast}$ is a Galois
étale morphism with automorphism group
$\operatorname{Aut}_{\Delta}(\Delta(m))\cong\mu_{m}$; we denote by
$0_{m}\in\Delta(m)$ the ramified point, omitting the index $m$ if no confusion
with $0\in\Delta$ may arise.

#### 3.1.3

We use the standard language of monoids and logarithmic geometry, as in [9] or
in [1] and we refer to [19] for the material concerning toric schemes. We only
recall here the notions that will be used later in this section:

#### 3.1.4

Let $V^{\dagger}=(V,\pazocal{M}_{V})$ be a locally Noetherian fs logarithmic
scheme and $x\in V$ be a point. We say that $V^{\dagger}$ is log regular at
$x$ if the following:

  * •

The ring $\pazocal{O}_{V,x}/\pazocal{M}_{V,x}\pazocal{O}_{V,x}$ is regular;

  * •

$\dim\pazocal{O}_{V,x}=\dim\pazocal{O}_{V,x}/\pazocal{M}_{V,x}\pazocal{O}_{V,x}+\dim\pazocal{M}_{V,x}$.

are satisfied. A log structure $V^{\dagger}$ is logarithmically regular if it
is log regular at each of its points.

#### 3.1.5

A _(sharp) monoidal space_ is a couple $(T,\pazocal{M}_{T})$ consisting of a
topological space $T$ and a sheaf of (sharp) monoids over $T$. A morphism of
monoidal spaces
$(T^{\prime},\pazocal{M}_{T^{\prime}})\rightarrow(T,\pazocal{M}_{T})$ is the
datum of a continuous map $f\colon T^{\prime}\rightarrow T$ and a map of
sheaves of monoids $h\colon
f^{-1}(\pazocal{M}_{T^{\prime}})\rightarrow\pazocal{M}_{T}$. If
$(T,\pazocal{M}_{T})$ is a monoidal space, one can define its sharpification
$(T,\pazocal{M}_{T})^{\sharp}\coloneqq(T,\pazocal{M}_{T}^{\sharp})$, where
$\pazocal{M}_{T}^{\sharp}$ is the sheafification of
$U\mapsto\pazocal{M}_{T}(U)^{\sharp}$.

#### 3.1.6

Let $P$ be a monoid, consider the set $\operatorname{Spec}P$ of its prime
ideals, i.e. its submonoids whose complement in $P$ is again a submonoid of
$P$, endowed with the topology generated by $\\{D(f)\colon f\in P\\}$, where
$D(f)=\\{\mathfrak{p}\in\operatorname{Spec}P\colon f\notin\mathfrak{p}\\}$.
Let $\pazocal{M}_{P}$ be the sheaf of monoids such that
$\pazocal{M}_{P}(D(f))=P_{f}$. The space
$(\operatorname{Spec}P,\pazocal{M}_{P})$, or simply $\operatorname{Spec}P$, is
the _spectrum_ of $P$. Moreover $(SpecP)^{\sharp}$ is called the _sharp
spectrum_ of $P$.

#### 3.1.7

A _fan_ is a sharp monoidal space $(F,\pazocal{M}_{F})$, that can be covered
by open subsets isomorphic to sharp spectra of monoids. A fan is _locally fine
and saturated_ (or locally fs) if it can be covered by spectra of fs monoids.
The category of fans is a full subcategory of the category of sharp monoidal
spaces, i.e. a morphism of fans is just a morphism of monoidal spaces between
fans.

#### 3.1.8

Let $(F,\pazocal{M}_{F})$ be a fan. A subdivision is a morphism of fans
$\varphi\colon(F^{\prime},\pazocal{M}_{F^{\prime}})\rightarrow(F,\pazocal{M}_{F})$
such that:

  * •

For every $t\in F^{\prime}$, the map induced on stalks
$\pazocal{M}_{F,\varphi(t)}^{\operatorname{grp}}\rightarrow\pazocal{M}_{F,t}^{\operatorname{grp}}$
is surjective;

  * •

The composition with $\varphi$ induces a bijection
$\operatorname{Hom}(\operatorname{Spec}\mathbb{N},F^{\prime})\rightarrow\operatorname{Hom}(\operatorname{Spec}\mathbb{N},F)$.

#### 3.1.9

Let $V^{\dagger}=(V,\pazocal{M}_{V})$ be a log regular scheme. There is a fan
associated to $V$ whose underlying topological space is the topological
subspace of $V$ whose points are $F(V)\coloneqq\\{x\in
V\colon\mathfrak{m}_{x}\mbox{ is generated by
}\pazocal{M}_{V,x}\backslash\pazocal{O}_{V,x}^{\times}\\}$, where
$\mathfrak{m}_{x}$ is the maximal ideal of $\pazocal{O}_{V,x}$, and whose
structural sheaf $\pazocal{M}_{F(V)}$ is the restriction of $\pazocal{M}_{V}$.

###  3.2 The construction of a semistable model

#### 3.2.1

Let $\Delta^{\dagger}$ be the log scheme supported on $\Delta$ with the
divisorial log structure associated to $0\in\Delta$. Similarly
$\Delta^{\dagger}(m)$ shall denote the scheme $\Delta(m)$ introduced above
endowed with the divisorial log structure for $0\in\Delta(m)$. Let
$S\rightarrow\Delta$ a flat morphism of relative dimension 2 with smooth
generic fibre and let $S^{\dagger}$ be the divisorial logarithmic structure
associated to the central fibre $S_{k}$. Assume
$S^{\dagger}\rightarrow\Delta^{\dagger}$ to be logarithmically smooth. Let us
fix a positive integer $N$ such that the multiplicity of any irreducible
component of $S_{0}$ divides $N$ and let $m$ be an arbitrary positive integer;
in case $\operatorname{char}K=p>0$ let us assume that the multiplicities of
the irreducible components of $S_{0}$ are coprime with $p$, choose $N,m$
coprime with $p$. Let $\Gamma$ be the fan associated to the logarithmic
structure of $S^{\dagger}$.

#### 3.2.2

The fs-basechange
$S^{\dagger}\times_{\Delta^{\dagger}}^{\operatorname{fs}}\Delta(mN)^{\dagger}\rightarrow\Delta^{\dagger}$
which is a normal space, yet not necessarily regular, is canonically endowed
with an equivariant $\mu_{mN}-$action which induces an action on the fan of
$S^{\dagger}\times_{\Delta^{\dagger}}^{\operatorname{fs}}\Delta(mN)^{\dagger}$,
which we call $\Gamma(mN)$.

#### 3.2.3

We define an equivariant resolution of
$S^{\dagger}\times_{\Delta^{\dagger}}^{\operatorname{fs}}\Delta(mN)^{\dagger}\rightarrow\Delta^{\dagger}$
as follows:

  1. _Step i._

_Choosing the locus for the blow-up._ We choose any conic cell of $\Gamma(mN)$
and we choose a simplicial subdivision of such cell; we copy this subdivision
in all the cells lying in the same orbit of the action of $\mu_{mN}$. We
repeat this step until we obtain a simplicial cone complex, which we name
$\widetilde{\Gamma}(mN)$.

  2. _Step ii._

_The blow-up gives a semistable model._ Because of [20, Lemma 2.4 (2) and
Lemma 2.7 (2)], the simplicial subdivision in step 1 induces a resolution
$p\colon S(mN)^{\dagger}\rightarrow
S^{\dagger}\times_{\Delta^{\dagger}}^{\operatorname{fs}}\Delta(mN)^{\dagger}$
whose underlying scheme $S(mN)$ is semistable.

  3. _Step iii._

_Description of the action of $\mu_{mN}$._ Let $\zeta\in\mu_{mN}$ a generator
and let $\zeta\colon
S^{\dagger}\times_{\Delta^{\dagger}}^{\operatorname{fs}}\Delta(mN)^{\dagger}\rightarrow
S^{\dagger}\times_{\Delta^{\dagger}}^{\operatorname{fs}}\Delta(mN)^{\dagger}$
its induced isomorphism. By composition we obtain a birational morphism
$S(mN)\rightarrow
S^{\dagger}\times_{\Delta^{\dagger}}^{\operatorname{fs}}\Delta(mN)^{\dagger}$
and, thence, a birational map $S(mN)\dashrightarrow S(mN)$.This is actually an
isomorphism because the subdivision of the fan is equivariant and so is the
ideal involved in the blow-up.

#### 3.2.4

We conclude the section with the following lemma which grasp the most
important, at least for our purpose, property of the action of $\mu_{mN}$ on
$S(mN)$; in fact this is the only reason why we did perform the construction
in this way:

######  Definition 3.1.

For each point $p\in S(mN)_{k}$ let $\operatorname{Stab}_{p}\subseteq\mu_{mN}$
be the subgroup consisting of the elements that fix $p$.

######  Lemma 3.2.

$\operatorname{Stab}_{-}$ is locally constant on
$S(mN)_{k,\operatorname{sm}}$. In particular if a point $p\in
S(mN)_{k,\operatorname{sm}}$ is fixed under the action of $\mu_{N}$, then the
whole connected component of $S(mN)_{k,\operatorname{sm}}$ containing $p$ is
fixed under such action.

###### Proof.

Each point of $S$ admits an étale neighbourhood $U\twoheadrightarrow
W\subseteq S$ (with $W$ being a Zariski open subset of $S$) such that the map
$U\rightarrow\Delta$ splits through a smooth map $U\rightarrow
V\subseteq\operatorname{Spec}R[P]$, where $P$ is a torsion-free monoid giving
a local chart $P\rightarrow\pazocal{M}$ for the logarithmic structure on $U$.
The embedding $V\hookrightarrow\operatorname{Spec}R[P]$ is given by the ideal
$(\chi^{v}-\pi)\subseteq R[P]$, where $v\in P$ is the image of $1$ under the
map of monoids $\mathbb{N}\rightarrow P$ giving the chart for
$U^{\dagger}\rightarrow\Delta^{\dagger}$.

Let $U(mN)\rightarrow W(mN)\subseteq S(mN)$ be the base change of
$U\rightarrow W\subseteq S$ with respect to the map $S(mN)\rightarrow S$
arising from the construction above and let $V(mN)\rightarrow
V\times_{\Delta}\Delta(mN)$ be the toric resolution arising from the same
subdivision that we performed above; in particular there is a smooth map
$\psi\colon U(mN)\rightarrow V(mN)$ which is equivariant with respect to the
natural action of $\mu_{mN}=\operatorname{Gal}(K(mN)|K)$.

Let us omit the symbol $\dagger$ from the log-schemes, all the object we deal
with are interpreted in the category of log-schemes unless differently stated.
Consider the following equivariant map:

$V(mN)\rightarrow
V\times^{\operatorname{fs}}_{\Delta}\Delta(mN)\subset\operatorname{Spec}R{\left[{\left(P\oplus_{\mathbb{N}}\frac{1}{mN}\mathbb{N}\right)}^{\operatorname{sat}}\right]}\eqqcolon\operatorname{Spec}R[P]\times^{\operatorname{fs}}_{\Delta}\Delta(mN)\,\text{,
}$

where the amalgamated sum is taken with respect to the obvious inclusion
$\mathbb{N}\subseteq\frac{1}{mN}\mathbb{N}$ and $\mathbb{N}\rightarrow P$,
$1\mapsto v\in P$ defined above. This map is an isomorphism outside the
central fibre of the two schemes.

There is an integer $d$ and a primitive element $v_{1}\in P$ such that
$v=d\cdot v_{1}$ (in our case, i.e. of a log structure arising from an _snc_
divisor, $d$ coincides with the _g.c.d._ of the multiplicities of the
components containing $p$, in general it was defined as _root index_ in [2]);
by our assumption on $N$, we have that $d|mN$, thus the following identity
holds:

$V\times^{\operatorname{fs}}_{\Delta}\Delta(mN)=(V\times^{\operatorname{fs}}_{\Delta}\Delta(d))\times^{\operatorname{fs}}_{\Delta(d)}\Delta(mN)\,\text{.
}$

We can, thus, consider the base-changes separately; the undelying scheme of
$V\times^{\operatorname{fs}}_{\Delta}\Delta(d)$ is the normalization of the
base-change in the category of schemes [2, Proposition 3.7.1]. We have that
$V\times_{\Delta}\Delta(d)\cong\operatorname{Spec}R(d)\left[{P}\right]/(\chi^{dv_{1}}-\varpi^{d})$,
where $\varpi\in R(d)$ is a uniformizer such that $\varpi^{d}=\pi$. The chart
of $\operatorname{Spec}R(d)[P]/(\chi^{dv_{1}}-\varpi^{d})$ is given by the map
of monoids

$\displaystyle P\oplus_{\mathbb{N}}\frac{1}{d}\mathbb{N}$
$\displaystyle\rightarrow R(d)\left[{P}\right]/(\chi^{dv_{1}}-\varpi^{d})$
$\displaystyle\left(0,\frac{1}{d}\right)$ $\displaystyle\mapsto\varpi\,\text{.
}$

One checks that
$\displaystyle{\left(P\oplus_{\mathbb{N}}\frac{1}{d}\mathbb{N}\right)}^{\operatorname{sat}}\cong
P\oplus\mathbb{Z}/d\mathbb{Z}$, where the latter factor is generated by
$\displaystyle\left(v_{1},-\frac{1}{d}\right)$. Then
$V\times^{\operatorname{fs}}_{\Delta}\Delta(d)\cong\coprod_{i=0}^{d-1}\operatorname{Spec}R(d)\left[{P}\right]/(\chi^{v_{1}}-\zeta_{d}^{i}\varpi)$,
and $\operatorname{Gal}(K(d)|K)$ acts on it via a cyclic permutation of the
components.

We now assume that $d=1$, i.e. that $v$ is primitive. In this case the monoid
$(P\oplus_{\mathbb{N}}\frac{1}{mN}\mathbb{N})^{\operatorname{sat}}$ is sharp
[2, Proposition 2.2.2 (3)] and the inclusion
$P^{\operatorname{grp}}\subseteq(P\oplus_{\mathbb{N}}\frac{1}{mN}\mathbb{N})^{\operatorname{grp}}$
induces an étale map of tori
$T(mN)\coloneqq\operatorname{Spec}\mathbb{Z}\left[(P\oplus_{\mathbb{N}}\frac{1}{mN}\mathbb{N})^{\operatorname{grp}}\right]\rightarrow
T\coloneqq\operatorname{Spec}\mathbb{Z}[P^{\operatorname{grp}}]$ of degree
$mN$, which is the quotient with respect to the action of $\mu_{mN}$; in
particular the group $\mu_{mN}$ acts freely and transitively on the kernel of
such map and there is an exact sequence of group schemes over
$\operatorname{Spec}\mathbb{Z}$:

$1\rightarrow\mu_{mN}\rightarrow T(mN)\rightarrow T\rightarrow 1\,\text{. }$
(3.1)

On the other hand, let us consider the $1-$codimensional subtorus
$T^{\prime}\subseteq T(mN)$ corresponding to the quotient
$P(mN)^{\operatorname{grp}}\rightarrow P(mN)^{\operatorname{grp}}/\langle
u\rangle$, where $u$ is the image of the generator of $\frac{1}{mN}\mathbb{N}$
in $P(mN)$.

######  Claim 3.3.

The action of $T(mN)_{R}$ on $\operatorname{Spec}R[P(mN)]$ induces an action

$T^{\prime}_{R}\times_{\Delta}(V\times_{\Delta}^{\operatorname{fs}}\Delta(mN))\rightarrow
V\times_{\Delta}^{\operatorname{fs}}\Delta(mN)$

making $V\times_{\Delta}^{\operatorname{fs}}\Delta(mN)\rightarrow\Delta$ a
toric scheme with respect to the torus $T^{\prime}$.

The equivariant toric resolution $V(mN)\rightarrow
V\times^{\operatorname{fs}}_{\Delta}\Delta(mN)$ is an isomorphims over an open
set containing the dense toric orbit, hence $\operatorname{Gal}(K(mN)|K)$ acts
also on $V(mN)$ as a subgroup of $T^{\prime}$.

Let $O\subseteq V(mN)$ be a locally closed stratum in the canonical
stratification of $V(mN)$, i.e. an orbit for the action of $T^{\prime}$, as
described in [19, 2.1.13]; then a suitable quotient
$T^{\prime}\twoheadrightarrow\overline{T}$ acts freely on $O$ and, thus, the
image of $\mu_{mN}$ in $\overline{T}$ acts freely on $O$, hence the stabilizer
of an arbitrary point $q\in O$ acts trivially on the whole orbit.

In the general case, we may apply the above argument to the map
$\Delta(mN)\rightarrow\Delta(d)$, obtaining the following chain of maps:

$V(mN)\rightarrow V\times^{\operatorname{fs}}_{\Delta}\Delta(mN)\rightarrow
V\times^{\operatorname{fs}}_{\Delta}\Delta(d)\rightarrow V\,\text{, }$

where the first map is a map of toric schemes (over $\Delta(d)$) whose main
torus fits in the sequence

$1\rightarrow\mu_{\frac{mN}{d}}\rightarrow T(mN)\rightarrow T(d)\rightarrow 1$

and the last map is just the collapse of $d$ copies of $V$. Since the
generator of $\mu_{mN}$ acts on $V(mN)$ by permuting the $d$ connected
components, then the stabilizer of each point is contained in the subgroup
generated by its $d-$th power, i.e. $\mu_{\frac{mN}{d}}$, in particular the
fact that the stabilizer of a point is locally constant on each orbit follows
from what said for the $d=1$ case.

Let $q\in U(mN)_{k,\operatorname{sm}}$ be a closed point and let $O$ be the
stratum of $V(mN)$ containing $\psi(q)$; and let
$Q\coloneqq\psi^{-1}(O)\subseteq U(mN)$. Since $q\in
U(mN)_{k,\operatorname{sm}}$, then $O$ is a connected component of
$V(mN)_{k,\operatorname{sm}}$, therefore $Q$ is open in
$U(mN)_{k,\operatorname{sm}}$. Since the map $U(mN)\rightarrow V(mN)$ is
equivariant, we have that
$\operatorname{Stab}_{q}\subseteq\operatorname{Stab}_{\psi(q)}$, thus
$\operatorname{Stab}_{q}$ acts trivially on $O$. The étale map $Q\rightarrow
O$ induces a $\operatorname{Stab}_{q}-$equivariant étale map of the complete
local rings
$\widehat{\pazocal{O}_{O,\psi(q)}}\rightarrow\widehat{\pazocal{O}_{Q,q}}$,
which is an isomorphism since $k$ is algebraically closed. In particular
$\operatorname{Stab}_{q}$ acts trivially on $\widehat{\pazocal{O}_{Q,q}}$
which is the formal completion of the local ring $\pazocal{O}_{Q,q}$, in
particular $\operatorname{Stab}_{q}$ acts trivially in a neighbourhood of $q$.
Since the fixed locus of $\operatorname{Stab}_{q}$ is also closed, it acts
trivially on the whole connected component containing $q$. It follows that
$\operatorname{Stab}_{-}$ is locally constant on
$U(mN)_{k,\operatorname{sm}}$.

In order to conclude that the stabilizer is locally constant on
$W(mN)_{k,\operatorname{sm}}$ as well, we shall prove that each point $q\in
U(mN)_{k,\operatorname{sm}}$ has the same stabilizer as its image $p\in
W(mN)$. Let $q_{0}\in U$ be the image of $q$ under $U(mN)\rightarrow U$ and
similarly let $p_{0}\in W$ be the image of $p$ under $W(mN)\rightarrow W$. Up
to replacing $U\rightarrow V$ with an open subset $U^{\prime}\subseteq
U\rightarrow V$ we can assume that $q_{0}$ is the only preimage of $p_{0}$
under $U\rightarrow W$; in particular the orbit of $q\in U(mN)$ is sent
bijectively onto the orbit of $p\in W(mN)$ under the $\mu_{mN}-$equivariant
map $U(mN)\rightarrow W(mN)$, so their stabilizers must coincide. ∎

###### Proof of Claim 3.3.

Consider the composition of maps of affine schemes

$T^{\prime}_{R}\times_{R}(V\times_{\Delta}^{\operatorname{fs}}\Delta(mN))\rightarrow
T_{R}\times_{R}\operatorname{Spec}R[P(mN)]\rightarrow\operatorname{Spec}R[P(mN)]\,\text{,
}$

corresponding to the following composition of maps of rings

$\displaystyle R[P(mN)]$ $\displaystyle\rightarrow$ $\displaystyle
R[P^{\operatorname{grp}}(mN)]\otimes_{R}R[P(mN)]$ $\displaystyle\rightarrow$
$\displaystyle R[P(mN)^{\operatorname{grp}}/\langle
u\rangle]\otimes_{R}R[P(mN)]/(\chi^{u}-\pi)$ $\displaystyle\chi^{x}$
$\displaystyle\mapsto$ $\displaystyle\chi^{x}\otimes\chi^{x}$
$\displaystyle\mapsto$
$\displaystyle\chi^{\overline{x}}\otimes\overline{\chi^{x}}\,\text{, }$

where $\overline{x}\in P^{\operatorname{grp}}/\langle u\rangle$ denotes the
projection of $x\in P(mN)$ and $\overline{\chi^{x}}$ denotes the the
projection of $\chi^{x}\in R[P(mN)]$ into $R[P(mN)]/(\chi^{u}-\pi)$. Since
$\overline{u}=0$, one sees that $\chi^{u}-\pi$ is sent to
$1\otimes\overline{\chi^{u}}-1\otimes\pi=0$, thus the map factors throug

$\displaystyle R[P(mN)]/(\chi^{u}-\pi)$ $\displaystyle\rightarrow
R[P(mN)^{\operatorname{grp}}/\langle
u\rangle]\otimes_{R}R[P(mN)]/(\chi^{u}-\pi)$
$\displaystyle\overline{\chi^{x}}$
$\displaystyle\mapsto\chi^{\overline{x}}\otimes\overline{\chi^{x}}\,\text{, }$

giving an action
$T^{\prime}_{R}\times_{\Delta}V\times_{\Delta}^{\operatorname{fs}}\Delta(mN)\rightarrow
V\times_{\Delta}^{\operatorname{fs}}\Delta(mN)$.

We conclude by showing that $V\times_{\Delta}^{\operatorname{fs}}\Delta(mN)$
admit a dense orbit with respect to the action of $T^{\prime}_{K}$.

Let $y\in(V\times_{\Delta}^{\operatorname{fs}}\Delta(mN))\cap T(mN)_{K}$ be a
closed point. Since $\operatorname{codim}_{T(mN)_{K}}T^{\prime}_{K}=1$, then
$T^{\prime}_{K}\cdot y\subseteq T(mN)_{K}\cdot y=T(mN)_{K}$ is a closed
subscheme of codimension at most 1. Density of $T^{\prime}_{K}\cdot y$ follows
from the fact that $(V\times_{\Delta}^{\operatorname{fs}}\Delta(mN))\cap
T(mN)_{K}$ is irreducible and has codimension 1 in $T(mN)_{K}$. ∎

##  4 The monodromy conjecture

###  4.1 Motivic integration

#### 4.1.1

Let us fix a complete DVR $R$ and let $K$ denote its fraction field, while $k$
its residue field, which we assume to be algebraically closed.

#### 4.1.2

Let $Y\rightarrow\Delta^{\ast}(m)$ be a smooth Calabi Yau variety and let
$\omega$ be a volume form on $Y$. Fix a weak Néron model
$\mathfrak{Y}\rightarrow\Delta(m)$ of $Y$. For a connected component
$C\in\pi_{0}(\mathfrak{Y}_{0})$ let $\operatorname{ord}_{C}(\omega)$ be the
order of $\omega$, considered as a meromorphic function, on the generic point
of $C$.

#### 4.1.3

It follows from a result of Loeser and Sebag (see [13, Proposition 4.3.1]),
that the following definition does not depend on the choice of the weak Néron
model of $Y$:

######  Definition 4.1.

(Motivic integral) With the same notation introduced in this paragraph, we
call motivic integral of the volume form $\omega$ on $Y$ the element of
$\mathscr{M}_{k}$ given by the following sum:

$\int_{Y}\omega
d\mu=\sum_{C\in\pi_{0}\mathfrak{Y}_{0}}[C]\mathbb{L}^{-\operatorname{ord}_{C}(\omega)}\,\text{.
}$

#### 4.1.4

Now fix a Calabi Yau variety $X\rightarrow\Delta^{\ast}$, together with a
volume form $\omega\in\omega_{X}(X)$.

For every positive integer $m$, define $X(m)\coloneqq
X\times_{\Delta^{\ast}}\Delta^{\ast}(m)$. Since the map
$\Delta^{\ast}(m)\rightarrow\Delta$ is an étale map, the basechange map
$X(m)\rightarrow X$ is étale as well. The pull-back of $\omega$ through that
map is thus a volume form on $X(m)$, which we denote by $\omega(m)$. Using
this construction, a formal series with coefficients in $\mathscr{M}_{k}$ is
defined:

######  Definition 4.2 (Motivic Zeta Function).

Keep the notation of this paragraph. The _Motivic Zeta Function_ of $X$ with
respect to the volume form $\omega$ is the formal series

$Z_{X,\omega}(T)\coloneqq\left(\sum_{\stackrel{{\scriptstyle m\geq
1}}{{\operatorname{char}k\nmid
m}}}\int_{X(m)}\omega(m)d\mu\right)T^{m}\,\text{. }$

#### 4.1.5

The motivic integral and the Zeta function have an equivariant counterpart in
$\mathscr{M}_{k}^{G}[[T]]$, provided that the volume form of the Calabi-Yau
variety is chosen to be $G-$equivariant.

###  4.2 Rational functions in $\mathscr{M}_{k}[[T]]$

#### 4.2.1

The notions discussed in this paragraph are introduced, for instance, in [16].

######  Definition 4.3.

Let $F\in\mathscr{M}_{k}[[T]]$, we say that $F$ is rational if there is a
finite set $S\subseteq\mathbb{N}\times\mathbb{N}_{+}$ such that $\displaystyle
F\in\mathscr{M}_{k}\left[T,\frac{1}{1-\mathbb{L}^{-a}T^{b}}\colon(a,b)\in
S\right]$.

In such case, we say that $F$ has a pole of order at most $n\in\mathbb{N}$ in
$q\in\mathbb{Q}$ if there exist a finite set
$S^{\prime}\in\mathbb{N}\times\mathbb{N}_{+}$ such that
$\displaystyle\frac{a}{b}=q(a,b)\notin S^{\prime}$ and a positive integer $N$
such that

$(1-\mathbb{L}^{-qN}T^{N})^{n}F\in\mathscr{M}_{k}\left[T,\frac{1}{1-\mathbb{L}^{-a}T^{b}}\colon(a,b)\in
S^{\prime}\right]\,\text{. }$

We say that $F$ has a pole of order $n\geq 1$ in $q\in\mathbb{Q}$ if $F$ has a
pole of order at most $n$, but not a pole of order at most $n-1$.

#### 4.2.2

The definition can be simplified, provided that we work on a ring
$\mathscr{R}$ endowed with a map $\mathscr{M}_{k}\rightarrow\mathscr{R}$ such
that all the elements of the form $\mathbb{L}^{r}-1$, with
$r\in\mathbb{N}\backslash\\{0\\}$, are invertible. The minimal such choice for
$\mathscr{R}$ is, clearly, the localization in $\mathscr{M}_{k}$ with respect
to that set of elements, i.e.
$\displaystyle\mathscr{M}_{k}\left[(\mathbb{L}^{r}-1)^{-1}\colon
0<r\in\mathbb{N}\right]$. Another natural choice for $\mathscr{R}$ is the
completed Grothendieck ring of varietes: $\widehat{\mathscr{M}}_{k}$, where
the inverse of $1-\mathbb{L}^{r}$ is
$1+\mathbb{L}^{r}+\mathbb{L}^{2r}+\cdots$. The following lemma clarifies why
in this case it is easier to define a pole:

######  Lemma 4.4.

Let $\mathscr{R}$ be a ring as above and $F\in\mathscr{R}[[T]]$ any rational
function. Then $\exists N>0$ a positive integer and a finite set
$S\subseteq\mathbb{Q}$ such that:

$F(T)=p(T)+\sum_{q\in
S}\frac{f_{q}(T)}{(1-\mathbb{L}^{-qN}T^{N})^{a_{q}}}\,\text{, }$ (4.1)

for some polynomials $p,f_{q}\in\mathscr{R}[T]$ and positive integers $a_{q}$.

###### Proof.

We begin by noticing that, given $\mu>\nu$ positive integers, one has that

$\frac{1}{(1-\mathbb{L}^{\mu}T^{N})(1-\mathbb{L}^{\nu}T^{N})}=\frac{1}{1-\mathbb{L}^{\mu-\nu}}\left(\frac{1}{1-\mathbb{L}^{\nu}T^{N}}-\frac{\mathbb{L}^{\mu-\nu}}{1-\mathbb{L}^{\mu}T^{N}}\right)\,\text{,
}$

where $N\in\mathbb{N}$.

By repeatedly applying this step, one obtains the following identity

$\frac{1}{(1-\mathbb{L}^{\mu}T^{N})^{a}(1-\mathbb{L}^{\nu}T^{N})^{b}}=\frac{p(T^{N})}{(1-\mathbb{L}^{\mu}T^{N})^{a}}+\frac{q(T^{N})}{(1-\mathbb{L}^{\nu}T^{N})^{b}}\,\text{,
}$

for positive integers $a,b$ and polynomials $\displaystyle
p,q\in\mathbb{Z}\left[\mathbb{L},\frac{1}{1-\mathbb{L}^{\mu-\nu}},t\right]$
such that $\deg_{t}(p)<a$, $\deg_{t}(q)<b$. In particular any rational
function in $\mathscr{R}[[T]]$ with two poles is the sum of two functions with
a single pole. We conclude the proof by induction on the number of poles of
$F$. ∎

######  Definition 4.5.

If $F$ is written as in (4.1) and $q\in S$, then we say that $F$ has a pole of
order at most $a_{q}$ in $q$.

If, moreover, there is no integer $N^{\prime}\in\mathbb{N}_{+}$ such that
$f_{q}\in(1-\mathbb{L}^{-qN^{\prime}}T^{N^{\prime}})\mathscr{R}[T]$, then $F$
has a pole of order exactly $a_{q}$ in $q$.

######  Remark 4.6.

Let $F\in\mathscr{M}_{k}[[T]]$ and let $\widetilde{F}\in\mathscr{R}[[T]]$ be
the image of $F$ under the completion map (or localisation map).

Because of Lemma 4.4, any pole $q$ of $\widetilde{F}$ of order
$a\in\mathbb{N}$ is also a pole of $F$ of order greater or equal than $a$.

###  4.3 Statement of the conjecture

#### 4.3.1

We conclude this section by explaining the main problem we are going to face.
Let $\mathscr{R}$ be one of the three rings
$\mathscr{M}_{k},\mathscr{M}_{k}\left[(\mathbb{L}^{r}-1)^{-1}\colon
0<r\in\mathbb{N}\right]$ or $\widehat{\mathscr{M}}_{k}$. We give a statement
of the monodromy conjecture that depends on how the ring of coefficients for
the zeta function is interpreted:

######  Conjecture 4.7 (Monodromy conjecture in $\mathscr{R}$).

Let $X\rightarrow\operatorname{Spec}K$ be a Calabi-Yau variety and let
$\omega$ be a volume form on it. Let $q\in\mathbb{Q}$ be a pole of
$Z_{X,\omega}(T)\in\mathscr{R}[[T]]$, then $e^{2\pi iq}$ is an eigenvalue for
the monodromy operator associated to $\operatorname{Gal}(\overline{K}|K)$.

#### 4.3.2

The monodromy conjecture in $\mathscr{M}_{k}$ implies the monodromy conjecture
in
$\mathscr{M}_{k}\left[(\mathbb{L}^{r}-1)^{-1}\colon 0<r\in\mathbb{N}\right]$
by the Remark 4.6. In turn, the monodromy conjecture in
$\mathscr{M}_{k}\left[(\mathbb{L}^{r}-1)^{-1}\colon 0<r\in\mathbb{N}\right]$
implies the version in $\widehat{\mathscr{M}}_{k}$.

##  5 Formal series in the motivic rings

In this section, the symbol $\mathscr{R}$ shall denote one of the rings
$\mathscr{M}_{k},\mathscr{M}_{k}\left[(\mathbb{L}^{r}-1)^{-1}\colon
0<r\in\mathbb{N}\right]$ or $\widehat{\mathscr{M}_{k}}$, unless differently
specified.

We will discuss some properties of power series with coefficients in
$\mathscr{R}$, then we will describe some operations that shall be useful for
computing the Motivic Zeta Function in our case.

###  5.1 Quotient by a group action

#### 5.1.1

Let us fix a finite group $G$, let $N\trianglelefteq G$ and let $H=G/N$ be its
quotient. Consider the equivariant versions of the ring $\mathscr{R}$, which
we call $\mathscr{R}^{G}$ and $\mathscr{R}^{H}$, as in §1.4. For an arbitrary
variety $X$ endowed with an action of $G$, we consider the quotient $X/N$
which is again an algebraic space endowed with an action of the group $H$,
namely the quotient action.

#### 5.1.2

The _example of Hironaka_ shows that $X/N$ is not necessarily a scheme. In any
case, there exists an open affine subscheme $\operatorname{Spec}A\subseteq X$
which is invariant under the action of $G$, so that
$\operatorname{Spec}A^{N}\subseteq X/N$ is a scheme. We may, thus, repeat the
argument for the closed $G-$invariant subscheme
$X\backslash\operatorname{Spec}A$ and, by Noetherian induction, we stratify
$X$ as a union of $G-$schemes whose quotients with respect to the action of
$N$ are $H-$schemes; moreover a $G-$invariant stratification of each stratum
induces an $H-$invariant stratification of its quotient. Thus there is a well
defined map of groups $\mathscr{R}^{G}\rightarrow\mathscr{R}^{H}$ by
$[X]\mapsto[X/N]$; this map does not preserve the products.

#### 5.1.3

In the following definition we extend the map above to a map
$\mathscr{R}^{G}[[T]]\rightarrow\mathscr{R}^{H}[[T]]$ and in the subsequent
proposition we show that rationality of any power series is well behaved under
this map.

######  Definition 5.1.

If $\displaystyle F=\sum_{n}A_{n}T^{n}\in\mathscr{R}^{G}[[T]]$, we define the
_series of the quotients_ associated to $F$ as

$(F/N)(T)=\sum_{n}(A_{n}/N)T^{n}\in\mathscr{R}^{H}[[T]]\,\text{. }$

######  Proposition 5.2.

For $f,g\in\mathscr{R}^{G}[T]$, with $g$ being of the form $\prod_{j\in
J}(1-\mathbb{L}^{a_{j}}T^{b_{j}})$, let $F$ be the rational function
$\displaystyle F(T)\coloneqq\frac{f(T)}{g(T)}$. Then $\displaystyle
F/N=\frac{f/N}{g}$.

In particular all the poles of $F/N$ belong to the set of poles of $F$.

###### Proof.

It suffices to show the statement in the case when
$f=\alpha\in\mathscr{R}^{G}$ is a constant. Let
$\displaystyle\frac{1}{g(T)}=\sum_{n}A_{n}T^{n}$, where $A_{n}\in\mathscr{R}$
has a trivial $G-$action. Then

$\left.\left(\frac{\alpha}{g(T)}\right)\right/N=\sum_{n}(\alpha
A_{n})/NT^{n}=\sum_{n}\alpha/NA_{n}T^{n}=\frac{\alpha/N}{g(T)}\,\text{. }$

∎

###  5.2 Power structures

#### 5.2.1

We need to define a map on the Grothendieck rings which extends the symmetric
product of a variety, allowing us to talk about the symmetric product of a
”difference of varieties”; in order to do so, we need to use a power structure
on $K_{0}(\operatorname{Var}_{k})$, thus we begin by recalling what a power
structure is, as introduced in [8]:

######  Definition 5.3.

Let $A$ be a ring. A power structure on $A$ is a map

$\displaystyle(1+tA[[t]])\times A$ $\displaystyle\rightarrow 1+tA[[t]]$
$\displaystyle(F(t),X)$ $\displaystyle\mapsto F(t)^{X}$

satisfying the following conditions for all $F,G\in 1+tA[[t]]$ and $X,Y\in A$:

  * •

$F(t)^{0}=1$;

  * •

$F(t)^{1}=F(t)$;

  * •

$(F(t)G(t))^{X}=F(t)^{X}\cdot G(t)^{X}$;

  * •

$F(t)^{X+Y}={\left(F(t)\right)}^{X}{\left(F(t)\right)}^{Y}$;

  * •

$F(t)^{XY}={\left(F(t)^{X}\right)}^{Y}$;

  * •

$(1+t)^{X}\in 1+Xt+t^{2}A[[t]]$;

  * •

$F(t)^{X}|_{t\rightarrow t^{n}}=F(t^{n})^{X}$.

In fact, the last two properties are not part of the original definition, but
other authors include them in their definition.

#### 5.2.2

In the rest of the section, for $F=\sum_{n}F_{n}T^{n}\in
K_{0}(\operatorname{Var}_{k})[[T]]$, with $F_{0}=1$, and for $X\in
K_{0}(\operatorname{Var}_{k})$ we denote by $F(T)^{X}$ the power structure
introduced by Gusein-Zade, Luengo and Melle-Hernandez in [8].

###  5.3 Symmetic powers

#### 5.3.1

Keep the notation introduced in the previous paragraph. Let us begin with the
following definition:

######  Definition 5.4 (Symmetric power in GRV).

Let $\alpha\in K_{0}(\operatorname{Var}_{k})$ and let $r\in\mathbb{N}$. The
_$r-$ th symmetric power of $\alpha$_ is the element
$\operatorname{Sym}^{r}(\alpha)\in K_{0}(\operatorname{Var}_{k})$ defined as

$\operatorname{Sym}^{r}(\alpha)\coloneqq[t^{r}](1-t)^{-\alpha}\,\text{. }$

#### 5.3.2

In particular, if $\alpha=[U]$, we have that
$\operatorname{Sym}^{r}([U])=[\operatorname{Sym}^{r}(U)]$; in this sense
$\operatorname{Sym}^{\bullet}$ exends the notion of symmetric power of an
algebraic variety. In general, for $\alpha=[U]-[V]$ one gets an explicit
formula by analyzing the coefficients of

$(1-t)^{[V]-[U]}=(1-t)^{[V]}\cdot(1+t+t^{2}+\cdots)^{[U]}\,\text{. }$

######  Example 5.5.

In order to show how this computation can be handled, we compute explicitly
the coefficient of $t^{2}$, that is, the expression for
$\operatorname{Sym}^{2}([U]-[V])$. We know, from [8, Theorem 1], that

$(1+t+t^{2}+\cdots)^{[U]}=1+[U]t+[\operatorname{Sym}^{2}(U)]t^{2}+o(t^{2})\,\text{,
}$

thus we get

$(1-t)^{[V]}=(1+[V]t+[\operatorname{Sym}^{2}(V)]t^{2}+o(t^{2}))^{-1}=1-[V]t+([V^{2}]-[\operatorname{Sym}^{2}(V)])t^{2}+o(t^{2})\,\text{.
}$

It follows that

$\operatorname{Sym}^{2}([U]-[V])=[\operatorname{Sym}^{2}(U)]-[\operatorname{Sym}^{2}(V)]+[V]^{2}-[U][V]\,\text{.
}$

#### 5.3.3

It is possible to extend the map $\operatorname{Sym}^{r}\colon
K_{0}(\operatorname{Var}_{k})\rightarrow K_{0}(\operatorname{Var}_{k})$ to a
map $\operatorname{Sym}^{r}\colon\mathscr{M}_{k}\rightarrow\mathscr{M}_{k}$ by
$\operatorname{Sym}^{r}{\mathbb{L}^{-s}\alpha}\coloneqq\mathbb{L}^{-rs}\operatorname{Sym}^{r}(\alpha)$.
In order to ensure that this map is well defined, we only need to show that,
for $\alpha\in K_{0}(\operatorname{Var}_{k})$ and for $s\in\mathbb{N}$, we
have
$\mathbb{L}^{-rs}\operatorname{Sym}^{r}(\alpha)=\mathbb{L}^{-r(s+1)}\operatorname{Sym}^{r}(\mathbb{L}\alpha)$.

Indeed, recalling that
$\operatorname{Sym}^{r}(\mathbb{A}^{n})\cong\mathbb{A}^{nr}$, we get

$(1-t)^{\mathbb{L}\alpha}={\left((1-t)^{-\mathbb{L}}\right)}^{-\alpha}=(1+\mathbb{L}t+\mathbb{L}^{2}t^{2}+\cdots)^{-\alpha}=(1-\mathbb{L}t)^{\alpha}\,\text{,
}$

which implies that $\forall\alpha\in K_{0}(\operatorname{Var}_{k})$,
$\operatorname{Sym}^{r}(\mathbb{L}\alpha)=\mathbb{L}^{r}\operatorname{Sym}^{r}(\alpha)$.

#### 5.3.4

The map $\operatorname{Sym}^{r}$ can also be defined at the level of
$\widehat{\mathscr{M}_{k}}$; indeed, for $\alpha,\beta\in
K_{0}(\operatorname{Var}_{k})$, we have that:

$\forall
p\in\mathbb{N},\quad(1-t)^{\alpha+\mathbb{L}^{p}\beta}-(1-t^{\alpha})=(1-t)^{\alpha}\cdot\left((1-\mathbb{L}^{p}t)^{\beta}-1\right)\in\mathbb{L}^{p}K_{0}(\operatorname{Var}_{k})[[t]]\,\text{,
}$

thus
$\operatorname{Sym}^{r}(\alpha+\mathbb{L}^{p}\beta)\equiv\operatorname{Sym}^{r}(\alpha)\pmod{\mathbb{L}^{p}}$.

#### 5.3.5

For the sake of completeness, we shall also define a version of
$\operatorname{Sym}^{r}$ defined over
$\mathscr{M}_{k}\left[(\mathbb{L}^{n}-1)^{-1}\colon 0<n\in\mathbb{N}\right]$.
Since $\operatorname{Sym}^{1}$ is already defined as the identity map we can
proceed inductively on $r$. Let us assume that all the maps
$\operatorname{Sym}^{i}$ are defined for $1\leq i\leq r-1$. Let us first check
that for $\alpha\in\mathscr{M}_{k}$, the value of

$\operatorname{Sym}^{r}\left(\frac{\alpha}{1}\right)\coloneqq\frac{\operatorname{Sym}^{r}(\alpha)}{1}\in\mathscr{M}_{k}\left[(\mathbb{L}^{n}-1)^{-1}\colon
0<n\in\mathbb{N}\right]$

is well defined, i.e. that if $\displaystyle\frac{\alpha}{1}=\frac{\beta}{1}$,
then
$\displaystyle\frac{\operatorname{Sym}^{r}(\alpha)}{1}=\frac{\operatorname{Sym}^{r}(\beta)}{1}$.
If $\gamma\in\mathscr{M}_{k}$ is such that $(\mathbb{L}^{n}-1)\gamma=0$ for
some $n\in\mathbb{N}$, then

$\operatorname{Sym}^{r}(\mathbb{L}^{n}\gamma)=\operatorname{Sym}^{r}(\gamma+(\mathbb{L}^{n}-1)\gamma)=\sum_{j=0}^{r}\operatorname{Sym}^{j}(\gamma)\operatorname{Sym}^{r-j}(0)=\operatorname{Sym}^{r}(\gamma)\,\text{,
}$

thus $(\mathbb{L}^{nr}-1)\operatorname{Sym}^{r}(\gamma)=0$. By an inductive
argument one proves that if $\displaystyle\frac{\gamma}{1}=0$, then also
$\displaystyle\frac{\operatorname{Sym}^{r}(\gamma)}{1}=0$. It follows that,
whenever $\displaystyle\frac{\alpha}{1}=\frac{\beta}{1}$, then

$\frac{\operatorname{Sym}^{r}(\beta)}{1}=\frac{\operatorname{Sym}^{r}(\alpha+(\beta-\alpha))}{1}=\sum_{j=1}^{r}\frac{\operatorname{Sym}^{j}(\alpha)}{1}\frac{\operatorname{Sym}^{r-j}(\beta-\alpha)}{1}=\frac{\operatorname{Sym}^{r}(\alpha)}{1}\,\text{.
}$

Then we define, recursively:

$\operatorname{Sym}^{r}\left(\frac{\alpha}{\mathbb{L}^{n}-1}\right)\coloneqq(\mathbb{L}^{nr}-1)^{-1}\sum_{i=1}^{r}\operatorname{Sym}^{i}(\alpha)\operatorname{Sym}^{r-i}\left(\frac{\alpha}{\mathbb{L}^{n}-1}\right)\,\text{.
}$

######  Example 5.6.

We show how to compute this map in a specific case. For
$\alpha=\displaystyle\frac{[U]}{1-\mathbb{L}^{n}}$, we have that
$\operatorname{Sym}^{2}(\alpha)=\displaystyle\frac{[\operatorname{Sym}^{2}(U)]}{1-\mathbb{L}^{2n}}+\frac{\mathbb{L}^{n}\cdot[U]^{2}}{(1-\mathbb{L}^{n})(1-\mathbb{L}^{2n})}$.

#### 5.3.6

For a power series $F(T)=\sum A_{n}T^{n}\in\mathscr{R}[[T]]$, let us consider
the power series obtained by plugging each coefficient of $F$ into the above
mentioned maps $\operatorname{Sym}^{r}$:

$\operatorname{Sym}^{r}(F)(T)\coloneqq\sum\operatorname{Sym}^{r}(A_{n})T^{n}\,\text{.
}$

These maps have very interesting properties when the function $F$ is rational,
indeed in this case we are able to control the poles of
$\operatorname{Sym}^{r}$ thanks to the upcoming results:

######  Lemma 5.7.

Let $\displaystyle F=\frac{\alpha
T^{h}}{(1-\mathbb{L}^{-qN}T^{N})^{e}}\in\mathscr{R}[[T]]$. Then, for $r>0$, we
have that $\operatorname{Sym}^{r}(F)$ has at most one pole of order (at most)
$r(e-1)+1$ in $rq$.

###### Proof.

Let $\displaystyle F=\sum_{m\geq 0}A_{m}T^{mN+h}$; then $\displaystyle
A_{m}={m+e-1\choose e-1}\alpha\mathbb{L}^{-mqN}$. It follows that

$\operatorname{Sym}^{r}(A_{m})=\sum_{\beta\dashv r}{{m+e-1\choose
e-1}\choose\beta_{1},\dots,\beta_{r}}\alpha^{\beta_{1}}\cdot(\operatorname{Sym}^{2}\alpha)^{\beta_{2}}\cdots(\operatorname{Sym}^{r}\alpha)^{\beta_{r}}\mathbb{L}^{-rmqN}\,\text{;
}$

once $\beta$ is fixed, $\displaystyle{{m+e-1\choose
e-1}\choose\beta_{1},\dots,\beta_{r}}$ is either $0$ $\forall m\in\mathbb{Z}$
(this happens only if $e=1$ and $\beta_{1}+\cdots+\beta_{r}>1$) or a
polynomial in $m$ of degree $(e-1)(\beta_{1}+\dots+\beta_{r})$.

It follows (from the fact that
$\displaystyle\frac{1}{(1-x)^{n}}=\sum_{m}p_{n}(m)x^{m}$, where
$p_{n}\in\mathbb{Q}[t]$ is a polynomial such that $\deg p_{n}=n-1$) that

$\sum_{m\geq 0}{{m+e-1\choose
e-1}\choose\beta_{1},\dots,\beta_{r}}\alpha^{\beta_{1}}\cdot(\operatorname{Sym}^{2}\alpha)^{\beta_{2}}\cdots(\operatorname{Sym}^{r}\alpha)^{\beta_{r}}\mathbb{L}^{-rmqN}T^{mN+h}$

is a suitable combination with rational coefficients of

$\left\\{\frac{\alpha^{\beta_{1}}\cdot(\operatorname{Sym}^{2}\alpha)^{\beta_{2}}\cdots(\operatorname{Sym}^{r}\alpha)^{\beta_{r}}T^{h}}{(1-\mathbb{L}^{-rqN}T^{N})^{j}}\right\\}_{j=0}^{(e-1)(\beta_{1}+\cdots+\beta_{r})+1}\,\text{.
}$

Among all the partitions of $r$, the one which gives the highest possible
order of the pole is the one maximizing $\beta_{1}+\cdots+\beta_{r}$, namely
$\beta=(r,0,0,\dots)$, which gives a pole of order at most $r(e-1)+1$. ∎

#### 5.3.7

For the rest of the section, we denote by $\mathscr{R}$ one of the two rings
$\mathscr{M}_{k}\left[(\mathbb{L}^{n}-1)^{-1}\colon 0<n\in\mathbb{N}\right]$
or $\widehat{\mathscr{M}_{k}}$: the proofs we shall present do not hold for
functions with coefficients in $\mathscr{M}_{k}$.

######  Proposition 5.8.

For $i=1,\dots,s$ let $F_{i}=\sum_{m\geq
0}A_{m}^{[i]}T^{m}\in\mathscr{R}[[T]]$ be rational functions and let
$\pazocal{Q}_{i}\subseteq\mathbb{Q}$ be the set of poles of $F_{i}$. Let
$F=\sum_{m\geq 0}A_{m}^{[1]}\cdots A_{m}^{[s]}T^{m}\in\mathscr{R}[[T]]$. Then
$F$ is also rational and its set of poles, $\pazocal{Q}$, is contained in
$\pazocal{Q}_{1}+\pazocal{Q}_{2}+\cdots+\pazocal{Q}_{s}$.

Moreover, for each $q\in\pazocal{Q}$, we have that

$\operatorname{ord}_{q}(F)\leq\max\left\\{1-s+\sum_{i=1}^{s}\operatorname{ord}_{q_{i}}(F_{i})\colon
q_{i}\in\pazocal{Q}_{i}\mbox{ and }\sum q_{i}=q\right\\}\,\text{. }$

######  Remark 5.9.

This statement holds, with the same proof, also if we consider functions
$F_{i}\in\mathscr{M}_{k}[[T]]$, provided that each of them is sum of functions
with a single pole.

###### Proof.

Let us assume for a moment that, $\forall i$,
$F_{i}=\frac{\alpha_{i}T^{h_{i}}}{(1-\mathbb{L}^{-q_{i}N}T^{N})^{e_{i}}}$, for
some $\alpha_{i}\in\mathscr{R}$, $0\leq h_{i}<N$ integers,
$q_{i}\in\mathbb{Q}$, $0<e_{i}\in\mathbb{N}$. In such case
$A_{mN+h_{i}}^{[i]}={m+e_{i}-1\choose
e_{i}-1}\alpha_{i}\mathbb{L}^{-mq_{i}N}$. Thus $F=0$ unless
$h_{i}=h\leavevmode\nobreak\ \forall i$, while in this case we have that

$\prod_{i=1}^{s}A_{mN+h}^{[i]}=\left(\prod_{i=1}^{s}{m+e_{i}-1\choose
e_{i}-1}\alpha_{i}\right)\mathbb{L}^{-mqN}\,\text{, }$

where $q=q_{1}+\cdots+q_{s}$. The degree of $\prod_{i=1}^{s}{m+e_{i}-1\choose
e_{i}-1}$, seen as a polynomial in $m$, is $\sum e_{i}-s$, thus we get the
desired result in this case.

For the general case it is enough to consider
$F_{1}=F_{1}^{\prime}+F_{1}^{\prime\prime}$, where $F_{1}^{\prime}=\sum
B_{m}T^{m}$ and $F_{1}^{\prime\prime}=\sum C_{m}T^{m}$; then, setting
$F^{\prime}\coloneqq\sum B_{m}A_{m}^{[2]}\cdots A_{m}^{[s]}T^{m}$ and
$F^{\prime\prime}\coloneqq\sum C_{m}A_{m}^{[2]}\cdots A_{m}^{[s]}T^{m}$, we
have that $F=F^{\prime}+F^{\prime\prime}$ and if our statement holds for both
$F^{\prime}$ and $F^{\prime\prime}$ then it holds also for $F$, in particular
writing all the $F_{i}$ as in Equation (4.1), the proposition follows by an
induction on the number of their summands. ∎

######  Lemma 5.10.

Let $F\in\mathscr{R}[[T]]$ be a rational function whose set of poles is
$\pazocal{Q}\subseteq\mathbb{Q}$, or let $F\in\mathscr{M}_{k}[[T]]$ be the sum
of functions with at most one pole. For all $r\in\mathbb{N}$, let
$\Sigma^{r}\pazocal{Q}$ be the set of rational numbers that are sum of $r$
elements of $\pazocal{Q}$. Then $\operatorname{Sym}^{r}F$ is also rational and
its set of poles is contained in $\Sigma^{r}\pazocal{Q}$.

Moreover, for each $q\in\pazocal{Q}$, we have that

$\operatorname{ord}_{q}(\operatorname{Sym}^{r}F)\leq\max\left\\{1-r+\sum_{i=1}^{s}\operatorname{ord}_{q_{i}}(F)\colon
q_{i}\in\pazocal{Q}\mbox{ and }\sum q_{i}=q\right\\}\,\text{. }$

###### Proof.

If the Lemma holds for $F$ and $G$, then by Proposition 5.8, it holds also for
$F+G$, since
$\operatorname{Sym}^{r}(F+G)=(\operatorname{Sym}^{r}F)+(\operatorname{Sym}^{r-1}F)G+\cdots+(\operatorname{Sym}^{r}G)$.
Thus it is enough to write $F$ as in Equation $\eqref{eqn:zetaK3form}$ and
notice that for each addendum the statement coincides with Lemma 5.7. ∎

##  6 Hilbert schemes of points on a surface

###  6.1 Construction of a weak Néron model

#### 6.1.1

Let $X\rightarrow\operatorname{Spec}K$ be a smooth surface with trivial
canonical divisor and let $\omega\in\omega_{X/K}(X)$ be a volume form on it.
Let $\mathfrak{X}\rightarrow\Delta$ be a regular model whose central fibre
$\mathfrak{X}_{k}$ is a strict normal crossing divisor of $\mathfrak{X}$. Let
us keep the notation of §3 concerning the field extension over $K$ and the
corresponding base-changes. If $\operatorname{char}k=p>0$, we add the further
assumptions that $k$ is separably closed and that the central fibre
$\mathfrak{X}_{k}$ has no components with multiplicity divisible by $p$.

#### 6.1.2

The aim of this section is to provide a closed formula for the zeta function
of $\operatorname{Hilb}^{n}(X)$; generalizing the results developed in the
previous sections, one expects to be able to understand the value of the zeta
function in terms of the zeta functions of $X(i)$ for $1\leq i\leq n$.

#### 6.1.3

Let $a$ be the _lcm_ of the multiplicities of the irreducible components of
$\mathfrak{X}_{k}$. For $n\in\mathbb{N}$, let $\tilde{n}\coloneqq
a\operatorname{lcm}(1,2,\dots,n)$ and let $K(\tilde{n})$ be the unique totally
ramified extension of $K$ whose degree is $\tilde{n}$, so that
$\operatorname{Gal}(K(\tilde{n})/K)=\mu_{\tilde{n}}$.

#### 6.1.4

Denote by $\mathfrak{X}(m\tilde{n})$ be the semistable model of
$X(m\tilde{n})$ obtained from $\mathfrak{X}$ using the construction of §3.2.
As usual we denote by $\mathfrak{X}(m\tilde{n})_{\operatorname{sm}}$ the
smooth locus of $\mathfrak{X}(m\tilde{n})\rightarrow\Delta(m\tilde{n})$. Since
$\operatorname{Hilb}^{n}(\mathfrak{X}(m\tilde{n})_{\operatorname{sm}}/\Delta(m\tilde{n}))\rightarrow\Delta(m\tilde{n})$
is a smooth model of $\operatorname{Hilb}^{n}(X(m\tilde{n})$, we have that

$\mathfrak{X}^{[n]}(m)\coloneqq{\left(\operatorname{Res}_{\Delta(m\tilde{n})/\Delta(m)}\operatorname{Hilb}^{n}(\mathfrak{X}(m\tilde{n})_{\operatorname{sm}}/\Delta(m\tilde{n}))\right)}^{\mu_{\tilde{n}}}\rightarrow\Delta(m)$

is a smooth model of $\operatorname{Hilb}^{n}(X(m))$.

######  Proposition 6.1.

Assume that either $K$ is perfect or $\operatorname{char}K>n$, then
$\mathfrak{X}^{[n]}(m)\rightarrow\Delta(m)$ is a weak Néron model of
$\operatorname{Hilb}^{n}(X(m))$.

###### Proof.

We assume that $a=1$, i.e. that $X$ has semistable reduction on $K$. The proof
in the general case will descend from Proposition 2.7 and Lemma 2.8. We just
need to show that every point
$\operatorname{Spec}K(m)\rightarrow\operatorname{Hilb}^{n}(X(m))\subseteq\mathfrak{X}^{[n]}(m)$
extends to a morphism $\Delta(m)\rightarrow\mathfrak{X}^{[n]}(m)$.

Consider a point
$\operatorname{Spec}K(m)\rightarrow\operatorname{Hilb}^{n}(X(m))$ and let
$Z\subseteq X(m\tilde{n})$ the ($\mu_{\tilde{n}}-$invariant) subscheme
representing such point. Either the closure of $Z$ in
$\mathfrak{X}(m\tilde{n})$ is contained in
$\mathfrak{X}(m\tilde{n})_{\operatorname{sm}}$ or at least one point
$P\in\operatorname{supp}Z$ specializes to the singular locus
$\mathfrak{X}(m\tilde{n})_{k,\operatorname{sing}}$. In the first case
$\overline{Z}$ represents a morphism
$\Delta(m)\rightarrow\mathfrak{X}^{[n]}(m)$ extending the given
$\operatorname{Spec}K(m)\rightarrow\operatorname{Hilb}^{n}(X(m))$. If
$P\in\operatorname{supp}Z$ specializes to
$\mathfrak{X}(m\tilde{n})_{k,\operatorname{sing}}$, then its residue field
$k(P)$ contains strictly $K(m\tilde{n})$, for $\mathfrak{X}(m\tilde{n})$ is a
regular model of $X(m\tilde{n})$.

Let $Q\in X(m)$ be the image of $P$ under the map
$\operatorname{Spec}k(P)\hookrightarrow X(m\tilde{n})\rightarrow X(m)$. The
degree $[k(Q):K(m)]$ cannot divide $\tilde{n}$, otherwise $k(P)$ would be
$K(m\tilde{n})$, thus $[k(Q)\colon K(m)]\geq n+1$.

Since $Z$ is $\mu_{\tilde{n}}-$invariant, then it contains the whole orbit of
$P$ which is the reduced scheme associated to the preimage of $Q$ under the
map $\pi\colon X(m\tilde{n})\rightarrow X(m\tilde{n})/\mu_{\tilde{n}}\cong
X(m)$. We have that
$\pi^{-1}(Q)=\operatorname{Spec}(k(Q)\otimes_{K(m)}K(m\tilde{n}))$, which is a
reduced $K(m\tilde{n})-$algebra of dimension $[k(Q):K(m)]>n$.

This contradicts the fact that $Z$ is a subscheme of length $n$, thus this
case cannot occur and we are done. ∎

#### 6.1.5

Our goal is to study the motivic integral of $\operatorname{Hilb}^{n}(X(m))$
using the models $\mathfrak{X}^{[n]}(m)$ constructed above. In order to
simplify notation, we perform the following computations only for $m=1$,
working with $\mathfrak{X}^{[n]}=\mathfrak{X}^{[n]}(1)$; similar arguments
apply when $m>1$. Even though
$\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{\operatorname{sm}}/\Delta(\tilde{n}))$
is not a weak Néron model of $X(\tilde{n})$, it is a smooth model and the
results of §2.4 apply. It follows that the connected components of
$\mathfrak{X}^{[n]}_{k}$ are in bijection with the connected components of
$\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}})^{\mu_{\tilde{n}}}$,
moreover if $C\subseteq\mathfrak{X}^{[n]}_{k}$ is sent to $C^{\prime}$ via
this bijection, then $[C]=\mathbb{L}^{2n-\dim C^{\prime}}[C^{\prime}]$.

#### 6.1.6

For any $d\in\mathbb{N}$ dividing $\tilde{n}$ we denote by
$Y(d)\subseteq\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}$ the subscheme
consisting of the points whose stabilizer is exacty
$\mu_{\nicefrac{{\tilde{n}}}{{d}}}$. Then $Y(d)$ is
$\mu_{\tilde{n}}-$invariant, since $\mu_{\tilde{n}}$ is an abelian group and
points in the same orbit have the same the stabilizer. Because of Lemma 3.2,
$Y(d)$ is at the same time an open and closed subscheme of
$\mathfrak{X}_{k,\operatorname{sm}}$ and we have that:

$\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}^{\mu_{\tilde{n}/d}}=\bigsqcup_{d^{\prime}|d}Y(d^{\prime})\,\text{.
}$

######  Remark 6.2.

Since $\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}^{\mu_{\tilde{n}/d}}$ is a
scheme of pure dimension 2, it can be identified with the central fibre of
$\operatorname{Res}_{\Delta(\tilde{n})/\Delta(d)}{(\mathfrak{X}(\tilde{n})_{\operatorname{sm}})}^{\mu_{\tilde{n}/d}}$,
which is a weak Néron model of $X(d)$, via the map $h_{k}$ described in
§2.4.2. Let
$C\subseteq\operatorname{Res}_{\Delta(\tilde{n})/\Delta(d)}{(\mathfrak{X}(\tilde{n})_{\operatorname{sm}})}^{\mu_{\tilde{n}/d}}$
be a connected component. It follows from Lemma 2.11 and from the fact that
$\operatorname{Gal}(K(\tilde{n})|K(d))$ acts trivially on $T_{h_{k}(C)}$ that,
,

$\operatorname{ord}_{h_{k}(C)}(\omega(\tilde{n}))=\frac{\tilde{n}}{d}\operatorname{ord}_{C}(\omega(d))\,\text{.
}$

######  Definition 6.3.

Let $n\geq 0$ be an arbitrary natural number and
$\alpha=(\alpha_{1},\alpha_{2},\dots)\in\mathbb{N}^{\mathbb{N}_{>0}}$. We say
that $\alpha$ is a _partition_ of $n$ if
$\displaystyle\sum_{i>0}i\alpha_{i}=n$. In this case we write $\alpha\dashv
n$.

#### 6.1.7

The following statement gives a decomposition of the central fibre as a union
of closed and open subschemes; it is an _ad hoc_ partition that is more
suitable, for our computation, than the ”canonical” stratification of the
Hilbert schemes:

######  Proposition 6.4.

$\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}})^{\mu_{\tilde{n}}}$
admits the following decomposition as disconnected union of subschemes:

$\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}})^{\mu_{\tilde{n}}}\cong\bigsqcup_{\alpha\dashv
n}\prod_{j=1}^{n}\operatorname{Hilb}^{\alpha_{j}}(Y(j)/\mu_{j})\,\text{. }$

Moreover, for a fixed partition $\alpha\dashv n$, the isomorphism above
restricts to an isomorphim between
$\prod_{j=1}^{n}\operatorname{Hilb}^{\alpha_{j}}(Y(j)/\mu_{j})$ and a finite
union of connected components of
$\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}})^{\mu_{\tilde{n}}}$.

###### Proof.

Fix a $k-$scheme $S$. For any closed subscheme
$Z\subseteq\mathfrak{X}(\tilde{n})_{S}$ endowed with a finite map
$Z\rightarrow S$, set $Z_{j}\coloneqq Z\cap Y(j)_{S}$. If $Z$ is stable under
the action of $\mu_{\tilde{n}}$, then every $Z_{j}$, which is the intersection
of two stable schemes, is stable as well; moreover the induced action of
$\mu_{j}=\mu_{\tilde{n}}/\mu_{\nicefrac{{\tilde{n}}}{{j}}}$ on $Y(j)_{S}$ is
free, thus the induced maps $\pi_{j}\colon Y(j)_{S}\rightarrow
Y(j)_{S}/\mu_{j}$ and $Z_{j}\rightarrow\pi_{j}(Z_{j})$ are étale of degree
$j$.

In this way, from any invariant finite $S-$subscheme of
$\mathfrak{X}(\tilde{n})_{S}$ we construct a finite subscheme in each
$Y(j)_{S}/\mu_{j}$; on the other hand given a sequence of finite
$S-$subschemes of $Y(j)_{S}/\mu_{j}$ of length $\alpha_{j}$ we get a unique
$\mu_{\tilde{n}}-$stable $S-$subscheme of $\mathfrak{X}(\tilde{n})_{S}$ whose
length is $\displaystyle\sum_{j}j\alpha_{j}$.

The last statement follows directly from the fact that the $Y(j)-$s are
themselves open and closed subschemes of
$\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}$. ∎

#### 6.1.8

Thus, recalling that $\operatorname{Hilb}^{\alpha_{j}}(Y(j)/\mu_{j})$ is pure
of dimension $2\alpha_{j}$, we conclude that
$\displaystyle\mathfrak{X}^{[n]}_{k}=\bigsqcup_{\alpha\dashv
n}\mathfrak{X}^{[n]}_{k,\alpha}$, where $\mathfrak{X}^{[n]}_{k,\alpha}$ is an
affine bundle of rank $\displaystyle\sum_{j}2(j-1)\alpha_{j}$ on

$\operatorname{Hilb}^{\alpha_{1}}(Y(1))\times\dots\times\operatorname{Hilb}^{\alpha_{n}}(Y(n)/\mu_{n})\,\text{.
}$

We thus have the following:

######  Corollary 6.5.

The following equation holds in the Grothendieck ring of varieties:

$\left[\mathfrak{X}^{[n]}_{k}\right]=\sum_{\alpha\dashv
n}\prod_{j=1}^{n}\mathbb{L}^{2(j-1)\alpha_{j}}[\operatorname{Hilb}^{\alpha_{j}}(Y(j)/\mu_{j})]\,\text{.
}$

###  6.2 The volume form on $\mathfrak{X}^{[n]}$

#### 6.2.1

There is a volume form, $\omega^{[n]}$, on $\operatorname{Hilb}^{n}(X)$ that
naturally arises from the given $\omega\in\omega_{X/K}$. In this paragraph we
shall recall its construction and compute its zeroes and poles on
$\mathfrak{X}^{[n]}$.

#### 6.2.2

Let $\operatorname{pr}_{i}\colon X^{n}\rightarrow X$, for
$i\in\\{1,\dots,n\\}$, denote the projections on the factors. Then
$\operatorname{pr}_{1}^{\ast}\omega\wedge\cdots\wedge\operatorname{pr}_{n}^{\ast}\omega$
is a global section of $\omega_{X^{n}/K}$ which, being invariant under the
permutation of coordinates, descends to a global section of
$\omega_{\operatorname{Sym}^{n}X/K}$, which we denote by $\varphi$. Let
finally $\omega^{[n]}$ be the pull-back of $\varphi$ through the Hilbert-Chow
morphism, thus $\omega^{[n]}\in
H^{0}(\operatorname{Hilb}^{n}(X),\omega_{\operatorname{Hilb}^{n}(X)})$ is a
volume form on $\operatorname{Hilb}^{n}(X)$.

#### 6.2.3

Now we shall compute the zeroes and poles of $\omega^{[n]}$ seen as a rational
section of $\omega_{\mathfrak{X}^{[n]}/\Delta}$. Since it is a volume form on
the generic fibre of $\mathfrak{X}^{[n]}$, its zeroes or poles are all
irreducible components of the central fibre. Let us fix a connected component
$C\subseteq\mathfrak{X}_{k}^{[n]}$ and let us denote by $C^{\prime}$ the
connected component of
$\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}})^{\mu_{\tilde{n}}}$
such that $C\rightarrow C^{\prime}$ is the affine bundle described in §2.4.5.
In the following lemma we compute the conductor of the action of
$\mu_{\tilde{n}}$ at points of $C^{\prime}$ in terms of the partition of $n$
corresponding to the stratum of
$\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})/\Delta(\tilde{n}))^{\mu_{\tilde{n}}}$
containing $C^{\prime}$.

######  Lemma 6.6.

Consider the decomposition of
$\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}})^{\mu_{\tilde{n}}}$
of Proposition 6.4 and fix a point $[Z]$ lying inside the stratum
corresponding to $\alpha\dashv n$. Then:

  1. 1.

The conductor of the action of $\mu_{\tilde{n}}$ at $[Z]$ is

$c\left(\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}),[Z]\right)=\tilde{n}\sum_{j=1}^{n}(j-1)\alpha_{j}\,\text{.
}$

  2. 2.

If we denote by $[Z_{j}]$ the point of
$\operatorname{Hilb}^{\alpha_{j}}(Y(j)/\mu_{j})$ corresponding to
$Z_{j}/\mu_{j}$, then one has that:

$\operatorname{ord}_{[Z]}(\omega^{[n]}(\tilde{n}))=\tilde{n}\sum_{j=1}^{n}\operatorname{ord}_{[Z_{j}/\mu_{j}]}(\omega^{[\alpha_{j}]}(j))\,\text{,
}$

where $Y(j)$ are considered as part of the central fibre of a weak Néron model
for $X(j)$ as in Remark 6.2.

###### Proof.

Since the values of
$c\left(\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}),[Z]\right)$
and $\operatorname{ord}_{[Z]}(\omega^{[n]}(\tilde{n}))$ depend only on the
connected component containing $[Z]$ and since the generic point of each
connected component corresponds to a reduced scheme, we may compute them with
the additional assumption that $Z$ is a reduced subscheme of
$\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}$. In particular $Z$ is the
disjoint union of $\alpha_{1}$ orbits of length 1, $\alpha_{2}$ orbits of
length 2 and so on. There is an equivariant isomorphism

$T_{[Z]}\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}})\cong\bigoplus_{p\in\operatorname{supp}Z}T_{p}\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}\,\text{.
}$

Let $\zeta=\zeta_{\tilde{n}}$ be a primitive root of unity and let $\sigma$ be
the unique generator of $\mu_{\tilde{n}}$ such that $\sigma$ acts on
$R(\tilde{n})$ by multiplying the uniformizing parametre by $\zeta$. Consider
an orbit of points $p_{0},\dots,p_{j-1}\in Z$, let $e_{1},e_{2}$ be two
generators of $T_{p_{0}}\mathfrak{X}(\tilde{n})_{k}$, so that
$\sigma^{l}(e_{1}),\sigma^{l}(e_{2})$ will give a basis of
$T_{p_{l}}\mathfrak{X}(\tilde{n})_{k}$ for each $l=0,\dots,j-1$. Notice that
$\sigma^{j}(e_{h})=e_{h}$ for $h=1,2$ since
$\mu_{\nicefrac{{\tilde{n}}}{{j}}}$ acts trivially on the whole connected
component cointaining $p_{0}$.

For $i=0,\dots,j-1$, $h=1,2$ we have that

$(e_{h},\zeta^{\nicefrac{{i\tilde{n}}}{{j}}}\sigma(e_{h}),\zeta^{\nicefrac{{2i\tilde{n}}}{{j}}}\sigma^{2}(e_{h}),\dots,\zeta^{\nicefrac{{(j-1)i\tilde{n}}}{{j}}}\sigma^{j-1}(e_{h}))\in
T_{p_{0}}\mathfrak{X}(\tilde{n})_{k}\oplus\cdots\oplus
T_{p_{j-1}}\mathfrak{X}(\tilde{n})$

is an eigenvector with eigenvalue $\zeta^{-\nicefrac{{i\tilde{n}}}{{j}}}$. In
total there are $2j$ of such eigenvectors, which constitute a basis for
$T_{p_{0}}\mathfrak{X}(\tilde{n})\oplus\cdots\oplus
T_{p_{j-1}}\mathfrak{X}(\tilde{n})_{k}$.

The sum of the exponents of this base is

$2\sum_{i=0}^{j-1}-\frac{i\tilde{n}}{j}=-(j-1)\tilde{n}$

We construct eigenvectors of
$T_{[Z]}\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k})$ by putting a
vector such as the above one at the coordinates corresponding to an orbit and
0 at the other coordinates. Running through all the possible orbits, we get a
base of eigenvectors of
$T_{[Z]}\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})/\Delta(\tilde{n}))$.
Thus summing the exponents among all the eigenvectors will lead to the desired
result for the conductor.

Concerning the order of the volume form, we have that

$\displaystyle\operatorname{ord}_{[Z]}(\omega^{[n]}(\tilde{n}))$
$\displaystyle=\sum_{p\in\operatorname{supp}Z}\operatorname{ord}_{p}(\omega(\tilde{n}))$
$\displaystyle=\sum_{j=1}^{n}\frac{\tilde{n}}{j}\sum_{p\in\operatorname{supp}Z_{j}}\operatorname{ord}_{p}(\omega(j))$
$\displaystyle=\sum_{j=1}^{n}\tilde{n}\sum_{p\in
Z_{j}/\mu_{j}}\operatorname{ord}_{p}(\omega(j))$
$\displaystyle=\tilde{n}\sum_{j=1}^{n}\operatorname{ord}_{Z_{j}/\mu_{j}}(\omega(j)^{[\alpha_{j}]})\,\text{.
}$

Where the first and the last equality follow from the fact that the stalk of
the canonical bundle at a point with reduced support of an Hilbert scheme are
the tensor products of the stalks of the canonical bundle of the surface at
every point in the support. The second equality follows from Remark 6.2. The
third equality follow from the fact that $\omega(j)$ has the same order on all
the $j$ points of an orbit of $\mu_{j}$. ∎

#### 6.2.4

We are able, now, to compute the order of $\omega^{[n]}$ at any point
$z\in\mathfrak{X}^{[n]}_{k}$:

######  Corollary 6.7.

Let
$\pi\colon\mathfrak{X}^{[n]}_{k}\rightarrow\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}})$
and assume that

$\displaystyle\pi(z)$ $\displaystyle=(\pi^{1}(z),\pi^{2}(z),\dots,\pi^{n}(z))$
$\displaystyle\in\operatorname{Hilb}^{\alpha_{1}}(Y(1))\times\operatorname{Hilb}^{\alpha_{2}}(Y(2)/\mu_{2})\times\cdots\times\operatorname{Hilb}^{n}(Y(n)/\mu_{n})\,\text{.
}$

We have that
$\displaystyle\operatorname{ord}_{z}(\omega^{[n]})=\sum_{j=1}^{n}\left((j-1)\alpha_{j}+\operatorname{ord}_{\pi^{j}(z)}(\omega(j)^{[\alpha_{j}]})\right)$.

###### Proof.

As a direct consequence of Lemma 2.11 and of the previous lemma we get:

$\displaystyle\operatorname{ord}_{z}(\omega^{[n]})=\frac{\operatorname{ord}_{\pi(z)}(\omega(\tilde{n})^{[n]})-c\left(\operatorname{Hilb}^{n}(\mathfrak{X}(\tilde{n})_{k,\operatorname{sm}}),[Z]\right)}{\tilde{n}}=\sum_{j=1}^{n}\left(-(j-1)\alpha_{j}+\operatorname{ord}_{\pi^{j}(z)}(\omega(j)^{[\alpha_{j}]})\right)\,\text{.
}$

∎

###  6.3 Motivic integral

#### 6.3.1

We keep the convention on $\mathscr{R}$ being one of the three rings
$\mathscr{M}_{k}$,
$\mathscr{M}_{k}\left[(\mathbb{L}^{r}-1)^{-1}\colon 0<r\in\mathbb{N}\right]$
or $\widehat{\mathscr{M}_{k}}$. We are now ready to perform the main
computation of the manuscript; by using the models we constructed above we are
able to compute a generating function for the motivic integrals of all the
Hilbert schemes of points of a surface with trivial canonical bundle. More
precisely the formula we are going to prove is the content of the following
proposition:

######  Theorem 6.8.

The following identity holds true in $\mathscr{R}[[q]]$ if
$\operatorname{char}k=0$, while it holds true in $\mathscr{R}[q]/(q^{p})$ if
$\operatorname{char}k=p>0$:

$\sum_{n\geq
0}\left(\int_{\operatorname{Hilb}^{n}(X)}\omega^{[n]}q^{n}\right)=\prod_{m\geq
1}\left(\left(1-\mathbb{L}^{m-1}q^{m}\right)^{-\left(\int_{X(m)}\omega(m)\right)/\mu_{m}}\right)\,\text{.
}$

######  Corollary 6.9.

Assume that either $\operatorname{char}k=0$ or $\operatorname{char}k>n$, then
the following equation holds:

$\int_{\operatorname{Hilb}^{n}(X)}\omega^{[n]}=\sum_{\alpha\dashv
n}\prod_{j=1}^{\infty}\left(\mathbb{L}^{(j-1)\alpha_{j}}\operatorname{Sym}^{\alpha_{j}}\left(\left(\int_{X(j)}\omega(j)\right)/\mu_{j}\right)\right)\,\text{.
}$

###### Proof.

Since

$\left(1-\mathbb{L}^{m-1}q^{m}\right)^{-\left(\int_{X(m)}\omega(m)\right)/\mu_{m}}=\sum_{l=0}^{\infty}\mathbb{L}^{(m-1)l}\operatorname{Sym}^{l}\left(\left(\int_{X(m)}\omega(m)\right)/\mu_{m}\right)q^{ml}\,\text{,
}$

it follows that, for $\alpha=(\alpha_{1},\alpha_{2},\dots)$ sequence such that
$\alpha_{j}=0$ for $j\gg 1$, one has that

$\deg_{q}\left(\prod_{j=1}^{\infty}\left(\mathbb{L}^{(j-1)\alpha_{j}}\operatorname{Sym}^{\alpha_{j}}\left(\left(\int_{X(j)}\omega(j)\right)/\mu_{j}\right)q^{j\alpha_{i}}\right)\right)=\sum_{j=1}^{\infty}j\alpha_{j}\,\text{.
}$

We get the desired result after identifying the coefficients of $q^{n}$ from
Theorem 6.8. ∎

###### Proof of Theorem 6.8.

We begin our computation using some identities we proved in the previous
section.

$\displaystyle\sum_{n\geq
0}\left(\int_{\operatorname{Hilb}^{n}(X)}\omega^{[n]}q^{n}\right)=$
$\displaystyle\sum_{n\geq
1}\left(\int_{\mathfrak{X}^{[n]}_{k}}\mathbb{L}^{-\operatorname{ord}_{z}(\omega^{[n]})}dzq^{n}\right)$
$\displaystyle=$ $\displaystyle\sum_{n\geq 1}\left(\sum_{\alpha\dashv
n}\left(\int_{\mathfrak{X}^{[n]}_{k,\alpha}}\mathbb{L}^{-\operatorname{ord}_{z}(\omega^{[n]})}dz\right)q^{n}\right)$
$\displaystyle=$ $\displaystyle\sum_{n\geq 1}\left(\sum_{\alpha\dashv
n}\left(\int_{\mathfrak{X}^{[n]}_{k,\alpha}}\prod_{j\geq
1}\left(\mathbb{L}^{(j-1)\alpha_{j}-\operatorname{ord}_{\pi^{j}(z)}(\omega^{[\alpha_{j}]}(j))}q^{j\alpha_{j}}\right)dz\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{\alpha\in\mathbb{N}^{\oplus\mathbb{N}_{\geq
1}}}\left(\prod_{j\geq
1}\left(\mathbb{L}^{(j-1)\alpha_{j}}q^{j\alpha_{j}}\int_{\operatorname{Hilb}^{\alpha_{j}}(Y(j)/\mu_{j})}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(j)^{[\alpha_{j}]})}dz\right)\right)$
$\displaystyle=$ $\displaystyle\prod_{j\geq 1}\left(\sum_{\alpha_{j}\geq
0}\left(\mathbb{L}^{(j-1)\alpha_{j}}q^{j\alpha_{j}}\int_{\operatorname{Hilb}^{\alpha_{j}}(Y(j)/\mu_{j})}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(j)^{[\alpha_{j}]})}dz\right)\right)\,\text{.
}$

######  Lemma 6.10.

Let $Y\rightarrow\operatorname{Spec}k$ be a smooth surface endowed with a
locally constant function $\nu\colon|Y|\rightarrow\mathscr{R}$. Suppose that
functions $\nu^{[n]}\colon|\operatorname{Hilb}^{n}(Y)|\rightarrow\mathscr{R}$
and $\nu^{\prime[n]}\colon|\operatorname{Sym}^{n}(Y)|\rightarrow\mathscr{R}$
are defined in such a way that, for a given subscheme $Z\subseteq Y$ of length
$n$ we have
$\nu^{[n]}(Z)=\prod_{p\in\operatorname{supp}(Z)}\nu(p)^{\operatorname{length}(\mathscr{O}_{Z,p})}$
and $\nu^{[n]}=\nu^{\prime[n]}\circ\pi_{n}$, where
$\pi_{n}\colon\operatorname{Hilb}^{n}(Y)\rightarrow\operatorname{Sym}^{n}(Y)$
is the Hilbert-Chow morphism. Then, for an arbitrary natural number
$\alpha\in\mathbb{N}$, the following identity holds:

$\int_{\operatorname{Hilb}^{\alpha}(Y)}\nu^{[\alpha]}(z)dz=\sum_{\beta\dashv\alpha}\left(\prod_{l\geq
1}\left(\mathbb{L}^{(l-1)\beta_{l}}\int_{\operatorname{Sym}^{\beta_{l}}(Y)}\nu^{\prime[l\beta_{l}]}(z)dz\right)\right)\,\text{.
}$

###### Proof.

We first suppose that $Y$ is connected and, thus,
$\nu\equiv\lambda\in\mathscr{R}$ is constant. Thus we simply have that

$\int_{\operatorname{Hilb}^{\alpha}(Y)}\nu^{[\alpha]}(z)dz=\lambda^{\alpha}[\operatorname{Hilb}^{\alpha}(Y)]\,\text{.
}$

It follows from a well known result in DT theory, for instance [21, §2.2.3],
that

$[\operatorname{Hilb}^{\alpha}(Y)]=\sum_{\beta\dashv\alpha}\left(\prod_{l\geq
1}\left(\mathbb{L}^{(l-1)\beta_{l}}[\operatorname{Sym}^{\beta_{l}}(Y)]\right)\right)\,\text{,
}$

thus we deduce the desired statement, at least when $Y$ is connected.

Now suppose that $C\subseteq Y$ is a connected component and that the
statement holds for $Y\backslash C$.

Using the fact that
$\operatorname{Hilb}^{\alpha}(Y)=\bigsqcup_{j=0}^{\alpha}\left(\operatorname{Hilb}^{\alpha-j}(Y\backslash
C)\times\operatorname{Hilb}^{j}(C)\right)$, we deduce that

$\displaystyle\int_{\operatorname{Hilb}^{\alpha}(Y)}\nu^{[\alpha]}(z)dz=$
$\displaystyle\sum_{j=0}^{\alpha}\left(\int_{\operatorname{Hilb}^{\alpha-j}(Y\backslash
C)}\nu^{[\alpha-j]}(z)dz\right)\cdot\left(\int_{\operatorname{Hilb}^{j}(C)}\nu^{[j]}dz\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\alpha}\left(\sum_{\delta^{j}\dashv\alpha-j}\left(\prod_{l\geq
1}\left(\mathbb{L}^{(l-1)\delta^{j}_{l}}\int_{\operatorname{Sym}^{\delta^{j}_{l}}(Y\backslash
C)}\nu^{\prime l[\delta^{j}_{l}]}(z)dz\right)\right)\cdot\right.$
$\displaystyle\cdot\left.\sum_{\gamma^{j}\dashv j}\left(\prod_{l\geq
1}\left(\mathbb{L}^{(l-1)\gamma^{j}_{l}}\int_{\operatorname{Sym}^{\gamma^{j}_{l}}(C)}\nu^{\prime
l[\gamma^{j}_{l}]}(z)dz\right)\right)\right)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\alpha}\left(\sum_{\begin{subarray}{c}{\delta^{j}\dashv\alpha-j}\\\
{\gamma^{j}\dashv j}\end{subarray}}\left(\prod_{l\geq
1}\left(\mathbb{L}^{(l-1)(\delta^{j}_{l}+\gamma^{j}_{l})}\int_{\operatorname{Sym}^{\beta^{j}_{l}}(Y\backslash
C)\times\operatorname{Sym}^{\gamma^{j}_{l}}(C)}\nu^{\prime
l[\delta^{j}_{l}+\gamma^{j}_{l}]}(z)dz\right)\right)\right)$ $\displaystyle=$
$\displaystyle\sum_{\beta\dashv\alpha}\left(\prod_{l\geq
1}\left(\mathbb{L}^{(l-1)\beta_{l}}\sum_{i=0}^{\beta_{l}}\int_{\operatorname{Sym}^{\beta_{l}-i}(Y\backslash
C)\times\operatorname{Sym}^{i}(C)}\nu^{\prime
l[\beta_{l}]}(z)dz\right)\right)$ $\displaystyle=$
$\displaystyle\sum_{\beta\dashv\alpha}\left(\prod_{l\geq
1}\left(\mathbb{L}^{(l-1)\beta_{l}}\int_{\operatorname{Sym}^{\beta_{l}}(Y)}\nu^{\prime
l[\beta_{l}]}(z)dz\right)\right)\,\text{, }$

which concludes the proof. ∎

We plug the lemma above in our chain of equalities using $Y=Y(j)/\mu_{j}$,
$\nu\coloneqq\mathbb{L}^{-\operatorname{ord}_{z}(\omega(j))}$, recalling also
that $l\operatorname{ord}_{z}(\omega(j))=\operatorname{ord}_{z}(\omega(jl))$
and that $Y(j)$ naturally embeds in the central fibre of some weak Néron model
of $X(lj)$ (as in Remark 6.2 with $d=\nicefrac{{\tilde{n}}}{{lj}}$), we get:

$\displaystyle\sum_{n\geq 0}$
$\displaystyle\left(\int_{\operatorname{Hilb}^{n}(X)}\omega^{[n]}q^{n}\right)=$
$\displaystyle=\prod_{j\geq 1}\left(\sum_{\alpha_{j}\geq
0}\left(\mathbb{L}^{(j-1)\alpha_{j}}\sum_{\beta^{j}\dashv\alpha_{j}}\left(q^{j\alpha_{j}}\prod_{l\geq
1}\left(\mathbb{L}^{(l-1)\beta^{j}_{l}}\int_{\operatorname{Sym}^{\beta^{j}_{l}}(Y(j)/\mu_{j})}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(lj)^{[\beta_{l}^{j}]})}dz\right)\right)\right)\right)$
$\displaystyle=\prod_{j\geq
1}\left(\sum_{\beta^{j}\in\mathbb{N}^{\oplus\mathbb{N}_{\geq
1}}}\left(\prod_{l\geq
1}\left(\mathbb{L}^{(jl-1)\beta_{l}^{j}}q^{jl\beta_{l}^{j}}\int_{\operatorname{Sym}^{\beta^{j}_{l}}(Y(j)/\mu_{j})}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(lj)^{[\beta_{l}^{j}]})}dz\right)\right)\right)$
$\displaystyle=\prod_{j,l\geq 1}\left(\sum_{\beta^{j}_{l}\geq
0}\left(\mathbb{L}^{(jl-1)\beta_{l}^{j}}q^{jl\beta_{l}^{j}}\int_{\operatorname{Sym}^{\beta^{j}_{l}}(Y(j)/\mu_{j})}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(lj)^{[\beta_{l}^{j}]})}dz\right)\right)\,\text{.
}$

Recalling that, in the sum above, there is only a finite number of
nonvanishing coefficients of $q^{n}$, for every positive integer $n$, we are
allowed to group such summands in a different order; since the map

$\displaystyle\mathbb{N}_{+}\times\mathbb{N}_{+}$
$\displaystyle\rightarrow\mathbb{N}_{+}\times\mathbb{N}_{+}$
$\displaystyle(j,l)$ $\displaystyle\mapsto(j\cdot l,j)$

is injective and its image is $\\{(m,j)\colon j|m\\}$, after the substitution
$\lambda^{m}_{j}\coloneqq\beta^{j}_{l}$, we get the equivalent expression:

$\displaystyle\sum_{n\geq 0}$
$\displaystyle\left(\int_{\operatorname{Hilb}^{n}(X)}\omega^{[n]}q^{n}\right)=$
$\displaystyle=\prod_{m\geq 1}\left(\prod_{j|m}\left(\sum_{\lambda^{m}_{j}\geq
0}\left(\mathbb{L}^{(m-1)\lambda_{j}^{m}}q^{m\lambda_{j}^{m}}\int_{\operatorname{Sym}^{\lambda_{j}^{m}}(Y(j)/\mu_{j})}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(m)^{[\lambda_{j}^{m}]})}dz\right)\right)\right)$
$\displaystyle=\prod_{m\geq
1}\left(\sum_{\lambda^{m}\in\mathbb{N}^{\operatorname{Div}(m)}}\left(\left(\mathbb{L}^{(m-1)}q^{m}\right)^{\sum_{j|m}\lambda_{j}^{m}}\prod_{j|m}\left(\int_{\operatorname{Sym}^{\lambda_{j}^{m}}(Y(j)/\mu_{j})}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(m)^{[\lambda_{j}^{m}]})}dz\right)\right)\right)$
$\displaystyle=\prod_{m\geq 1}\left(\sum_{r_{m}\geq
0}\left(\left(\mathbb{L}^{(m-1)}q^{m}\right)^{r_{m}}\int_{\operatorname{Sym}^{r_{m}}\left(\left(\sqcup_{j|m}Y(j)\right)/\mu_{m}\right)}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(m)^{[r_{m}]})}dz\right)\right)$
$\displaystyle=\prod_{m\geq 1}\left(\sum_{r_{m}\geq
0}\left(\left(\mathbb{L}^{(m-1)}q^{m}\right)^{r_{m}}\operatorname{Sym}^{r_{m}}\left(\int_{\left(\sqcup_{j|m}Y(j)\right)/\mu_{m}}\mathbb{L}^{-\operatorname{ord}_{z}(\omega(m))}dz\right)\right)\right)$
$\displaystyle=\prod_{m\geq 1}\left(\sum_{r_{m}\geq
0}\left(\left(\mathbb{L}^{(m-1)}q^{m}\right)^{r_{m}}\operatorname{Sym}^{r_{m}}\left(\nicefrac{{\left(\int_{X(m)}\omega(m)\right)}}{{\mu_{m}}}\right)\right)\right)$
$\displaystyle=\prod_{m\geq
1}\left(\left(1-\mathbb{L}^{m-1}q^{m}\right)^{-\left(\int_{X(m)}\omega(m)\right)/\mu_{m}}\right)\,\text{.
}$ (6.1)

∎

#### 6.3.2

Applying this identity to every coefficient of the zeta function
$Z_{X,\omega}(T)$, we obtain a formula for the motivic zeta function for its
Hilbert schemes of points:

######  Theorem 6.11.

Assume that either $\operatorname{char}k=0$ or $\operatorname{char}k>n$, then
the following equation holds:

$Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}=\sum_{\alpha\dashv
n}\prod_{j=1}^{\infty}\left(\mathbb{L}^{(j-1)\alpha_{j}}\operatorname{Sym}^{\alpha_{j}}\left(Z_{X(j),\omega(j)}\right)\right)\,\text{.
}$ (6.2)

###### Proof.

It follows after an application of Corollary 6.9 to each coefficient of the
zeta function. ∎

##  7 Proof of the conjecture

###  7.1 Poles of the Zeta function

#### 7.1.1

Througout this section, we denote by $\overline{\mathscr{R}}$ one of the two
rings $\mathscr{M}_{k}\left[(\mathbb{L}^{r}-1)^{-1}\colon
0<r\in\mathbb{N}\right]$ or $\widehat{\mathscr{M}}_{k}$, while $\mathscr{R}$
will denote either $\overline{\mathscr{R}}$ or $\mathscr{M}_{k}$.

The aim of this section is to study the poles of
$Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}(T)$ in terms of those of
$Z_{X,\omega}$ and deduce the following:

######  Theorem 7.1 (Monodromy conjecture for Hilbert schemes).

Let $X$ be a surface with trivial canonical bundle satisfying the monodromy
conjecture in $\overline{\mathscr{R}}$.
If $\operatorname{char}k=0$, then the same holds for
$\operatorname{Hilb}^{n}(X)$, $\forall n\in\mathbb{N}$. If
$\operatorname{char}k=p>0$ and $X$ admits a model as in §6.1, then the
monodromy conjecture in $\overline{\mathscr{R}}$ holds for
$\operatorname{Hilb}^{n}(X)$, $\forall n<\operatorname{char}k$.

#### 7.1.2

Despite not being the aim of our discussion, we report here the following
statement, which can be obtained as a byproduct of the argments we have
developed so far:

######  Proposition 7.2.

Let $Y,Z$ be two Calabi-Yau varieties endowed with volume forms
$\omega_{1},\omega_{2}$ satisfying the monodromy conjecture in
$\overline{\mathscr{R}}$. Let $\omega$ be volume forms on $Y\times Z$ the
volume form
$\omega\coloneqq\operatorname{pr}_{Y}^{\ast}\omega_{1}\wedge\operatorname{pr}_{Z}^{\ast}\omega_{2}$.
Then also $Y\times Z$, endowed with the volume form $\omega$, satisfies the
monodromy conjecture in $\overline{\mathscr{R}}$.

#### 7.1.3

We start with a simple remark:

######  Remark 7.3.

For an arbitrary positive integer $l>0$ we have that

$l\cdot
Z_{X(l),\omega(l)}(T^{l})=\sum_{i=0}^{l-1}Z_{X,\omega}(\zeta_{l}^{i}T)\,\text{,
}$

where we consider the functions as power series with coefficient in an
algebraic extensions of $\mathscr{R}$ containing the $l-$th roots of unity
(though after the due cancellations, the above equation involves only elements
of $\mathscr{R}$). Thus, by writing $Z_{X,\omega}(T)$ in the form Equation
(4.1), with $N$ divisible by $l$, we see that the set of poles of
$Z_{X(l),\omega(l)}(T)$ is contained in $l\cdot\pazocal{P}$.

#### 7.1.4

Using this remark and the results from §5.3 we get an upper bound on the set
of poles of $Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}(T)$:

######  Corollary 7.4.

Let $X$ be a surface with trivial canonical bundle and $\omega$ a volume form
on it. Assume that $Z_{X,\omega}(T)\in\mathscr{R}[[T]]$ can be written as a
sum of functions with only one pole. Let $\pazocal{P}$ be the set of poles of
$Z_{X,\omega}$. Then all the poles of
$Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}(T)$ are contained in
$\Sigma^{n}\pazocal{P}$.

###### Proof.

Let us write $Z_{X(j),\omega(j)}/\mu_{j}(T)=\sum_{i\geq 0}A_{i}^{(j)}T^{i}$.
For each $\alpha\dashv n$, let

$F_{\alpha}(T)\coloneqq\sum_{i>0}\left(\operatorname{Sym}^{\alpha_{1}}A_{i}^{(1)}\right)\cdots\left(\operatorname{Sym}^{\alpha_{n}}A_{i}^{(n)}\right)T^{i}\,\text{.
}$

According to Equation (6.3.1), we have that
$Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}(T)=\sum_{\alpha\dashv
n}\mathbb{L}^{n-|\alpha|}F_{\alpha}(T)$, thus we only need to prove that
$F_{\alpha}$ has only poles inside $\Sigma^{n}\pazocal{P}$. Lemma 5.10 implies
that $(\operatorname{Sym}^{\alpha_{j}}Z_{X(j),\omega(j)}/\mu_{j})(T)$ has
poles in
$\Sigma^{\alpha_{j}}(j\pazocal{P})\subseteq\Sigma^{j\alpha_{j}}\pazocal{P}$;
our statement follows from Proposition 5.8 and from the identity

$\Sigma^{\alpha_{1}}\pazocal{P}+\Sigma^{2\alpha_{2}}\pazocal{P}+\cdots+\Sigma^{n\alpha_{n}}\pazocal{P}=\Sigma^{n}\pazocal{P}\,\text{.
}$

∎

#### 7.1.5

We are now ready to prove Theorem 7.1:

###### Proof of Theorem 7.1.

Let $q$ be a pole of $Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}(T)$ and let
$\sigma\in\operatorname{Gal}(\overline{K}|K)$ be a topological generator of
the tame Galois subgroup. Consider poles $q_{1},\dots,q_{n}$ of
$Z_{X,\omega}(T)$ such that $q=q_{1}+\cdots+q_{n}$. Since the monodromy
conjecture holds for $X$, there are elements $v_{1},v_{2},\dots,v_{n}\in
H^{\ast}(X_{\overline{K}},\mathbb{Q}_{l})$ such that $\sigma(v_{j})=e^{2\pi
iq_{j}}v_{j}$. Let us consider the Galois-equivariant isomorphism from [7,
Theorem 2]:

$H^{\ast}(\operatorname{Hilb}^{n}(X),\mathbb{Q}_{l})\cong\bigoplus_{\alpha\dashv
n}H^{\ast}(\operatorname{Sym}^{|\alpha|}(X),\mathbb{Q}_{l})(n-|\alpha|)\,\text{;
}$

focusing on the summand
$H^{\ast}(\operatorname{Sym}^{n}(X),\mathbb{Q}_{l})\cong
H^{\ast}(X^{n},\mathbb{Q}_{l})^{\Sigma_{n}}$, where the action of $\Sigma_{n}$
on
$H^{\ast}(X^{n},\mathbb{Q}_{l})\cong H^{\ast}(X,\mathbb{Q}_{l})^{\otimes n}$
is induced by the usual action $\Sigma_{n}\curvearrowright X^{n}$ given by
permutation of the factors. Thus the element

$\sum_{\rho\in\Sigma_{n}}v_{\rho(1)}\otimes\cdots\otimes v_{\rho(n)}$

is a non-zero eigenvector of
$H^{\ast}(\operatorname{Hilb}^{n}(X),\mathbb{Q}_{l})$ for the eigenvalue
$\prod_{j=1}^{n}e^{2\pi iq_{j}}$. ∎

#### 7.1.6

And similarly:

###### Proof of Proposition 7.2..

We have that $\displaystyle\int_{Y(n)\times
Z(n)}\omega(n)=\left(\int_{Y(n)}\omega_{1}(n)\right)\left(\int_{Z(n)}\omega_{2}(n)\right)$.

Hence Proposition 5.8 implies that $\forall q$ pole of $Z_{Y\times
Z,\omega}(T)$ there are a pole $q_{1}$ of $Z_{Y,\omega_{1}}(T)$ and a pole
$q_{2}$ of $Z_{Z,\omega_{2}}(T)$ such that $q=q_{1}+q_{2}$.

Since $Y$ and $Z$ satisfy the monodromy conjecture, there are nonzero
eigenvectors $v\in H^{\ast}(Y,\mathbb{Q}_{l})$ with eigenvalue $\exp(2\pi
iq_{1})$ and $w\in H^{\ast}(Z,\mathbb{Q}_{l})$ with eigenvalue $\exp(2\pi
iq_{2})$, so that the element $v\otimes w\in H^{\ast}(Y,\mathbb{Q}_{l})\otimes
H^{\ast}(Z,\mathbb{Q}_{l})\cong H^{\ast}(Y\times Z,\mathbb{Q}_{l})$ is an
eigenvector with eigenvalue $\exp(2\pi iq)$. ∎

#### 7.1.7

We are not able to say much about the monodromy conjecture in
$\mathscr{M}_{k}$ for all the Hilbert schemes of points on a surface, since it
is not always possible to write $Z_{X,\omega}(T)\in\mathscr{M}_{k}[[T]]$ as a
sum of functions with a single pole. However, there are a few remarkable
classes of surfaces whose zeta function has a unique pole. In these cases,
such a condition is automatically satisfied, so also
$Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}$ has a unique pole which is $n$
times the pole of $Z_{X,\omega}$ and $\operatorname{Hilb}^{n}(X)$ will then
satisfy the monodromy property.

######  Example 7.5.

We list a few classes of surfaces satisfying the property above:

  * •

Assume that $X$ is an abelian surface; according to [10], $Z_{X,\omega}$ has a
unique pole which coincides with Chai’s basechange conductor of $X$;

  * •

If $X\rightarrow\operatorname{Spec}K$ is a $K3$ surface admitting an
equivariant Kulikov model after a finite base change with respect to afinite
extension $F/K$, then Halle and Nicaise proved in [11] that $Z_{X,\omega}$ has
a unique pole;

  * •

Assume that $X$ is a Kummer surface constructed from an abelian surface $A$;
then Overkamp proved in [18] that $Z_{X,\omega}$ has a unique pole.

Moreover all the surfaces in this list satisfy the monodromy property.

######  Corollary 7.6.

Let $X$ be a surface in the list above, then the monodromy conjecture holds
for $\operatorname{Hilb}^{n}(X)$, provided that either
$\operatorname{char}k=0$ or $\operatorname{char}k>n$, with the usual
assumptions on the models of $X$.

Similarly, the monodromy conjecture holds for a product
$X_{1}\times\dots\times X_{n}$, where all the $X_{i}$ are surfaces in the list
above.

###### Proof.

The first statement follows from 7.1, while the latter follows from 7.2. ∎

###  7.2 Further remarks

#### 7.2.1

It is still an open question whether all the sums of $n$ poles of
$Z_{X,\omega}(T)$ are actually poles of
$Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}(T)$, or if cancellation might
occur. It is reasonable to expect that given a very general $K3$ surface $X$
and a positive integer $n$, if $\pazocal{P}$ is the set of poles of
$Z_{X,\omega}$, then the set of poles of $Z_{Hilb^{n}(X),\omega^{[n]}}$
coincides with $\Sigma^{n}\pazocal{P}$. We expect this in virtue of the fact
that the expression for $Z_{\operatorname{Hilb}^{n}(X),\omega^{[n]}}$,
obtained by following the algorithm of Corollary 7.4 and §5.3, contains terms
having a pole in each element of $\Sigma^{n}\pazocal{P}$ and the cancellation
among them ”should happen only exceptionally”.

######  Example 7.7.

Let $K\coloneqq k((t))$, $R\coloneqq k[[t]]$, where $\operatorname{char}k=0$.
Let $X\subseteq\mathbb{P}^{3}_{K}$ (with homogeneous coordinates $[w:x:y:z]$)
be the surface defined by the quartic polynomial:

$w^{2}x^{2}+w^{2}y^{2}+w^{2}z^{2}+x^{4}+y^{4}+z^{4}+tw^{4}\,\text{, }$

and let, finally, $\omega$ be an arbitrary volume form over it; this example
was already studied in [11]. It is possible to prove that $Z_{X,\omega}$ has
two poles and satisfies the monodromy conjecture in $\mathscr{M}_{k}$. A
direct computation (relying on a construction we will sketch later) shows that
the poles of $Z_{\operatorname{Hilb}^{2}(X),\omega^{[2]}}$ are actually the
three expected poles. Let $\mathfrak{Y}\subseteq\mathbb{P}^{3}_{R}$ the model
of $X$ obtained by the above equation considered as a polynomial with
coefficients in $R$. The model constructed in this way is a regular model
whose central fibre is an irreducible surface with only a singular point $O$
of type $A_{1}$. After blowing up $O\in\mathfrak{Y}$ one obtains a regular
model with strict normal crossing divisor, whose central fibre consists of two
components: a regular $K3$ surface (the strict transform of $\mathfrak{Y}_{k}$
and a copy of $\mathbb{P}^{2}_{k}$ with multiplicity 2, their intersection is
a rational curve which sits in $\mathbb{P}^{2}$ as a conic. After semistable
reduction, one gets a model $\mathfrak{X}(2)\rightarrow R(2)$ of $X(2)$ whose
central fibre consists of5 a smooth $K3$ surface intersecting
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ along its diagonal. Using the
construction of Nagai in [17] one gets a semistable model for
$\operatorname{Hilb}^{2}(X(2))$ over $R(2)$ and after basechanging such model
and Weil-restricting it is possible to compute the motivic integral of
$\operatorname{Hilb}^{2}(X(m))$ for all $m\in\mathbb{N}$ and thus
$Z_{\operatorname{Hilb}^{2}(X),\omega^{[2]}}\in\mathscr{M}_{k}[[T]]$. After
specializing the zeta function using the Poincaré polynomial one sees that all
the three possible poles are indeed poles for
$Z_{\operatorname{Hilb}^{2}(X),\omega^{[2]}}$.

## References

  * [1] E. Bultot, Motivic Integration and Logarithmic Geometry _Ph.D Thesis, Katholieke Universiteit Leuven,_ 2015
  * [2] E. Bultot, J. Nicaise, Computing motivic zeta functions on log smooth models _Math. Z., 295,_ 2019
  * [3] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models in _Ergebnisse der Mathematik und ihrer Grenzgebiete,_ 21, Springer-Verlag, Berlin, 1990
  * [4] B. Conrad, F. Gabber, G. Prasad, Pseudo-reductive Groups, _Cambridge University Press_ 2015
  * [5] M.A. De Cataldo, L. Migliorini, The Chow groups and the motive of the Hilbert scheme of points on a surface, _Journal of Algebra 251, 824-848,_ 2002
  * [6] B. Edixhoven, Néron models and tame ramification, _Comp. Math., 81, 291-306,_ 1992
  * [7] L. Göttsche, W. Sörgel, The cohomology of Hilbert schemata of smooth algebraic surfaces, _Math. Ann., 296, 235-245,_ 1993
  * [8] Gusein-Zade, Luengo, Melle-Hernandez, A power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points, _Michigan Math. J., 54, 2, 353-359,_ 2006
  * [9] O. Gabber, L. Ramero, Foundations for almost ring theory, Sixth release Preprint: arXiv:math/0409584v7 2012
  * [10] L.H. Halle, J. Nicaise, Motivic zeta functions of abelian varieties, and the monodromy conjecture, _Advances in Mathematics, 227, 1, 610-653,_ 2011
  * [11] L.H. Halle, J. Nicaise, Motivic zeta functions of degenerating Calabi-Yau varieties, _Math. Ann., 370, 3-4, 1277–1320,_ 2018
  * [12] A. Jaspers, The monodromy property for $K3$ surfaces allowing a triple-point-free model, _Ph.D Thesis, Katholieke Universiteit Leuven,_ 2017
  * [13] F. Loeser, J. Sebag, Motivic integration on smooth rigid varieties and invariants of degenerations, _Duke Math. J., 119, 2, 315-344,_ 2003
  * [14] J. Nicaise, J. Sebag, Motivic Serre invariants, ramification, and the analytic Milnor fiber, _Invent. Math., 168, 1, 133-173,_ 2007
  * [15] J. Nicaise, J. Sebag, The Grothendieck ring of varieties, in _Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry, 145-188,_ _Cambridge university press,_ 2011
  * [16] J. Nicaise, C. Xu, Poles of maximal order of motivic zeta functions, _Duke Math. J., 165, 2, 217–243_ 2016
  * [17] Y. Nagai, On monodromies of a degeneration of irreducible symplectic Kähler manifolds, _Math. Z. 258, 407–426,_ 2008
  * [18] O. Overkamp, Degeneration of Kummer surfaces, _Mathematical Proceedings of the Cambridge Philosophical Society, 4, 1-33,_ 2020
  * [19] Z. Qu, Toric schemes over a discrete valuation ring and tropical compactifications, _Ph.D. Thesis, The University of Texas at Austin,_ 2009
  * [20] T. Saito, Log smooth extension of a family of curves and semi-stable reduction, _J. Alg. Geom., 13, 287-321,_ 2004
  * [21] A. T. Ricolfi, Local Donaldson-Thomas invariants and their refinements _Ph.D. Thesis, University of Stavager,_ 2017

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