# Geometric Monodromies, Mixed Hodge Numbers of Motivic Milnor Fibers and
Newton Polyhedra 1112010 Mathematics Subject Classification: 14F05, 14M25,
32C38, 32S35, 32S40

Kiyoshi TAKEUCHI 222Mathematical Institute, Tohoku University, Aramaki Aza-
Aoba 6-3, Aobaku, Sendai, 980-8578, Japan. E-mail<EMAIL_ADDRESS>
###### Abstract

We introduce the theory of local and global monodromies of polynomials in
cohomology groups in various geometric situations, focusing on its relations
with toric geometry and motivic Milnor fibers, and moreover in the modern
languages of nearby and vanishing cycle functors. Equivariant mixed Hodge
numbers of motivic Milnor fibers will be described in terms of Newton
polyhedra of polynomials.

##  1 Introduction

Since Milnor’s discovery of Milnor fibers of complex hypersufaces in [75],
many mathematicians in different fields, such as singularity theory, algebraic
geometry, topology, complex analysis, number theory and D-modules, studied
them from various points of view. Moreover, after the two fundamental papers
[6] and [102], globalizing the situation, they started to study also the
singularities and the monodromies at infinity of polynomial maps. The results
thus obtained are surprisingly similar to the ones obtained in the local case.
Here let us call these monodromies in the both local and global cases
“geometric monodromies” for short, in order to distinguish them from the ones
of general constructible sheaves. The aim of this note is to introduce the
theory of such geometric monodromies, laying special emphasis on its relations
with toric geometry and motivic Milnor fibers, and moreover in the modern
languages of nearby and vanishing cycle functors.

To this end, we try to explain all the basic definitions and results on nearby
and vanishing cycle functors in a comprehensible way and show their geometric
applications at the same time as many as possible. By considering also the
superstructures of mixed Hodge modules of nearby cycle complexes, in [17]
Denef and Loeser introduced motivic Milnor fibers, which can be regarded as
the motivic reincarnations of the classical Milnor fibers. This motivic point
of view is very important in the recent progress of the theory of geometric
monodromies. In the last half of this note, we introduce the theory of motivic
Milnor fibers at infinity developed in Matsui-Takeuchi [71] and Raibaut [89],
[90], [91]. Combining it with some results in toric geometry such as the
Bernstein-Khovanskii-Kouchnirenko theorem in [54] and the results in [71,
Section 2], in the both local and global cases, one can now describe the
equivariant mixed Hodge numbers of motivic Milnor fibers in terms of Newton
polyhedra of polynomials. In this way, the readers will see how the
combinatorial expressions of the Jordan normal forms of various geometric
monodromies are obtained.

The organization of this note is as follows. In Section 2 we first recall the
classical definitions of Milnor fibers and Milnor monodromies and explain
their relations with the nearby and vanishing cycle functors introduced by
Deligne [15]. Then we show how the monodromy zeta functions associated to
nearby cycle complexes of constructible sheaves are calculated via resolutions
of singularities. In Section 3 we combine this method with some notions and
results in toric geometry, such as toric modifications and the Bernstein-
Khovanskii-Kouchnirenko theorem. Specifically, we thus prove the celebrated
theorems of Kouchnirenko, Varchenko and Oka which allow us to express the
monodromy zeta functions explicitly in terms of Newton polyhedra associated to
polynomials. Here as in [68], we prove them by using the theory of nearby
cycle functors explained in Section 2. By the clarity of its expression and
its applicability to generic situations, Varchenko’s theorem is extremely
useful in singularity theory, especially in the study of the monodromy
conjecture proposed in Denef-Loeser [16]. For the applications of Varchenko’s
theorem in this direction, see e.g. [30], [59], [63]. In Section 4 we take the
same strategy to introduce some basic results on singularities and monodromies
at infinity of polynomial maps. In particular, here we focus on Némethi-
Zaharia’s theorem in [76] and Libgober-Sperber’s one in [64]. Note that
Libgober-Sperber’s theorem can be considered as a global counterpart of
Varchenko’s one. In Sections 5 and 6, we prepare several notions and results
which will be used in subsequent sections. Then in Section 7, we explain the
theory of motivic Milnor fibers at infinity developed in Matsui-Takeuchi [71]
and Raibaut [89], [90], [91] independently. Combining it with the results in
Sections 5 and 6, then we see how the combinatorial expressions in [71] of the
Jordan normal forms of monodromies at infinity are obtained. In the course of
their proofs, we obtain also precise descriptions of the equivariant mixed
Hodge numbers of motivic Milnor fibers at infinity. As we will see in Section
8, similar results hold true also for (local) Milnor monodromies and (local)
motivic Milnor fibers. For the further applications of these methods, see
[11], [31], [82], [98] and [112]. In Section 9 we introduce the theory of
Katz-Stapledon [52], [53] and Stapledon [104] which impoved the results in
[31], [71], [72], [112]. Remarkably, by using also the theories of
intersection cohomology, tropical geometry, mirror symmetry and combinatorics,
in [104] Stapledon obtained the full combinatorial expressions of the Jordan
normal forms of various geometic monodromies. Finally in Section 10, we show
how these methods can be applied partially to Milnor fibers and Milnor
monodromies of meromorphic functions. For this purpose, we explain some basic
properties of the meromorphic nearby and vanishing cycle functors introduced
in [82] and [92].

Acknowledgement: The author is very grateful to Professor Pepe Seade and his
colleagues for suggesting him to write this expository paper in Volume VI of
the Handbook of Geometry and Topology of Singularities. He also thanks the
referee for many valuable suggestions and comments which improved this paper.

##  2 Milnor fibers and nearby cycle sheaves

In this section, we recall the classical definitions of Milnor fibers and
Milnor monodromies and explain their relations with the nearby and vanishing
cycle functors introduced by Deligne [15]. In this note, we essentially follow
the terminology of [20], [45], [50], [51] and [101]. For example, for a
topological space $X$ we denote by ${\bf D}^{b}(X)$ the derived category whose
objects are bounded complexes of sheaves of ${\mathbb{C}}_{X}$-modules on $X$.
For a complex analytic space $X$, let ${\bf D}_{c}^{b}(X)$ be the full
subcategory of ${\bf D}^{b}(X)$ consisting of constructible complexes of
sheaves. In what follows, let $X$ be a complex analytic space and $f\colon
X\longrightarrow{\mathbb{C}}$ a non-constant holomorphic function on it. The
problem being local, we may assume that $X$ is a complex analytic subset of a
complex manifold $M$ and for a non-constant holomorphic function
$\widetilde{f}\colon M\longrightarrow{\mathbb{C}}$ on it we have
$\widetilde{f}\big{|}_{X}\equiv f$. Set $X_{0}:=f^{-1}(0)\subsetneqq X$. First
of all, the following fundamental result was proved by Milnor [75] and Lê
[61]. See also Massey [67, Proposition 1.3].

######  Theorem 2.1.

Let $X\subset{\mathbb{C}}^{N}$ be a complex analytic subset of
${\mathbb{C}}^{N}$ containing the origin $0\in{\mathbb{C}}^{N}$ and $f\colon
X\longrightarrow{\mathbb{C}}$ a non-constant holomorphic function on it
satisfying the condition $f(0)=0$. Then for $0<\eta\ll\varepsilon\ll 1$ the
restriction

$X\cap B(0;\varepsilon)\cap f^{-1}(B(0;\eta)^{*})\longrightarrow
B(0;\eta)^{*}\subset{\mathbb{C}}^{*}$ (2.1)

of $f$ is a fiber bundle, where we set $B(0;\eta)^{*}:=\\{z\in{\mathbb{C}}\mid
0<|z|<\eta\\}\subset{\mathbb{C}}^{*}$ and
$B(0;\varepsilon):=\\{x=({x}_{1},\dots,x_{N})\in{\mathbb{C}}^{N}\mid||x||=\sqrt{\sum_{i=1}^{N}|x_{i}|^{2}}<\varepsilon\\}\subset{\mathbb{C}}^{N}$.

######  Definition 2.2.

In the situation of Theorem 2.1, for a point $z\in B(0;\eta)^{*}$ we call

$F_{0}:=X\cap B(0;\varepsilon)\cap f^{-1}(z)$ (2.2)

the Milnor-Lê fiber (or simply the Milnor fiber) of $f$ at the origin $0\in
X_{0}=f^{-1}(0)\subset X$.

In the situation of Theorem 2.1, for any $j\in{\mathbb{Z}}$ the $j$-th higher
direct image sheaf of the constant sheaf on $X\cap B(0;\varepsilon)\cap
f^{-1}(B(0;\eta)^{*})$ by $f$ is a local system on the punctured disk
$B(0;\eta)^{*}\subset{\mathbb{C}}^{*}$. Then by turning around the origin
$0\in{\mathbb{C}}$ in $B(0;\eta)^{*}$ we obtain automorphisms of their stalks,
i.e. ${\mathbb{C}}$-linear isomorphisms

$\Phi_{j,0}\colon
H^{j}(F_{0};{\mathbb{C}})\overset{\sim}{\longrightarrow}H^{j}(F_{0};{\mathbb{C}})\qquad(j\in{\mathbb{Z}}).$
(2.3)

We call them the Milnor-Lê monodromies (or simply the Milnor monodromies) of
$f$ at the origin $0\in X_{0}=f^{-1}(0)\subset X$. It is clear that the Milnor
fiber and the Milnor monodromies are defined similarly also at any point $x\in
X_{0}=f^{-1}(0)\subset X$ of $X_{0}$. We denote them $F_{x}$ and $\Phi_{j,x}$
($j\in{\mathbb{Z}}$) respectively. If $X={\mathbb{C}}^{n}$ and the complex
hypersurface $X_{0}=f^{-1}(0)\subset X$ of it has an isolated singular point
at the origin $0\in X_{0}=f^{-1}(0)\subset X$, we have the following beautiful
theorem due to Milnor [75]

######  Theorem 2.3.

(Milnor [75]) Let $f\colon X={\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}\
(n\geq 2)$ be a non-constant holomorphic function on $X={\mathbb{C}}^{n}$
satisfying the condition $f(0)=0$ and assume that the complex hypersurface
$X_{0}=f^{-1}(0)\subset X$ has an isolated singular point at the origin $0\in
X_{0}=f^{-1}(0)\subset X$. Then the Milnor fiber $F_{0}$ of $f$ at the origin
is homotopy equivalent to the bouquet of some $(n-1)$-dimensional spheres
$S^{n-1}$:

$F_{0}\sim S^{n-1}\vee\cdots\vee S^{n-1}.$ (2.4)

In particular, for its cohomology groups $H^{j}(F_{0};{\mathbb{C}})$
($j\in{\mathbb{Z}}$) we have

$H^{j}(F_{0};{\mathbb{C}})\simeq\begin{cases}\ {\mathbb{C}}&(j=0)\\\ &\\\ \
{\mathbb{C}}^{\mu}&(j=n-1)\\\ &\\\ \ 0&(j\neq 0,n-1),\end{cases}$ (2.5)

where $\mu>0$ is the number of spheres $S^{n-1}$ in the above bouquet
decomposition.

We call the number $\mu>0$ in Theorem 2.3 the Milnor number of $f$ (or of
$X_{0}=f^{-1}(0)$) at the origin $0\in X_{0}=f^{-1}(0)\subset X$. Note that
the Milnor fiber $F_{0}\subset{\mathbb{C}}^{n}$ in Theorem 2.3 is an
$(n-1)$-dimensional complex manifold i.e. a $(2n-2)$-dimensional real analytic
manifold. Theorem 2.3 asserts that it has the homotopy type of the real
$(n-1)$-dimensional topological space $S^{n-1}\vee\cdots\vee S^{n-1}$. If
$X_{0}=f^{-1}(0)\subset X={\mathbb{C}}^{n}$ for
$f:X={\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ does not have an isolated
singular point at the origin $0\in X_{0}=f^{-1}(0)$, it is in general very
hard to determine the Betti numbers of the Milnor fiber $F_{0}$. Nevertheless,
making use of Morse theory Lê [60] and Massey [66] obtained the handle
decompositions of $F_{0}$. Namely we can know at least the homotopy type of
it. From now, we shall explain how Milnor fibers and their monodromies can be
effectively studied by the more sophisticated machineries of constructible
sheaves. Let $i\colon X_{0}=f^{-1}(0)\hookrightarrow X$ and $j\colon
X\setminus X_{0}\hookrightarrow X$ be the inclusion maps. For
${\mathbb{C}}^{*}:={\mathbb{C}}\setminus\\{0\\}$ we regard the exponential map

$\exp:\widetilde{{\mathbb{C}}^{*}}:={\mathbb{C}}\longrightarrow{\mathbb{C}}^{*},\qquad(t\longmapsto\exp(t))$
(2.6)

as its universal covering. Then for the fiber product $\widetilde{X\setminus
X_{0}}$ of
$\exp:\widetilde{{\mathbb{C}}^{*}}={\mathbb{C}}\longrightarrow{\mathbb{C}}^{*}$
and $f|_{X\setminus X_{0}}\colon X\setminus
X_{0}\longrightarrow{\mathbb{C}}^{*}$ we obtain the following commutative
diagram.

---
$\textstyle{\,\widetilde{X\setminus
X_{0}}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j\circ\pi}$$\scriptstyle{\pi}$$\scriptstyle{\square}$$\textstyle{\,\widetilde{{\mathbb{C}}^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}}$$\textstyle{{\,\simeq\,{\mathbb{C}}}}$$\textstyle{X_{0}=f^{-1}(0)\,\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\,X\,}$$\textstyle{\,X\setminus
X_{0}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{f|_{X\setminus
X_{0}}}$$\textstyle{\,{\mathbb{C}}^{*}.}$ (2.7)

Here $\Box$ stands for the Cartesian diagram.

######  Definition 2.4.

(Deligne [15]) For ${\cal F}\in{\bf D}^{b}(X)$ we define its nearby cycle
complex $\psi_{f}({\cal F})\in{\bf D}^{b}(X_{0})$ along $f$ by

$\psi_{f}({\cal F})=i^{-1}{\rm R}(j\circ\pi)_{*}(j\circ\pi)^{-1}{\cal F}\
\in{\bf D}^{b}(X_{0}).$ (2.8)

Moreover, we define the vanishing cycle complex $\phi_{f}({\cal F})\in{\bf
D}^{b}(X_{0})$ of ${\cal F}\in{\bf D}^{b}(X)$ along $f$ so that it fits into
the distinguished triangle

$i^{-1}{\cal F}\longrightarrow\psi_{f}({\cal F})\longrightarrow\phi_{f}({\cal
F})\overset{+1}{\longrightarrow},$ (2.9)

where the morphism $i^{-1}{\cal F}\longrightarrow\psi_{f}({\cal F})$ is
induced by the one ${\rm id}\longrightarrow{\rm
R}(j\circ\pi)_{*}(j\circ\pi)^{-1}$ of functors (for the more precise
definition, see Kashiwara-Schapira [51, page 351]).

We thus obtain two functors

$\psi_{f}(\cdot),\ \phi_{f}(\cdot)\colon{\bf D}^{b}(X)\longrightarrow{\bf
D}^{b}(X_{0}).$ (2.10)

We call them the nearby and vanishing cycle functors respectively. Note that
for the universal covering $\widetilde{{\mathbb{C}}^{*}}={\mathbb{C}}$ of
${\mathbb{C}}^{*}$ we have a deck transformation

$\widetilde{{\mathbb{C}}^{*}}={\mathbb{C}}\overset{\sim}{\longrightarrow}\widetilde{{\mathbb{C}}^{*}}={\mathbb{C}}\qquad(z\longmapsto
z-2\pi\sqrt{-1}).$ (2.11)

Then by our construction of the covering space $\pi\colon\widetilde{X\setminus
X_{0}}\longrightarrow X\setminus X_{0}$ of $X\setminus X_{0}$, it induces an
automorphism

$T\colon\widetilde{X\setminus
X_{0}}\overset{\sim}{\longrightarrow}\widetilde{X\setminus X_{0}}$ (2.12)

of $\widetilde{X\setminus X_{0}}$. By $\pi\circ T=\pi$ there exists a morphism
of functors

${\rm R}\pi_{*}\pi^{-1}\longrightarrow{\rm R}\pi_{*}({\rm
R}T_{*}T^{-1})\pi^{-1}={\rm R}\pi_{*}\pi^{-1}.$ (2.13)

Then we obtain isomorphisms

$\Psi_{f}({\cal F})\colon\psi_{f}({\cal
F})\overset{\sim}{\longrightarrow}\psi_{f}({\cal F}),\qquad\Phi_{f}({\cal
F})\colon\phi_{f}({\cal F})\overset{\sim}{\longrightarrow}\phi_{f}({\cal F}).$
(2.14)

We call them the monodromy automorphisms. Now we recall the following basic
result.

######  Proposition 2.5.

(see Kashiwara-Schapira [51, Exercise VIII.15] and Dimca [20, Proposition
4.2.11]) Let $\rho\colon Y\longrightarrow X$ be a proper morphism of complex
analytic spaces and $f\colon X\longrightarrow{\mathbb{C}}$ a non-constant
holomorphic function on $X$. We set $g:=f\circ\rho\colon
Y\longrightarrow{\mathbb{C}}$ and

$X_{0}=f^{-1}(0)\subset X,\quad Y_{0}=g^{-1}(0)=\rho^{-1}(X_{0})\subset Y.$
(2.15)

Let $\rho|_{Y_{0}}\colon Y_{0}\longrightarrow X_{0}$ be the restriction of
$\rho$ to $Y_{0}\subset Y$, Then for ${\cal G}\in{\bf D}^{b}(Y)$ there exist
isomorphisms

$\psi_{f}({\rm R}\rho_{*}{\cal G})\simeq{\rm
R}(\rho|_{Y_{0}})_{*}\psi_{g}({\cal G}),\qquad\phi_{f}({\rm R}\rho_{*}{\cal
G})\simeq{\rm R}(\rho|_{Y_{0}})_{*}\phi_{g}({\cal G}).$ (2.16)

By the following basic results we can study Milnor fibers and their
monodromies via nearby and vanishing cycle functors.

######  Theorem 2.6.

(see e.g. Dimca [20, Proposition 4.2.2]) For any point $x\in
X_{0}=f^{-1}(0)\subset X$ of $X_{0}$ there exist isomorphisms

$H^{j}(F_{x};{\mathbb{C}})\simeq
H^{j}\psi_{f}({\mathbb{C}}_{X})_{x},\quad\widetilde{H}^{j}(F_{x};{\mathbb{C}})\simeq
H^{j}\phi_{f}({\mathbb{C}}_{X})_{x}\qquad(j\in{\mathbb{Z}}),$ (2.17)

where $\widetilde{H}^{j}(F_{x};{\mathbb{C}})$ stand for the reduced cohomology
groups of $F_{x}$. Moreover these isomorphisms are compatible with the
automorphisms of both sides induced by the monodromies.

######  Proof.

The problem being local, we may assume that $X\subset{\mathbb{C}}^{N}$ and the
point $x$ is the origin $0\in{\mathbb{C}}^{N}$ of ${\mathbb{C}}^{N}$. Set
$Y:=X\times{\mathbb{C}}_{t}$ and let

$g:X\hookrightarrow Y=X\times{\mathbb{C}}_{t}\qquad(x\longmapsto(x,\,f(x)))$
(2.18)

be the graph embedding of $X$ by $f$. By the projection
$Y=X\times{\mathbb{C}}_{t}\longrightarrow X$ we identify the graph
$\Gamma_{f}:=g(X)\subset Y$ of $f$ with $X$. Then for $0<\eta\ll\varepsilon\ll
1$

$\Gamma_{f}\cap(B(0;\varepsilon)\times\\{t_{0}\\})\qquad(0<|t_{0}|<\eta)$
(2.19)

is identified with the Milnor fiber $F_{0}$ of $f$ at the origin $0\in
X_{0}=f^{-1}(0)$. On the other hand, by applying Proposition 2.5 to the
constructible sheaves ${\mathbb{C}}_{X}\in{\bf D}_{c}^{b}(X)$ and
$g_{*}{\mathbb{C}}_{X}={\mathbb{C}}_{\Gamma_{f}}\in{\bf D}_{c}^{b}(Y)$ and the
function $t\colon Y=X\times{\mathbb{C}}\longrightarrow{\mathbb{C}}$, we obtain
an isomorphism

$\psi_{t}({\mathbb{C}}_{\Gamma_{f}})_{(0,0)}\simeq\psi_{t\circ
g}({\mathbb{C}}_{X})_{0}=\psi_{f}({\mathbb{C}}_{X})_{0}.$ (2.20)

Hence it suffices to show that for $0<\eta\ll\varepsilon\ll 1$ we have
isomorphisms

$H^{j}\psi_{t}({\mathbb{C}}_{\Gamma_{f}})_{(0,0)}\simeq
H^{j}(B(0,\varepsilon)\times\\{t_{0}\\};{\mathbb{C}}_{\Gamma_{f}})\qquad(j\in{\mathbb{Z}},0<|t_{0}|<\eta).$
(2.21)

Note also that by Theorem 2.1 for $0<\eta\ll\varepsilon\ll 1$ there exist
isomorphisms

$H^{j}(\overline{B(0,\varepsilon)}\times\\{t_{0}\\};{\mathbb{C}}_{\Gamma_{f}})\simeq
H^{j}(B(0,\varepsilon)\times\\{t_{0}\\};{\mathbb{C}}_{\Gamma_{f}})\qquad(j\in{\mathbb{Z}},0<|t_{0}|<\eta).$
(2.22)

Restricting the exponential map
$\exp\colon\widetilde{{\mathbb{C}}^{*}}={\mathbb{C}}\longrightarrow{\mathbb{C}}^{*}$
to the punctured disk $B(0;\eta)^{*}\subset{\mathbb{C}}^{*}$ we obtain a
covering maps

$\pi_{0}\colon\widetilde{B(0;\eta)^{*}}:=\\{z\in{\mathbb{C}}\mid{\rm
Re}\,z<\log\eta\\}\longrightarrow B(0;\eta)^{*}$ (2.23)

and

$\pi={\rm
id_{\overline{B(0;\varepsilon)}}}\times\pi_{0}\colon\overline{B(0;\varepsilon)}\times\widetilde{B(0;\eta)^{*}}\longrightarrow\overline{B(0;\varepsilon)}\times
B(0;\eta)^{*}.$ (2.24)

Then by the definition of the nearby cycle sheaf
$\psi_{t}({\mathbb{C}}_{\Gamma_{f}})$ for $0<\eta\ll\varepsilon\ll 1$ we
obtain isomorphisms

$H^{j}\psi_{t}({\mathbb{C}}_{\Gamma_{f}})_{(0,0)}\simeq
H^{j}(\overline{B(0;\varepsilon)}\times\widetilde{B(0;\eta)^{*}};\pi^{-1}{\mathbb{C}}_{\Gamma_{f}})\qquad(j\in{\mathbb{Z}}).$
(2.25)

Let $p\colon\overline{B(0;\varepsilon)}\times B(0;\eta)^{*}\longrightarrow
B(0;\eta)^{*}$ and
$q\colon\overline{B(0;\varepsilon)}\times\widetilde{B(0;\eta)^{*}}\longrightarrow\widetilde{B(0;\eta)^{*}}$
be the projections. As they are proper, we obtain an isomorphism

${\rm R}q_{*}(\pi^{-1}{\mathbb{C}}_{\Gamma_{f}})\simeq\pi_{0}^{-1}{\rm
R}p_{*}({\mathbb{C}}_{\Gamma_{f}}).$ (2.26)

Moreover by Theorem 2.1 the cohomology sheaves

$H^{j}{\rm R}q_{*}(\pi^{-1}{\mathbb{C}}_{\Gamma_{f}})\simeq
H^{j}\pi_{0}^{-1}{\rm
R}p_{*}({\mathbb{C}}_{\Gamma_{f}})\qquad(j\in{\mathbb{Z}})$ (2.27)

on $\widetilde{B(0;\eta)^{*}}$ are locally constant. Since
$\widetilde{B(0;\eta)^{*}}$ is homeomorphic to the complex plane
${\mathbb{C}}$, they are globally constant. Take a point
$z_{0}\in\widetilde{B(0;\eta)^{*}}$ and set
$t_{0}:=\pi_{0}(z_{0})=\exp(z_{0})\in B(0;\eta)^{*}$. Then we obtain
isomorphisms

$\displaystyle{\rm R}\Gamma(\widetilde{B(0;\eta)^{*}};{\rm
R}q_{*}(\pi^{-1}{\mathbb{C}}_{\Gamma_{f}}))\simeq{\rm
R}q_{*}(\pi^{-1}{\mathbb{C}}_{\Gamma_{f}})_{z_{0}}$ (2.28)
$\displaystyle\simeq{\rm R}p_{*}({\mathbb{C}}_{\Gamma_{f}})_{t_{0}}\simeq{\rm
R}\Gamma(\overline{B(0;\varepsilon)}\times\\{t_{0}\\};{\mathbb{C}}_{\Gamma_{f}}).$
(2.29)

We thus obtain the isomorphisms in (2.21). Following the above argument more
carefully, we can also show the assertion on the monodromy of
$\psi_{f}({\mathbb{C}}_{X})$. The remaining assertion for
$\phi_{f}({\mathbb{C}}_{X})$ follows from the cone theorem proved in Milnor
[75, Theorem 2.10] and Burghelea-Verona [10, Lemma 3.2]. This completes the
proof.

Similarly, we can prove the following more general result.

######  Theorem 2.7.

In the situation of Theorem 2.6, for any constructible sheaf ${\cal F}\in{\bf
D}_{c}^{b}(X)$ on $X$ and any point $x\in X_{0}=f^{-1}(0)\subset X$ of $X_{0}$
there exist isomorphisms

$H^{j}(F_{x};{\cal F})\simeq H^{j}\psi_{f}({\cal
F})_{x}\qquad(j\in{\mathbb{Z}}).$ (2.30)

Since for a point $x\in X_{0}=f^{-1}(0)$ the Milnor fiber $F_{x}$ of $f$ at
$x$ is a subset of $X\setminus X_{0}$, we obtain the following very simple
consequence of Theorem 2.7.

######  Corollary 2.8.

In the situation of Theorem 2.7, let $j:X\setminus X_{0}\hookrightarrow X$ be
the inclusion map. Then for any constructible sheaf ${\cal F}\in{\bf
D}_{c}^{b}(X)$ on $X$ and any point $x\in X_{0}=f^{-1}(0)\subset X$ of $X_{0}$
there exist isomorphisms

$\displaystyle\psi_{f}(j_{!}j^{-1}{\cal F})_{x}$ $\displaystyle=\psi_{f}({\cal
F}_{X\setminus X_{0}})_{x}\simeq\psi_{f}({\cal F})_{x},$ (2.31)
$\displaystyle\psi_{f}({\rm R}j_{*}j^{-1}{\cal F})_{x}$
$\displaystyle=\psi_{f}({\rm R}\Gamma_{X\setminus X_{0}}{\cal
F})_{x}\simeq\psi_{f}({\cal F})_{x},$ (2.32)

######  Lemma 2.9.

For $1\leq k<l\leq n$ set
$\Omega:=({\mathbb{C}}^{*})^{l}\times{\mathbb{C}}^{n-l}\subset{\mathbb{C}}^{n}$
and let $j:\Omega\hookrightarrow{\mathbb{C}}^{n}$ be the inclusion map and
$h:{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ the function on
${\mathbb{C}}^{n}$ defined by $h(z)=z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots
z_{k}^{m_{k}}$ ($m_{i}\in{\mathbb{Z}}_{>0}$) for
$z=(z_{1},z_{2},\ldots,z_{n})\in{\mathbb{C}}^{n}$. Then for the constructible
sheaf $j_{!}j^{-1}{\mathbb{C}}_{{\mathbb{C}}^{n}}\simeq
j_{!}{\mathbb{C}}_{\Omega}\in{\bf D}_{c}^{b}({\mathbb{C}}^{n})$ we have
$\psi_{h}(j_{!}{\mathbb{C}}_{\Omega})_{0}\simeq 0$.

######  Proof.

Set $D:={\mathbb{C}}^{n}\setminus\Omega=\\{z\in{\mathbb{C}}^{n}\ |\
z_{1}z_{2}\cdots z_{l}=0\\}$ and let $i:D\hookrightarrow{\mathbb{C}}^{n}$ be
the inclusion map. Then there exists a distinguished triangle

$\psi_{h}(j_{!}{\mathbb{C}}_{\Omega})_{0}\longrightarrow\psi_{h}({\mathbb{C}}_{{\mathbb{C}}^{n}})_{0}\longrightarrow\psi_{h}(i_{!}{\mathbb{C}}_{D})_{0}\overset{+1}{\longrightarrow}$
(2.33)

in ${\bf D}_{c}^{b}(\\{0\\})$. Note that by the condition $l>k$ the Milnor
fiber $F_{0}$ of $h$ at the origin $0\in h^{-1}(0)\subset{\mathbb{C}}^{n}$ is
homotopic to $F_{0}\cap D$. Then by Theorem 2.6 we obtain an isomorphism

$\psi_{h}({\mathbb{C}}_{{\mathbb{C}}^{n}})_{0}\overset{\sim}{\longrightarrow}\psi_{h}(i_{!}{\mathbb{C}}_{D})_{0}.$
(2.34)

From this the assertion immediately follows.

Now we shall introduce some basic properties of the two functors

$\psi_{f}(\cdot),\ \phi_{f}(\cdot)\colon{\bf D}^{b}(X)\longrightarrow{\bf
D}^{b}(X_{0}).$ (2.35)

######  Theorem 2.10.

(see Kashiwara-Schapira [51, Proposition 8.6.3], Dimca [20, page 103] and
Schürmann [101, Theorem 4.0.2 and Lemma 4.2.1]) Assume that ${\cal F}\in{\bf
D}^{b}(X)$ is constructible. Then $\psi_{f}({\cal F}),\phi_{f}({\cal
F})\in{\bf D}^{b}(X_{0})$ are also constructible. Namely there exist two
functors

$\psi_{f}(\cdot),\ \phi_{f}(\cdot)\colon{\bf D}_{c}^{b}(X)\longrightarrow{\bf
D}_{c}^{b}(X_{0}).$ (2.36)

Shifting the functors $\psi_{f}(\cdot),\phi_{f}(\cdot)$ by $-1\in{\mathbb{Z}}$
we set

${}^{p}\\!\psi_{f}(\cdot):=\psi_{f}(\cdot)[-1],\quad^{p}\\!\phi_{f}(\cdot):=\phi_{f}(\cdot)[-1].$
(2.37)

Then we have the following deep result proved first by Kashiwara in [49] using
D-modules. See also Brylinski [7, Theorem 1.2 and Corollary 1.7] and Mebkhout-
Sabbah [74, Chapter III, Section 4] for the details. Later, purely geometric
proofs to it were given by Goresky-MacPherson [36], Kashiwara-Schapira [51,
Corollary 10.3.11] and Schürmann [101, Chapter 6].

######  Theorem 2.11.

In the situation of Theorem 2.6, assume that ${\cal F}\in{\bf D}_{c}^{b}(X)$
is a perverse sheaf on $X$. Then ${}^{p}\\!\psi_{f}({\cal F}),\
^{p}\\!\phi_{f}({\cal F})\in{\bf D}_{c}^{b}(X_{0})$ are also perverse sheaves
on $X_{0}=f^{-1}(0)$.

By Theorem 2.11, even if the complex hypersurface $X_{0}=f^{-1}(0)\subset
X={\mathbb{C}}^{n}$ for $f:X={\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ does
not have an isolated singular point at the origin $0\in X_{0}=f^{-1}(0)$, we
can prove various vanishing theorems for the cohomology groups
$H^{j}(F_{0};{\mathbb{C}})$ of Milnor fiber $F_{0}$ (see e.g. Dimca [20,
Section 6.1] for the details). Similarly, by using D-modules we obtain also
the following refinement of Theorem 2.11.

######  Theorem 2.12.

In the situation of 2.6, assume that ${\cal F}\in{\bf D}_{c}^{b}(X)$ is a
perverse sheaf on $X$. Then in the abelian category ${\rm
Perv}({\mathbb{C}}_{X_{0}})$ of perverse sheaves on $X_{0}$ we have the direct
sum decompositions:

${}^{p}\\!\psi_{f}({\cal F})\simeq\bigoplus_{\lambda\in{\mathbb{C}}}\
^{p}\\!\psi_{f,\lambda}({\cal F}),\qquad^{p}\\!\phi_{f}({\cal
F})\simeq\bigoplus_{\lambda\in{\mathbb{C}}}\ ^{p}\\!\phi_{f,\lambda}({\cal
F}),$ (2.38)

where the right hand sides are finite sums and for $\lambda\in{\mathbb{C}}$
taking a large enough integer $m\gg 0$ we set

${}^{p}\\!\psi_{f,\lambda}({\cal F}):=$ $\displaystyle{\rm
Ker}\Bigl{[}\bigl{(}\Psi_{f}({\cal F})-\lambda\cdot{\rm id}\bigr{)}^{m}:\
^{p}\\!\psi_{f}({\cal F})\longrightarrow\ ^{p}\\!\psi_{f}({\cal
F})\Bigr{]}\subset\ ^{p}\\!\psi_{f}({\cal F}),$ (2.39)
${}^{p}\\!\phi_{f,\lambda}({\cal F}):=$ $\displaystyle{\rm
Ker}\Bigl{[}\bigl{(}\Phi_{f}({\cal F})-\lambda\cdot{\rm id}\bigr{)}^{m}:\
^{p}\\!\phi_{f}({\cal F})\longrightarrow\ ^{p}\\!\phi_{f}({\cal
F})\Bigr{]}\subset\ ^{p}\\!\phi_{f}({\cal F}).$ (2.40)

######  Corollary 2.13.

In the situation of Theorem 2.6, assume moreover that $X={\mathbb{C}}^{n}$.
Then for any point $x\in X_{0}=f^{-1}(0)$ of $X_{0}$, $\lambda\in{\mathbb{C}}$
and $j\in{\mathbb{Z}}$ there exist isomorphisms

$\displaystyle H^{j}(F_{x};{\mathbb{C}})_{\lambda}\simeq$ $\displaystyle
H^{j-n+1}\ {}^{p}\\!\psi_{f,\lambda}({\mathbb{C}}_{X}[n])_{x},$ (2.41)
$\displaystyle\widetilde{H}^{j}(F_{x};{\mathbb{C}})_{\lambda}\simeq$
$\displaystyle H^{j-n+1}\ {}^{p}\\!\phi_{f,\lambda}({\mathbb{C}}_{X}[n])_{x},$
(2.42)

where $H^{j}(F_{x};{\mathbb{C}})_{\lambda}\subset H^{j}(F_{x};{\mathbb{C}})$
and
$\widetilde{H}^{j}(F_{x};{\mathbb{C}})_{\lambda}\subset\widetilde{H}^{j}(F_{x};{\mathbb{C}})$
are the generalized eigenspaces of the Milnor monodromies for the eigenvalue
$\lambda$.

As was first observed by A’Campo [1], Milnor fibers and their monodromies can
be precisely studied by resolutions of singularities of
$X_{0}=f^{-1}(0)\subset X$. Let us explain his brilliant idea in the more
general framework of constructible sheaves. For this purpose, we define
monodromy zeta functions of $f$ as follows. Let
${\mathbb{C}}(t)^{*}={\mathbb{C}}(t)\setminus\\{0\\}$ be the multiplicative
group of the function field ${\mathbb{C}}(t)$ of one variable $t$.

######  Definition 2.14.

For a point $x\in X_{0}=f^{-1}(0)$ of $X_{0}$ we define the monodromy zeta
function $\zeta_{f,x}(t)\in{\mathbb{C}}(t)^{*}$ of $f$ at $x$ by

$\zeta_{f,x}(t):=\prod_{j=0}^{\infty}\ \Bigl{\\{}\det({\rm
id}-t\cdot\Phi_{j,x})\Bigr{\\}}^{(-1)^{j}}\quad\in{\mathbb{C}}(t)^{*},$ (2.43)

where the ${\mathbb{C}}$-linear maps $\Phi_{j,x}\colon
H^{j}(F_{x};{\mathbb{C}})\overset{\sim}{\longrightarrow}H^{j}(F_{x};{\mathbb{C}})$
are the Milnor monodromies at $x$.

The following lemma was obtained by A’Campo [1] (see also Oka [85, Chapter I,
Example (3.7)] for a precise explanation).

######  Lemma 2.15.

For $1\leq k\leq n$ let $h:{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ be the
function on ${\mathbb{C}}^{n}$ defined by
$h(z)=z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots z_{k}^{m_{k}}$
($m_{i}\in{\mathbb{Z}}_{>0}$) for
$z=(z_{1},z_{2},\ldots,z_{n})\in{\mathbb{C}}^{n}$. Then we have

$\zeta_{h,0}(t)=\begin{cases}\ 1-t^{m_{1}}&(k=1)\\\ &\\\ \ 1&(k>1)\\\
\end{cases}$ (2.44)

Note that if the complex hypersurface $X_{0}=f^{-1}(0)\subset
X={\mathbb{C}}^{n}$ for $f:X={\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ has
an isolated singular point at the origin $0\in X_{0}=f^{-1}(0)$ by Theorem 2.3
we have

$\zeta_{f,x}(t)=(1-t)\cdot\Bigl{\\{}\det({\rm
id}-t\cdot\Phi_{n-1,x})\Bigr{\\}}^{(-1)^{n-1}}.$ (2.45)

Hence in this particular case, we can know the characteristic polynomial
$\det(t\cdot{\rm id}-\Phi_{n-1,x})\in{\mathbb{C}}[t]$ of the “principal”
Milnor monodromy

$\Phi_{n-1,x}\colon
H^{n-1}(F_{x};{\mathbb{C}})\overset{\sim}{\longrightarrow}H^{n-1}(F_{x};{\mathbb{C}})$
(2.46)

from the monodromy zeta function $\zeta_{f,x}(t)\in{\mathbb{C}}(t)^{*}$. This
classical notion of monodromy zeta functions can be generalized as follows.

######  Definition 2.16.

For a constructible sheaf ${\cal F}\in{\bf D}_{c}^{b}(X)$ on $X$ and a point
$x\in X_{0}=f^{-1}(0)$ of $X_{0}$ we define the monodromy zeta function
$\zeta_{f,x}({\cal F})\in{\mathbb{C}}(t)^{*}$ of ${\cal F}$ along $f$ at $x$
by

$\zeta_{f,x}({\cal F})(t):=\prod_{j\in{\mathbb{Z}}}\ \Bigl{\\{}\det\left({\rm
id}-t\Phi({\cal
F})_{j,x}\right)\Bigr{\\}}^{(-1)^{j}}\quad\in{\mathbb{C}}(t)^{*},$ (2.47)

where the ${\mathbb{C}}$-linear maps $\Phi({\cal F})_{j,x}\colon
H^{j}(\psi_{f}({\cal
F}))_{x}\overset{\sim}{\longrightarrow}H^{j}(\psi_{f}({\cal F}))_{x}$ are
induced by $\Phi({\cal F})$.

As in A’Campo [1] (see also Oka [85, Chapter I]) we can easily prove the
following very useful result.

######  Lemma 2.17.

Let ${\cal F}^{\prime}\longrightarrow{\cal F}\longrightarrow{\cal
F}^{\prime\prime}\overset{+1}{\longrightarrow}$ be a distinguished triangle in
${\bf D}_{c}^{b}(X)$. Then for any point $x\in X_{0}=f^{-1}(0)$ of $X_{0}$ we
have

$\zeta_{f,x}({\cal F})(t)=\zeta_{f,x}({\cal
F}^{\prime})(t)\cdot\zeta_{f,x}({\cal F}^{\prime\prime})(t).$ (2.48)

For an abelian group $G$ and a complex analytic space $Z$, we shall say that a
$G$-valued function $\varphi:Z\longrightarrow G$ on $Z$ is constructible if
there exists a stratification $Z=\bigsqcup_{\alpha}Z_{\alpha}$ of $Z$ such
that $\varphi|_{Z_{\alpha}}$ is constant for any $\alpha$. We denote by ${\rm
CF}_{G}(Z)$ the abelian group of $G$-valued constructible functions on $Z$. In
this note, we consider ${\rm CF}_{G}(Z)$ only for $G={\mathbb{C}}(t)^{*}$.
Then for the complex analytic space $X$ and the non-constant holomorphic
function $f:X\longrightarrow{\mathbb{C}}$ on it, by Theorem 2.12 we can easily
see that the ${\mathbb{C}}(t)^{*}$-valued function $\zeta_{f}({\cal
F}):X_{0}\longrightarrow{\mathbb{C}}(t)^{*}$ on $X_{0}=f^{-1}(0)$ defined by

$\zeta_{f}({\cal F})(x):=\zeta_{f,x}({\cal
F})(t)\quad\in{\mathbb{C}}(t)^{*}\qquad(x\in X_{0}=f^{-1}(0))$ (2.49)

is constructible. For a $G$-valued constructible function $\varphi\colon
Z\longrightarrow G$ on a complex analytic space $Z$, by taking a
stratification $Z=\bigsqcup_{\alpha}Z_{\alpha}$ of $Z$ such that
$\varphi|_{Z_{\alpha}}$ is constant for any $\alpha$, we set

$\int_{Z}\varphi:=\sum_{\alpha}\chi(Z_{\alpha})\cdot\varphi(z_{\alpha})\quad\in
G,$ (2.50)

where $\chi(\cdot)$ stands for the topological Euler characteristic and
$z_{\alpha}$ is a reference point in the stratum $Z_{\alpha}$. By the
following lemma $\int_{Z}\varphi\in G$ does not depend on the choice of the
stratification $Z=\bigsqcup_{\alpha}Z_{\alpha}$ of $Z$. We call it the
topological (or Euler) integral of $\varphi$ over $Z$.

######  Lemma 2.18.

Let $Y$ be a complex analytic space and $Y=\bigsqcup_{\alpha}Y_{\alpha}$ a
stratification of $Y$. Then we have

$\chi_{{\rm c}}(Y)=\displaystyle\sum_{\alpha}\chi_{{\rm c}}(Y_{\alpha}),$
(2.51)

where $\chi_{{\rm c}}(\cdot)$ stands for the Euler characteristic with compact
supports. Moreover, for any $\alpha$ we have $\chi_{{\rm
c}}(Y_{\alpha})=\chi(Y_{\alpha})$.

######  Proof.

Let $n:={\rm dim}Y$ and for $0\leq i\leq n$ set

$Y_{i}:=\bigsqcup_{\alpha:\ {\rm dim}Y_{\alpha}\leq i}Y_{\alpha}\quad\subset
Y.$ (2.52)

Then $Y_{i}$ are closed subsets of $Y$ and $Y_{n}=Y$. Applying the functor
${\rm R}\Gamma_{{\rm c}}(Y;\ \cdot\ )$ to the exact sequence

$0\longrightarrow{\mathbb{C}}_{Y\setminus
Y_{n-1}}\longrightarrow{\mathbb{C}}_{Y}\longrightarrow{\mathbb{C}}_{Y_{n-1}}\longrightarrow
0,$ (2.53)

we obtain an equality

$\chi_{{\rm c}}(Y)=\chi_{{\rm c}}(Y_{n-1})+\chi_{{\rm c}}(Y\setminus
Y_{n-1}).$ (2.54)

Note also that we have

$Y\setminus Y_{n-1}=\bigsqcup_{\alpha:\ {\rm dim}Y_{\alpha}=n}Y_{\alpha}$
(2.55)

and hence

$\chi_{{\rm c}}(Y\setminus Y_{n-1})=\displaystyle\sum_{\alpha:\ {\rm
dim}Y_{\alpha}=n}\chi_{{\rm c}}(Y_{\alpha}).$ (2.56)

Repeating this argument, we finally obtain the first assertion. Since for any
$\alpha$ the stratum $Y_{\alpha}$ is a complex manifold, by the (generalized)
Poincaré duality isomorphisms

$\bigl{[}H^{j}_{{\rm c}}(Y_{\alpha};{\mathbb{C}})\bigr{]}^{*}\simeq H^{2{\rm
dim}Y_{\alpha}-j}(Y_{\alpha};{\mathbb{C}})\qquad(0\leq j\leq 2{\rm
dim}Y_{\alpha})$ (2.57)

we obtain also the second assertion.

More generally, for any morphism $\rho\colon Z\longrightarrow W$ of complex
analytic spaces and any $G$-valued constructible function $\varphi\in{\rm
CF}_{G}(Z)$ on $Z$, we define the push-forward $\int_{\rho}\varphi\in{\rm
CF}_{G}(W)$ of $\varphi$ by

$\Bigl{(}\int_{\rho}\varphi\Bigr{)}(w):=\int_{\rho^{-1}(w)}\varphi\qquad(w\in
W).$ (2.58)

We thus obtain a homomorphism

$\int_{\rho}:{\rm CF}_{G}(Z)\longrightarrow{\rm CF}_{G}(W)$ (2.59)

of abelian groups. By Theorem 2.12 and Proposition 2.5, we can easily prove
the following very useful result.

######  Theorem 2.19.

(Dimca [20, Proposition 4.2.11] and Schürmann [101, Chapter 2]) In the
situation of Proposition 2.5, assume that ${\cal G}\in{\bf D}^{b}(Y)$ is
constructible. Then we have

$\int_{\rho|_{Y_{0}}}\zeta_{g}({\cal G})=\zeta_{f}({\rm R}\rho_{*}{\cal G})$
(2.60)

in ${\rm CF}_{{\mathbb{C}}(t)^{*}}(X_{0})$, where
$\int_{\rho|_{Y_{0}}}\colon{\rm
CF}_{{\mathbb{C}}(t)^{*}}(Y_{0})\longrightarrow{\rm
CF}_{{\mathbb{C}}(t)^{*}}(X_{0})$ is the push-forward of
${\mathbb{C}}(t)^{*}$-valued constructible functions by $\rho|_{Y_{0}}\colon
Y_{0}\longrightarrow X_{0}$.

For the precise proof of this theorem, see for example, [20, p.170-173] and
[101, Chapter 2].

######  Corollary 2.20.

In the situation of Theorem 2.19, for a point $x\in X_{0}=f^{-1}(0)$ of
$X_{0}$ let $\rho^{-1}(x)=\sqcup_{\alpha}Z_{\alpha}$ be a stratification of
$Z:=\rho^{-1}(x)\subset Y_{0}=g^{-1}(0)$ such that for any $\alpha$ the
${\mathbb{C}}(t)^{*}$-valued constructible function $\zeta_{g}({\cal
G})\in{\rm CF}_{{\mathbb{C}}(t)^{*}}(Y_{0})$ is constant on $Z_{\alpha}$. Then
we have

$\zeta_{f,x}({\rm R}\rho_{*}{\cal
G})(t)=\displaystyle\prod_{\alpha}\Bigl{\\{}\zeta_{g,y_{\alpha}}({\cal
G})(t)\Bigr{\\}}^{\chi(Z_{\alpha})},$ (2.61)

where $y_{\alpha}$ is a reference point of $Z_{\alpha}$.

In the special but important case where ${\cal G}={\mathbb{C}}_{Y}$ and the
restriction $Y\setminus Y_{0}\longrightarrow X\setminus X_{0}$ of $\rho$ is an
isomorphism, this result was first obtained by Gusein-Zade-Luengo-Melle-
Hernández [38, Corollary 1]. The following two lemmas are useful to calculate
monodromy zeta functions of constructible sheaves by Corollary 2.20. Indeed,
by resolutions of singularities, we can reduce the problem to the situations
treated in them. For their applications to the monodromies of
$A$-hypergeometric functions, see [108] and [3].

######  Lemma 2.21.

For $1\leq k\leq n$ let ${\mathcal{L}}$ be a ${\mathbb{C}}$-local system of
rank $r>0$ on
$({\mathbb{C}}^{*})^{k}\times{\mathbb{C}}^{n-k}\subset{\mathbb{C}}^{n}$. Let
$j:({\mathbb{C}}^{*})^{k}\times{\mathbb{C}}^{n-k}\hookrightarrow{\mathbb{C}}^{n}$
be the inclusion map and $h:{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ the
function on ${\mathbb{C}}^{n}$ defined by $h(z)=z_{1}^{m}$
($m\in{\mathbb{Z}}_{>0}$) for
$z=(z_{1},z_{2},\ldots,z_{n})\in{\mathbb{C}}^{n}$.

  1. (i)

Assume moreover that $k>1$. Then we have
$\psi_{h}(j_{!}{\mathcal{L}})_{0}\simeq 0$ and hence
$\zeta_{h,0}(j_{!}{\mathcal{L}})(t)=1$.

  2. (ii)

Assume moreover that $k=1$ and let $A\in{\rm GL}_{r}({\mathbb{C}})$ be the
monodromy matrix of ${\mathcal{L}}$ along the loop
$C:=\\{(e^{\sqrt{-1}\theta},0,\ldots,0)\ |\ 0\leq\theta\leq 2\pi\\}$ in
$({\mathbb{C}}^{*})^{k}\times{\mathbb{C}}^{n-k}={\mathbb{C}}^{*}\times{\mathbb{C}}^{n-1}$
(defined up to conjugacy). Then we have

$\zeta_{h,0}(j_{!}{\mathcal{L}})(t)={\rm det}\bigl{(}{\rm
id}-t^{m}A\bigr{)}\quad\in{\mathbb{C}}(t)^{*}.$ (2.62)

######  Proof.

By Theorem 2.6 the assertion of (i) immediately follows from [68, Proposition
5.2 (ii)] and the Künneth formula. We also obtain the one in (ii) by [68,
Proposition 5.2 (iii)].

The following result, which is a generalization of A’Campo’s lemma i.e. Lemma
2.15 (see also Oka [85, Example (3.7)]) to constructible sheaves, was proved
in [68, Proposition 5.3].

######  Lemma 2.22.

For $0\leq k\leq n$ let ${\mathcal{L}}$ be a ${\mathbb{C}}$-local system of
rank $r>0$ on
$({\mathbb{C}}^{*})^{k}\times{\mathbb{C}}^{n-k}\subset{\mathbb{C}}^{n}$. Let
$j:({\mathbb{C}}^{*})^{k}\times{\mathbb{C}}^{n-k}\hookrightarrow{\mathbb{C}}^{n}$
be the inclusion map and $h:{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ the
function on ${\mathbb{C}}^{n}$ defined by
$h(z)=z_{1}^{m_{1}}z_{2}^{m_{2}}\cdot z_{n}^{m_{n}}$
($m_{i}\in{\mathbb{Z}}_{\geq 0}$) for
$z=(z_{1},z_{2},\ldots,z_{n})\in{\mathbb{C}}^{n}$. Assume that $\sharp\\{1\leq
i\leq n\ |\ m_{i}>0\\}\geq 2$. Then we have
$\zeta_{h,0}(j_{!}{\mathcal{L}})(t)=1$.

##  3 Theorems of Kouchnirenko, Varchenko and Oka

In this section, we introduce the beautiful theorems of Kouchnirenko,
Varchenko and Oka which express the Milnor numbers and the monodromy zeta
functions of holomorphic functions in terms of their Newton polyhedra. Here,
for simplicity, we first consider non-constant polynomials
$f(x)\in{\mathbb{C}}[x]={\mathbb{C}}[x_{1},\ldots,x_{n}]$ on
${\mathbb{C}}^{n}$ such that $f(0)=0$. We set
${\mathbb{Z}}_{+}^{n}:=({\mathbb{Z}}_{\geq 0})^{n}\subset{\mathbb{Z}}^{n}$ and
${\mathbb{R}}_{+}^{n}:=({\mathbb{R}}_{\geq 0})^{n}\subset{\mathbb{R}}^{n}$. We
shall freely use standard notions in toric geomety, for which we refer to
Fulton [32] and Oda [83]. First of all, we recall the simplest case of the
Bernstein-Khovanskii-Kouchnirenko theorem [54].

######  Definition 3.1.

Let
$g(x)=\sum_{v\in{\mathbb{Z}}^{n}}c_{v}x^{v}\in{\mathbb{C}}[x_{1}^{\pm},\ldots,x_{n}^{\pm}]$
($c_{v}\in{\mathbb{C}}$) be a Laurent polynomial on the algebraic torus
$T=({\mathbb{C}}^{*})^{n}$.

  1. (i)

We call the convex hull of ${\rm supp}(g):=\\{v\in{\mathbb{Z}}^{n}\ |\
c_{v}\neq 0\\}\subset{\mathbb{Z}}^{n}\subset{\mathbb{R}}^{n}$ in
${\mathbb{R}}^{n}$ the Newton polytope of $g$ and denote it by $NP(g)$.

  2. (ii)

For a face $\gamma\prec NP(g)$ of $NP(g)$, we define the $\gamma$-part
$g^{\gamma}$ of $g$ by
$g^{\gamma}(x):=\sum_{v\in\gamma\cap{\mathbb{Z}}^{n}}c_{v}x^{v}$.

######  Definition 3.2.

(see [85]) Let $g(x)\in{\mathbb{C}}[x_{1}^{\pm},\ldots,x_{n}^{\pm}]$ be a
Laurent polynomial on $T=({\mathbb{C}}^{*})^{n}$ and set
$\Delta:=NP(g)\subset{\mathbb{R}}^{n}$. Then we say that $g$ is non-degenerate
if for any face $\gamma\prec\Delta$ of $\Delta$ the $1$-form $dg^{\gamma}$
does not vanish on the complex hypersurface $\\{x\in T=({\mathbb{C}}^{*})^{n}\
|\ g^{\gamma}(x)=0\\}$.

######  Theorem 3.3.

(Bernstein-Khovanskii-Kouchnirenko theorem [54], see also Oka [85, Chapter IV,
Theorem (3.1)]) Let $g(x)\in{\mathbb{C}}[x_{1}^{\pm},\ldots,x_{n}^{\pm}]$ be a
Laurent polynomial on $T=({\mathbb{C}}^{*})^{n}$ and set
$\Delta:=NP(g)\subset{\mathbb{R}}^{n}$. Assume that $g$ is non-degenerate.
Then for the complex hypersurface $Z^{*}=\\{x\in T=({\mathbb{C}}^{*})^{n}\ |\
g(x)=0\\}$ of $T=({\mathbb{C}}^{*})^{n}$ we have

$\chi(Z^{*})=(-1)^{n-1}{\rm Vol}_{{\mathbb{Z}}}(\Delta),$ (3.1)

where ${\rm Vol}_{{\mathbb{Z}}}(\Delta)\in{\mathbb{Z}}_{+}$ is the normalized
$n$-dimensional volume (i.e. the $n!$ times the usual volume) of $\Delta$ with
respect to the lattice ${\mathbb{Z}}^{n}\subset{\mathbb{R}}^{n}$.

######  Definition 3.4.

Let $f(x)=\sum_{v\in{\mathbb{Z}}_{+}^{n}}a_{v}x^{v}\in{\mathbb{C}}[x]$
($a_{v}\in{\mathbb{C}}$) be a non-constant polynomial on ${\mathbb{C}}^{n}$
such that $f(0)=0$.

  1. (i)

We define the support ${\rm supp}(f)\subset{\mathbb{Z}}_{+}^{n}$ of $f$ by

${\rm supp}(f):=\\{v\in{\mathbb{Z}}^{n}_{+}\ |\ a_{v}\neq 0\\}\
\subset{\mathbb{Z}}_{+}^{n}.$ (3.2)

  2. (ii)

We define the Newton polyhedron $\Gamma_{+}(f)\subset{\mathbb{R}}_{+}^{n}$ of
$f$ at the origin $0\in{\mathbb{C}}^{n}$ of ${\mathbb{C}}^{n}$ to be the
convex hull of $\cup_{v\in{\rm supp}(f)}(v+{\mathbb{R}}_{+}^{n})$ in
${\mathbb{R}}_{+}^{n}$.

  3. (iii)

For a compact face $\gamma\prec\Gamma_{+}(f)$ of $\Gamma_{+}(f)$, we define
the $\gamma$-part $f^{\gamma}\in{\mathbb{C}}[x]$ of $f$ by
$f^{\gamma}(x):=\sum_{v\in\gamma\cap{\mathbb{Z}}_{+}^{n}}a_{v}x^{v}$.

  4. (iv)

We say that $f$ is convenient if $\Gamma_{+}(f)$ intersects the positive part
of each coordinate axis of ${\mathbb{R}}^{n}$.

######  Definition 3.5.

(Kouchnirenko [56]) Let
$f(x)=\sum_{v\in{\mathbb{Z}}_{+}^{n}}a_{v}x^{v}\in{\mathbb{C}}[x]$
($a_{v}\in{\mathbb{C}}$) be a non-constant polynomial on ${\mathbb{C}}^{n}$
such that $f(0)=0$. Then we say that $f$ is non-degenate (at the origin
$0\in{\mathbb{C}}$) if for any compact face $\gamma\prec\Gamma_{+}(f)$ of
$\Gamma_{+}(f)$ the complex hypersurface

$\\{x=(x_{1},\ldots,x_{n})\in({\mathbb{C}}^{*})^{n}\ |\ f^{\gamma}(x)=0\\}\
\subset({\mathbb{C}}^{*})^{n}$ (3.3)

of $({\mathbb{C}}^{*})^{n}$ is smooth.

Note that among the polynomials having the same Newton polyhedron this non-
degeneracy condition is satisfied by generic ones. For a subset
${S}\subset\\{1,2,\ldots,n\\}$, let us set

${\mathbb{R}}^{{S}}:=\\{v=(v_{1},v_{2},\ldots,v_{n})\in{\mathbb{R}}^{n}\ |\
\text{$v_{i}=0$ for $i\notin{S}$}\\}\simeq{\mathbb{R}}^{\sharp S}.$ (3.4)

Denote by $({\mathbb{R}}^{{S}})^{*}\simeq{\mathbb{R}}^{\sharp S}$ the dual
vector space of ${\mathbb{R}}^{{S}}\simeq{\mathbb{R}}^{\sharp S}$. We set also
$\Gamma_{+}^{{S}}(f)=\Gamma_{+}(f)\cap{\mathbb{R}}^{{S}}$. Note that if $f$ is
convenient we have $\dim\Gamma_{+}^{{S}}(f)=\sharp{{S}}$ for any non-empty
subset ${{S}}\subset\\{1,2,\ldots,n\\}$. Now let
$f(x)=\sum_{v\in{\mathbb{Z}}_{+}^{n}}a_{v}x^{v}\in{\mathbb{C}}[x]$
($a_{v}\in{\mathbb{C}}$) be a non-constant polynomial on ${\mathbb{C}}^{n}$
such that $f(0)=0$. Denote by
$({\mathbb{R}}^{n})^{*}_{+}\simeq{\mathbb{R}}^{n}_{+}$ the dual cone of
${\mathbb{R}}_{+}^{n}\subset{\mathbb{R}}^{n}$ in $({\mathbb{R}}^{n})^{*}$. For
$u\in({\mathbb{R}}^{n})^{*}_{+}$ we define the supporting face
$\gamma^{u}\prec\Gamma_{+}(f)$ of $u$ in $\Gamma_{+}(f)$ by

$\gamma^{u}:=\left\\{v\in\Gamma_{+}(f)\ |\ \langle
u,v\rangle=\min_{w\in\Gamma_{+}(f)}\langle u,w\rangle\right\\}.$ (3.5)

Then for a face $\gamma\prec\Gamma_{+}(f)$ of $\Gamma_{+}(f)$ the (closed)
polyhedral cone

$\sigma(\gamma):=\overline{\\{u\in({\mathbb{R}}^{n})^{*}_{+}\ |\
\gamma^{u}=\gamma\\}}\quad\subset({\mathbb{R}}^{n})^{*}_{+}$ (3.6)

in $({\mathbb{R}}^{n})^{*}_{+}$ is rational and strongly convex. Moreover, we
can easily see that the family
$\Sigma_{1}:=\\{\sigma(\gamma)\\}_{\gamma\prec\Gamma_{+}(f)}$ of cones in
$({\mathbb{R}}^{n})^{*}_{+}$ thus obtained satisfies the axioms of fans. We
call it the dual fan of $\Gamma_{+}(f)$ in $({\mathbb{R}}^{n})^{*}_{+}$.
Conversely, for a rational and strongly convex polyhedral cone
$\sigma\subset({\mathbb{R}}^{n})^{*}_{+}$ we set

$\gamma(\sigma):=\bigcap_{u\in\sigma}\gamma^{u}\quad\prec\Gamma_{+}(f).$ (3.7)

We call it the supporting face of $\sigma$ in $\Gamma_{+}(f)$. Then it is easy
to see that for any $u\in({\mathbb{R}}^{n})^{*}_{+}$ in the relative interior
${\rm rel.int}(\sigma)$ of $\sigma$ we have $\gamma(\sigma)=\gamma^{u}$. For
each non-empty subset ${{S}}\subset\\{1,2,\ldots,n\\}$, let
$n({{S}})\in{\mathbb{Z}}_{+}$ be the number of the
$(\sharp{{S}}-1)$-dimensional compact faces i.e. the compact facets of
$\Gamma_{+}^{{S}}(f)$ and
$\gamma_{1}^{{S}},\gamma_{2}^{{S}},\ldots,\gamma_{n({{S}})}^{{S}}$ be such
faces. For $1\leq i\leq n({S})$, let
$u_{i}^{{S}}\in({\mathbb{R}}^{{S}})^{*}\cap{\mathbb{Z}}_{+}^{{S}}$ be the
unique non-zero primitive vector which takes its minimum in
$\Gamma_{+}^{{S}}(f)$ on the whole $\gamma_{i}^{{S}}$ and set

$d_{i}^{{S}}:=\min_{v\in\Gamma_{+}^{{S}}(f)}\langle u_{i}^{{S}},v\rangle\
\in{\mathbb{Z}}_{>0}.$ (3.8)

We call $d_{i}^{{S}}$ the lattice distance of $\gamma_{i}^{{S}}$ from the
origin $0\in{\mathbb{R}}^{{S}}$. For each compact facet
$\gamma_{i}^{{S}}\prec\Gamma_{+}^{{S}}(f)$, let
${\mathbb{L}}(\gamma_{i}^{{S}})$ be the smallest affine linear subspace of
${\mathbb{R}}^{n}$ containing $\gamma_{i}^{{S}}$ and ${\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})\in{\mathbb{Z}}_{>0}$ the normalized
$(\sharp{S}-1)$-dimensional volume (i.e. the $(\sharp S-1)!$ times the usual
volume) of $\gamma_{i}^{{S}}$ with respect to the lattice
${\mathbb{Z}}^{n}\cap{\mathbb{L}}(\gamma_{i}^{{S}})$. Then we have the
following celebrated theorem of Varchenko [116].

######  Theorem 3.6.

(Varchenko [116, Theorem 4.1]) Assume that $f$ is non-degenerate at the origin
$0\in{\mathbb{C}}^{n}$. Then we have

$\zeta_{f,0}(t)=\prod_{{{S}}\neq\emptyset}\ \zeta_{f,0}^{S}(t),$ (3.9)

where for each non-empty subset ${S}\subset\\{1,2,\ldots,n\\}$ we set

$\zeta_{f,0}^{S}(t):=\prod_{i=1}^{n({{S}})}\
(1-t^{d_{i}^{{S}}})^{(-1)^{\sharp{{S}}-1}{\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})}.$ (3.10)

We shall give a short proof to Theorem 3.6 along the same line as in [68,
Section 4].

######  Proof.

For a subset ${S}\subset\\{1,2,\ldots,n\\}$, let us set

$T_{{S}}:=\\{x=(x_{1},x_{2},\ldots,x_{n})\in{\mathbb{C}}^{n}\ |\ x_{i}=0\
(i\notin{S}),\ x_{i}\not=0\ (i\in{S})\\}\subset{\mathbb{C}}^{n}.$ (3.11)

Then we have $T_{S}\simeq({\mathbb{C}}^{*})^{\sharp S}$. Moreover for the
closure $\overline{T_{S}}$ of $T_{S}$ in ${\mathbb{C}}^{n}$ we have
$\overline{T_{S}}\simeq{\mathbb{C}}^{\sharp S}$. Let
$i_{S}:\overline{T_{S}}\hookrightarrow{\mathbb{C}}^{n}$ be the inclusion map.
Then by Lemma 2.17 and the decomposition
${\mathbb{C}}^{n}\setminus\\{0\\}=\sqcup_{{S}\not=\emptyset}T_{S}$ of
${\mathbb{C}}^{n}\setminus\\{0\\}$ we obtain

$\displaystyle\zeta_{f,0}(t)$
$\displaystyle=\zeta_{f,0}({\mathbb{C}}_{{\mathbb{C}}^{n}})(t)=\zeta_{f,0}({\mathbb{C}}_{{\mathbb{C}}^{n}\setminus\\{0\\}})(t)$
(3.12) $\displaystyle=\prod_{S\not=\emptyset}\
\zeta_{f,0}({\mathbb{C}}_{T_{S}})(t)$ (3.13)
$\displaystyle=\prod_{S\not=\emptyset}\
\zeta_{f,0}(i_{S*}{\mathbb{C}}_{T_{S}})(t)$ (3.14)
$\displaystyle=\prod_{S\not=\emptyset}\ \zeta_{f\circ
i_{S},0}({\mathbb{C}}_{T_{S}})(t),$ (3.15)

where we used also Corollary 2.8 (resp. Theorem 2.19) in the second (resp.
last) equality. Hence it suffices to prove that for any non-empty subset
${S}\subset\\{1,2,\ldots,n\\}$ we have

$\zeta_{f\circ
i_{S},0}({\mathbb{C}}_{T_{S}})(t)=\zeta_{f,0}^{S}(t)=\prod_{i=1}^{n({{S}})}\
(1-t^{d_{i}^{{S}}})^{(-1)^{\sharp{{S}}-1}{\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})}.$ (3.16)

Here we shall prove it only for ${S}=\\{1,2,\ldots,n\\}$. The proof for the
other cases is similar. For ${S}=\\{1,2,\ldots,n\\}$ we set

$T:=T_{S}=T_{\\{1,2,\ldots,n\\}}=({\mathbb{C}}^{*})^{n}\subset{\mathbb{C}}^{n}.$
(3.17)

Let us consider ${\mathbb{C}}^{n}$ as a toric variety associated to the fan
$\Sigma_{0}$ in $({\mathbb{R}}^{n})^{*}$ formed by the all faces of the first
quadrant $({\mathbb{R}}^{n})^{*}_{+}\subset({\mathbb{R}}^{n})^{*}$. Then we
regard $T=({\mathbb{C}}^{*})^{n}$ as the open dense torus in it. Let $\Sigma$
be a smooth subdivision of $\Sigma_{0}$ and the dual fan $\Sigma_{1}$ of
$\Gamma^{S}_{+}(f)=\Gamma_{+}(f)$ in $({\mathbb{R}}^{n})^{*}_{+}$. Let
$X_{\Sigma}$ be the smooth toric variety associated to $\Sigma$ and
$T^{\prime}\subset X_{\Sigma}$ the open dense torus in it. Since $\Sigma$ is a
subdivision of $\Sigma_{0}$, there exists a proper morphism
$\pi:X_{\Sigma}\longrightarrow{\mathbb{C}}^{n}$ of toric varieties. We can
easily see that $\pi$ induces an isomorphism
$T^{\prime}\overset{\sim}{\longrightarrow}T$. So in what follows, we identify
$T^{\prime}$ with $T$ and denote it simply by $T$. Recall that
$T^{\prime}\simeq T$ acts on $X_{\Sigma}$ and the $T$-orbits are parametrized
by the cones in $\Sigma$. For a cone $\sigma\in\Sigma$ denote by
$T_{\sigma}\simeq({\mathbb{C}}^{*})^{n-\dim\sigma}$ the corresponding
$T$-orbit. We have also natural affine open subsets
${\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}$ of $X_{\Sigma}$ associated to
$n$-dimensional cones $\sigma$ in $\Sigma$. Now we apply Theorem 2.19 to the
proper morphism $\pi:X_{\Sigma}\longrightarrow{\mathbb{C}}^{n}$, the function
$f\circ i_{S}=f:{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ and the
constructible sheaf ${\mathbb{C}}_{T}\in{\bf D}_{c}^{b}(X_{\Sigma})$. Then for
$S=\\{1,2,\ldots,n\\}$ we obtain an equality

$\zeta_{f\circ
i_{S},0}({\mathbb{C}}_{T_{S}})=\zeta_{f}({\mathbb{C}}_{T})(0)=\zeta_{f}({\rm
R}\pi_{*}{\mathbb{C}}_{T})(0)=\int_{\pi^{-1}(0)}\zeta_{f\circ\pi}({\mathbb{C}}_{T})$
(3.18)

in ${\mathbb{C}}(t)^{*}$. Hence for the calculation of $\zeta_{f\circ
i_{S},0}({\mathbb{C}}_{T_{S}})\in{\mathbb{C}}(t)^{*}$ it suffices to calculate
the value of the constructible function $\zeta_{f\circ\pi}({\mathbb{C}}_{T})$
at each point of $\pi^{-1}(0)\subset X_{\Sigma}$. Note that
$\pi^{-1}(0)\subset X_{\Sigma}$ is the union of the $T$-orbits $T_{\tau}$ in
$X_{\Sigma}$ associated to the cones $\tau\in\Sigma$ such that ${\rm
rel.int}(\tau)\subset{\rm
Int}({\mathbb{R}}^{n})^{*}_{+}\simeq({\mathbb{R}}_{>0})^{n}$. Let
$\tau\in\Sigma$ be a cone such that ${\rm rel.int}(\tau)\subset{\rm
Int}({\mathbb{R}}^{n})^{*}_{+}$ and $\sigma\in\Sigma$ an $n$-dimensional cone
such that $\tau\prec\sigma$. Let
$w_{1},w_{2},\ldots,w_{n}\in{\mathbb{Z}}_{+}^{n}\setminus\\{0\\}$ be the
primitive vectors on the edges of the simplicial cone $\sigma$. Then there
exists an affine open subset ${\mathbb{C}}^{n}(\sigma)$ of $X_{\Sigma}$ such
that ${\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}_{y}$ and $f\circ\pi$ has
the following form on it:

$(f\circ\pi)(y)=\sum_{v\in{\mathbb{Z}}_{+}^{n}}a_{v}y_{1}^{\langle
w_{1},v\rangle}\cdots y_{n}^{\langle w_{n},v\rangle}=y_{1}^{b_{1}}\cdots
y_{n}^{b_{n}}\cdot f_{\sigma}(y),$ (3.19)

where we set

$b_{i}=\min_{v\in\Gamma_{+}(f)}\langle w_{i},v\rangle\geq
0\qquad(i=1,2,\ldots,n)$ (3.20)

and $f_{\sigma}(y)$ is a polynomial on
${\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}_{y}$. Set $d:={\rm dim}\tau$
and for simplicity, assume that $w_{1},\ldots,w_{d}$ generate the simplicial
cone $\tau$. Then in the affine chart
${\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}_{y}$ the $T$-orbit $T_{\tau}$
associated to $\tau$ is explicitly defined by

$T_{\tau}=\\{(y_{1},\ldots,y_{n})\in{\mathbb{C}}^{n}(\sigma)\ |\
y_{1}=\cdots=y_{d}=0,\ y_{d+1},\ldots,y_{n}\neq
0\\}\simeq({\mathbb{C}}^{*})^{n-d}.$ (3.21)

By the condition ${\rm rel.int}(\tau)\subset{\rm
Int}({\mathbb{R}}^{n})^{*}_{+}$ the supporting face
$\gamma(\tau)\prec\Gamma_{+}(f)$ of $\tau$ in $\Gamma_{+}(f)$ being compact,
it follows form the non-degeneracy of $f$ that the complex hypersurface
$\\{x\in({\mathbb{C}}^{*})^{n}\ |\
f^{\gamma(\tau)}(x)=0\\}\subset({\mathbb{C}}^{*})^{n}$ is smooth. As the
restriction of $f_{\sigma}$ to $T_{\tau}\subset{\mathbb{C}}^{n}(\sigma)$ is
naturally identified with $f^{\gamma(\tau)}$, this implies that the
hypersurface
$f_{\sigma}^{-1}(0)\subset{\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}_{y}$
intersects $T_{\tau}$ transversally. Since we have
$T_{\tau}\subset\pi^{-1}(0)\subset(f\circ\pi)^{-1}(0)$ and hence the function
$f\circ\pi$ is zero on $T_{\tau}$, we have $b_{i}>0$ for some $1\leq i\leq d$.
Then by Lemmas 2.9 and 2.15, if $d\geq 2$ for any point $y\in T_{\tau}$ we
have
$\zeta_{f\circ\pi}({\mathbb{C}}_{T})(y)=\zeta_{f\circ\pi,y}({\mathbb{C}}_{T})(t)=1$.
By Lemma 2.15, if $d=1$ for any point $y\in T_{\tau}$ we have

$\zeta_{f\circ\pi}({\mathbb{C}}_{T})(y)=\zeta_{f\circ\pi,y}({\mathbb{C}}_{T})(t)=\begin{cases}1-t^{b_{1}}&(y\in
T_{\tau}\setminus f_{\sigma}^{-1}(0)),\\\ &\\\ 1&(y\in T_{\tau}\cap
f_{\sigma}^{-1}(0)).\end{cases}$ (3.22)

Moreover, if $d=1$ and the dimension of the supporting face
$\gamma(\tau)\prec\Gamma_{+}(f)$ of $\tau$ in $\Gamma_{+}(f)$ is less than
$n-{\rm dim}\tau=n-1$, then we have $T_{\tau}\cap f_{\sigma}^{-1}(0)\simeq
A\times({\mathbb{C}}^{*})^{n-1-{\rm dim}\gamma(\tau)}$ for some set $A$ and
hence we get

$\chi(T_{\tau}\setminus f_{\sigma}^{-1}(0))=\chi(T_{\tau})-\chi(T_{\tau}\cap
f_{\sigma}^{-1}(0))=0-0=0.$ (3.23)

This implies that $\int_{T_{\tau}}\zeta_{f\circ\pi}({\mathbb{C}}_{T})$ is not
trivial only in the case where $\gamma(\tau)\prec\Gamma_{+}(f)$ is equal to
the facet $\gamma_{i}^{S}\prec\Gamma_{+}(f)$ for some $1\leq i\leq n(S)$. In
such a case, we have $d=1$, $b_{1}=d_{i}^{S}>0$ and moreover

$\chi(T_{\tau}\setminus f_{\sigma}^{-1}(0))=0-(-1)^{n-2}\ {\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})=(-1)^{\sharp S-1}\ {\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})$ (3.24)

by Theorem 3.3. Then we obtain the desired equality

$\int_{\pi^{-1}(0)}\zeta_{f\circ\pi}({\mathbb{C}}_{T})=\prod_{i=1}^{n({{S}})}\
(1-t^{d_{i}^{{S}}})^{(-1)^{\sharp{{S}}-1}{\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})}.$ (3.25)

This completes the proof.

The following lemma is well-known to the specialists. For the convenience of
the readers, here we give a proof to it.

######  Lemma 3.7.

Assume that $f$ is non-degenerate at the origin $0\in{\mathbb{C}}^{n}$ and
convenient. Then the complex hypersurface $f^{-1}(0)\subset{\mathbb{C}}^{n}$
is smooth or has an isolated singular point at the origin
$0\in{\mathbb{C}}^{n}$.

######  Proof.

Let us consider ${\mathbb{C}}^{n}$ as the toric variety associated to the fan
$\Sigma_{0}$ formed by the all faces of the first quadrant
$({\mathbb{R}}^{n})^{*}_{+}\subset({\mathbb{R}}^{n})^{*}$. Recall that we
denote by $\Sigma_{1}$ the dual fan of $\Gamma_{+}(f)$ in
$({\mathbb{R}}^{n})^{*}_{+}$. Then by our assumptions, we can easily see that
any cone $\sigma\in\Sigma_{0}$ in $\Sigma_{0}$ such that
$\sigma\subset\partial({\mathbb{R}}^{n})^{*}_{+}\simeq({\mathbb{R}}^{n}_{+}\setminus{\rm
Int}({\mathbb{R}}^{n}_{+}))$ is an element of $\Sigma_{1}$. Hence we can
construct a smooth subdivision $\Sigma$ of $\Sigma_{1}$ without subdividing
the cones $\sigma\in\Sigma_{1}$ such that
$\sigma\subset\partial({\mathbb{R}}^{n})^{*}_{+}$ (see e.g. [85, Chapter II,
Lemma (2.6)]). Let $X_{\Sigma}$ be the smooth toric variety associated to
$\Sigma$. Then there exists a morphism
$\pi_{0}:X_{\Sigma}\rightarrow{\mathbb{C}}^{n}$ of toric varieties inducing an
isomorphism
$X_{\Sigma}\setminus\pi_{0}^{-1}(\\{0\\})\overset{\sim}{\longrightarrow}{\mathbb{C}}^{n}\setminus\\{0\\}$.
Let $\rho_{1},\rho_{2},\ldots,\rho_{m}\in\Sigma$ be the $1$-dimensional cones
in $\Sigma$ not contained in $\partial({\mathbb{R}}^{n})^{*}_{+}$. For each
$\rho_{i}$ denote by $T_{i}\simeq({\mathbb{C}}^{*})^{n-1}$ the $T$-orbit in
$X_{\Sigma}$ and set $D_{i}:=\overline{T_{i}}\subset X_{\Sigma}$. Then $D_{i}$
are smooth and we have $\pi_{0}^{-1}(\\{0\\})=D_{1}\cup D_{2}\cup\cdots\cup
D_{m}$. Moreover by the non-degeneracy of $f$ at $0\in{\mathbb{C}}^{n}$, for
any non-empty subset $I\subset\\{1,2,\ldots,m\\}$ the smooth subvariety
$D_{I}:=\cap_{i\in I}D_{i}$ of $X_{\Sigma}$ intersects the proper transform of
$f^{-1}(0)\subset{\mathbb{C}}^{n}$ in $X_{\Sigma}$ transversally. This in
particular implies that the proper transform itself is smooth. Then the
assertion is very clear.

For a non-empty subset ${{S}}\subset\\{1,2,\ldots,n\\}$ we set

$E_{S}:=\displaystyle\sum_{i=1}^{n(S)}d_{i}^{S}\cdot{\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})\quad\geq 0.$ (3.26)

Moreover for the empty set $S=\emptyset$ we set $E_{\emptyset}:=1$. Then by
Theorem 3.6 we obtain the following fundamental theorem of Kouchnirenko [56].

######  Theorem 3.8.

(Kouchnirenko [56, Theorem 1.10]) Assume that $f$ is non-degenerate at the
origin $0\in{\mathbb{C}}^{n}$. Then we have

$\chi(F_{0})=\displaystyle\sum_{{{S}}\neq\emptyset}(-1)^{\sharp{{S}}-1}E_{S}$
(3.27)

In particular, if moreover $f$ is convenient and $f^{-1}(0)$ is singular at
$0\in{\mathbb{C}}^{n}$, then $f^{-1}(0)$ has an isolated singular point at
$0\in{\mathbb{C}}^{n}$ and the Milnor number $\mu>0$ there is given by

$\mu=\displaystyle\sum_{{{S}}\subset\\{1,2,\ldots,n\\}}(-1)^{n-\sharp{{S}}}E_{S}\quad=E_{\\{1,2,\ldots,n\\}}+\cdots\cdots+(-1)^{n}>0.$
(3.28)

From now on, we shall introduce Oka’s theorem proved in [84] and [85] (see
also Kirillov [55]). As we will see below, it is a wide generalization of
Theorem 3.6. Since some technical assumptions imposed in Oka’s theorem in [84]
and [85] were removed later in Matsui-Takeuchi [68, Theorem 3.12], here we
present it in its final form in [68, Theorem 3.12]. Note that in [68, Theorem
3.12] more general results on the Milnor monodromies of regular functions on
complete intersection subvarieties of (possibly non-normal) affine toric
varieties were obtained. In [70], they were used to obtain formulas for the
degrees and the dimensions of the $A$-discriminant varieties introduced by
Gelfand-Kapranov-Zelevinsky [34]. However, for the sake of simplicity, here we
present only the results in the simplest case of regular functions on complete
intersection subvarieties of ${\mathbb{C}}^{n}$. First, let us recall the most
general form of the Bernstein-Khovanskii-Kouchnirenko theorem [54]. Let
$({\mathbb{R}}^{n})^{*}\simeq{\mathbb{R}}^{n}$ be the dual vector space of
${\mathbb{R}}^{n}$ and

$\langle\cdot,\cdot\rangle:({\mathbb{R}}^{n})^{*}\times{\mathbb{R}}^{n}\longrightarrow{\mathbb{R}},\quad((u,v)\longmapsto\langle
u,v\rangle:=\sum_{i=1}^{n}u_{i}v_{i})$ (3.29)

the canonical pairing. Let $\Delta\subset{\mathbb{R}}^{n}$ be a lattice
polytope in ${\mathbb{R}}^{n}$. For
$u\in({\mathbb{R}}^{n})^{*}\simeq{\mathbb{R}}^{n}$ we define the supporting
face $\gamma^{u}\prec\Delta$ of $u$ in $\Delta$ by

$\gamma^{u}=\left\\{v\in\Delta\ |\ \langle u,v\rangle=\min_{w\in\Delta}\langle
u,w\rangle\right\\}.$ (3.30)

For a face $\gamma$ of $\Delta$ set

$\sigma(\gamma):=\overline{\\{u\in({\mathbb{R}}^{n})^{*}\ |\
\gamma^{u}=\gamma\\}}\subset({\mathbb{R}}^{n})^{*}.$ (3.31)

Then $\sigma(\gamma)$ is an $(n-\dim\gamma)$-dimensional rational convex
polyhedral cone in $({\mathbb{R}}^{n})^{*}$. Moreover the family
$\\{\sigma(\gamma)\ |\ \gamma\prec\Delta\\}$ of cones in
$({\mathbb{R}}^{n})^{*}$ thus obtained is a subdivision of
$({\mathbb{R}}^{n})^{*}$. We call it the dual subdivision of
$({\mathbb{R}}^{n})^{*}$ by $\Delta$. If $\dim\Delta=n$, the cones
$\sigma(\gamma)$ are strongly convex and it satisfies the axiom of fans. Then
we call it the dual fan of $\Delta$. More generally, let
$\Delta_{1},\ldots,\Delta_{p}\subset{\mathbb{R}}^{n}$ be lattice polytopes in
${\mathbb{R}}^{n}$ and
$\Delta=\Delta_{1}+\cdots+\Delta_{p}\subset{\mathbb{R}}^{n}$ their Minkowski
sum. Then for a face $\gamma\prec\Delta$ of $\Delta$, by taking a point
$u\in({\mathbb{R}}^{n})^{*}$ in the relative interior of its dual cone
$\sigma(\gamma)$ we define the supporting face $\gamma_{i}\prec\Delta_{i}$ of
$u$ in $\Delta_{i}$ so that we have $\gamma=\gamma_{1}+\cdots+\gamma_{p}$.

######  Definition 3.9.

(see [85]) Let $g_{1},g_{2},\ldots,g_{p}$ be Laurent polynomials on
$T=({\mathbb{C}}^{*})^{n}$. Set $\Delta_{i}=NP(g_{i})$ $(i=1,\ldots,p)$ and
$\Delta=\Delta_{1}+\cdots+\Delta_{p}$. Then we say that the subvariety
$Z^{*}=\\{x\in T=({\mathbb{C}}^{*})^{n}\ |\
g_{1}(x)=g_{2}(x)=\cdots=g_{p}(x)=0\\}$ of $T=({\mathbb{C}}^{*})^{n}$ is a
non-degenerate complete intersection if for any face $\gamma\prec\Delta$ of
$\Delta$ the $p$-form $dg_{1}^{\gamma_{1}}\wedge
dg_{2}^{\gamma_{2}}\wedge\cdots\wedge dg_{p}^{\gamma_{p}}$ does not vanish on
$\\{x\in T=({\mathbb{C}}^{*})^{n}\ |\
g_{1}^{\gamma_{1}}(x)=\cdots=g_{p}^{\gamma_{p}}(x)=0\\}$.

######  Definition 3.10.

Let $\Delta_{1},\ldots,\Delta_{n}$ be lattice polytopes in ${\mathbb{R}}^{n}$.
Then their normalized $n$-dimensional mixed volume ${\rm
Vol}_{{\mathbb{Z}}}(\Delta_{1},\ldots,\Delta_{n})\in{\mathbb{Z}}$ is defined
by the formula

${\rm
Vol}_{{\mathbb{Z}}}(\Delta_{1},\ldots,\Delta_{n})=\frac{1}{n!}\displaystyle\sum_{k=1}^{n}(-1)^{n-k}\sum_{\begin{subarray}{c}I\subset\\{1,\ldots,n\\}\\\
|I|=k\end{subarray}}{\rm Vol}_{{\mathbb{Z}}}\left(\displaystyle\sum_{i\in
I}\Delta_{i}\right)$ (3.32)

where ${\rm Vol}_{{\mathbb{Z}}}(\cdot)=n!\cdot{\rm
Vol}(\cdot)\in{\mathbb{Z}}_{+}$ is the normalized $n$-dimensional volume with
respect to the lattice ${\mathbb{Z}}^{n}\subset{\mathbb{R}}^{n}$.

######  Theorem 3.11.

(Bernstein-Khovanskii-Kouchnirenko theorem [54], see also Oka [85, Chapter IV,
Theorem (3.1)]) Let $g_{1},g_{2},\ldots,g_{p}$ be Laurent polynomials on
$T=({\mathbb{C}}^{*})^{n}$. Assume that the subvariety $Z^{*}=\\{x\in
T=({\mathbb{C}}^{*})^{n}\ |\ g_{1}(x)=g_{2}(x)=\cdots=g_{p}(x)=0\\}$ of
$T=({\mathbb{C}}^{*})^{n}$ is a non-degenerate complete intersection. Set
$\Delta_{i}=NP(g_{i})$ $(i=1,\ldots,p)$. Then we have

$\chi(Z^{*})=(-1)^{n-p}\displaystyle\sum_{\begin{subarray}{c}m_{1},\ldots,m_{p}\geq
1\\\ m_{1}+\cdots+m_{p}=n\end{subarray}}{\rm
Vol}_{{\mathbb{Z}}}(\underbrace{\Delta_{1},\ldots,\Delta_{1}}_{\text{
$m_{1}$-times}},\ldots,\underbrace{\Delta_{p},\ldots,\Delta_{p}}_{\text{$m_{p}$-times}}),$
(3.33)

where ${\rm
Vol}_{{\mathbb{Z}}}(\underbrace{\Delta_{1},\ldots,\Delta_{1}}_{\text{$m_{1}$-times}},\ldots,\underbrace{\Delta_{p},\ldots,\Delta_{p}}_{\text{$m_{p}$-times}})\in{\mathbb{Z}}_{+}$
is the normalized $n$-dimensional mixed volume with respect to the lattice
${\mathbb{Z}}^{n}\subset{\mathbb{R}}^{n}$.

Now let
$f_{1},f_{2},\ldots,f_{k}\in{\mathbb{C}}[x]={\mathbb{C}}[x_{1},x_{2},\ldots,x_{n}]$
be polynomials on ${\mathbb{C}}^{n}$ and consider the following two
subvarieties of ${\mathbb{C}}^{n}$:

$V:=\\{f_{1}=f_{2}=\cdots=f_{k}=0\\}\ \subset
W:=\\{f_{1}=f_{2}=\cdots=f_{k-1}=0\\}.$ (3.34)

Assume that $V\not=W$ and $0\in V$. Then for the non-constant regular function
$g:=f_{k}|_{W}:W\longrightarrow{\mathbb{C}}$ on $W\subset{\mathbb{C}}^{n}$,
let us consider the Milnor fiber $F_{0}\subset(W\setminus V)$ and the Milnor
monodromies

$\Phi_{j,0}\colon
H^{j}(F_{0};{\mathbb{C}})\overset{\sim}{\longrightarrow}H^{j}(F_{0};{\mathbb{C}})\qquad(j\in{\mathbb{Z}})$
(3.35)

of $g$ at the origin $0\in{\mathbb{C}}^{n}$. Our objective here is to describe
the monodromy zeta function $\zeta_{g,0}(t)\in{\mathbb{C}}(t)^{*}$ of $g$ in
terms of the Newton polyhedra $\Gamma_{+}(f_{j})\subset{\mathbb{R}}^{n}_{+}$
of $f_{j}$ ($1\leq j\leq k$). For a subset ${{S}}\subset\\{1,2,\ldots,n\\}$
such that
$\Gamma_{+}^{S}(f_{k})=\Gamma_{+}(f_{k})\cap{\mathbb{R}}^{S}\not=\emptyset$ we
set

$I(S):=\\{1\leq j\leq k-1\ |\
\Gamma_{+}^{S}(f_{j})\not=\emptyset\\}\quad\subset\\{1,2,\ldots,k-1\\}$ (3.36)

and $m(S):=\sharp I(S)+1$. Moreover, for such $S$, $u\in{\rm
Int}({\mathbb{R}}^{S})^{*}_{+}\subset({\mathbb{R}}^{S})^{*}$ and $j\in
I(S)\sqcup\\{k\\}$ we define the supporting face
$\gamma_{j}^{u}\prec\Gamma_{+}^{S}(f_{j})\subset{\mathbb{R}}_{+}^{S}$ of $u$
in $\Gamma_{+}^{S}(f_{j})$ as before and set

$f_{j}^{u}(x):=\displaystyle\sum_{v\in\gamma_{j}^{u}\cap{\mathbb{Z}}^{n}_{+}}a_{j,v}x^{v}\quad\in{\mathbb{C}}[x],$
(3.37)

where $f_{j}(x)=\sum_{v\in{\mathbb{Z}}^{n}_{+}}a_{j,v}x^{v}$
($a_{j,v}\in{\mathbb{C}}$).

######  Definition 3.12.

(Oka [84], [85], see also [68, Definition 3.9]) We say that
$f=(f_{1},f_{2},\ldots,f_{k})$ is non-degenerate (at the origin
$0\in{\mathbb{C}}^{n}$) if for any subset ${{S}}\subset\\{1,2,\ldots,n\\}$
such that $\Gamma_{+}^{S}(f_{k})\not=\emptyset$ and any $u\in{\rm
Int}({\mathbb{R}}^{S})^{*}_{+}$ the following two subvarieties of
$({\mathbb{C}}^{*})^{n}$ are non-degenerate complete intersections:

$\displaystyle\\{x=(x_{1},\ldots,x_{n})\in({\mathbb{C}}^{*})^{n}\ |\
f_{j}^{u}(x)=0\ (j\in I(S)\sqcup\\{k\\})\\},$ (3.38)
$\displaystyle\\{x=(x_{1},\ldots,x_{n})\in({\mathbb{C}}^{*})^{n}\ |\
f_{j}^{u}(x)=0\ (j\in I(S))\\}.$ (3.39)

For each subset ${{S}}\subset\\{1,2,\ldots,n\\}$ such that
$\Gamma_{+}^{S}(f_{k})\not=\emptyset$, let us consider the Minkowski sum

$\Gamma_{+}^{S}(f):=\displaystyle\sum_{j\in I(S)\sqcup\\{k\\}}\
\Gamma_{+}^{S}(f_{j})\ \subset{\mathbb{R}}^{S}_{+}.$ (3.40)

Then, let $\gamma_{1}^{{S}},\gamma_{2}^{{S}},\ldots,\gamma_{n({{S}})}^{{S}}$
be the compact facets of $\Gamma_{+}^{{S}}(f)$. For $1\leq i\leq n({S})$, let
$u_{i}^{{S}}\in({\mathbb{R}}^{{S}})^{*}\cap{\mathbb{Z}}_{+}^{{S}}$ be the
unique non-zero primitive vector which takes its minimum in
$\Gamma_{+}^{{S}}(f)$ on the whole $\gamma_{i}^{{S}}$. Then for any $1\leq
i\leq n({S})$ the supporting faces

$\gamma_{i,j}^{S}:=\gamma_{j}^{u_{i}^{S}}\prec\Gamma_{+}^{S}(f_{j})\qquad(j\in
I(S)\sqcup\\{k\\})$ (3.41)

of $u_{i}^{S}$ in $\Gamma_{+}^{S}(f_{j})$ satisfy the condition

$\gamma_{i}^{S}=\displaystyle\sum_{j\in I(S)\sqcup\\{k\\}}\ \gamma_{i,j}^{S}.$
(3.42)

For $1\leq i\leq n({S})$ we set also

$d_{i}^{{S}}:=\min_{v\in\Gamma_{+}^{{S}}(f_{k})}\langle u_{i}^{{S}},v\rangle\
\in{\mathbb{Z}}_{>0}$ (3.43)

and

$K_{i}^{S}:=\displaystyle\sum_{\begin{subarray}{c}\alpha_{1}+\cdots+\alpha_{m(S)}=\sharp
S-1,\\\ \alpha_{q}\geq 1\ (q\leq m(S)-1),\ \alpha_{m(S)}\geq 0\end{subarray}}\
{\rm
Vol}_{{\mathbb{Z}}}(\underbrace{\gamma_{i,j_{1}}^{S},\ldots,\gamma_{i,j_{1}}^{S}}_{\text{
$\alpha_{1}$-times}},\ldots,\underbrace{\gamma_{i,j_{m(S)}}^{S},\ldots,\gamma_{i,j_{m(S)}}^{S}}_{\text{$\alpha_{m(S)}$-times}}),$
(3.44)

where we set
$I(S)\sqcup\\{k\\}=\\{j_{1},j_{2},\ldots,j_{m(S)-1},k=j_{m(S)}\\}$ and

${\rm
Vol}_{{\mathbb{Z}}}(\underbrace{\gamma_{i,j_{1}}^{S},\ldots,\gamma_{i,j_{1}}^{S}}_{\text{
$\alpha_{1}$-times}},\ldots,\underbrace{\gamma_{i,j_{m(S)}}^{S},\ldots,\gamma_{i,j_{m(S)}}^{S}}_{\text{$\alpha_{m(S)}$-times}})\
\in{\mathbb{Z}}_{+}$ (3.45)

is the normalized $(\sharp S-1)$-dimensional mixed volume with respect to the
lattice
${\mathbb{Z}}^{n}\cap{\mathbb{L}}(\gamma_{i}^{{S}})\simeq{\mathbb{Z}}^{\sharp
S-1}$ in ${\mathbb{L}}(\gamma_{i}^{{S}})\simeq{\mathbb{R}}^{\sharp S-1}$.

######  Remark 3.13.

If $\sharp S-1=0$ and $\sharp I(S)=m(S)-1=0$, we set

$K_{i}^{S}={\rm
Vol}_{{\mathbb{Z}}}(\underbrace{\gamma_{i,k}^{S},\ldots,\gamma_{i,k}^{S}}_{\text{
$0$-times}})=1$ (3.46)

(in this case $\gamma_{i,k}^{S}$ is a point).

######  Theorem 3.14.

(Oka [84], [85, Chapter IV], see also [68, Theorem 3.12]) Assume that
$f=(f_{1},f_{2},\ldots,f_{k})$ is non-degenerate at the origin
$0\in{\mathbb{C}}^{n}$. Then the monodromy zeta function
$\zeta_{g,0}(t)\in{\mathbb{C}}(t)^{*}$ of
$g=f_{k}|_{W}:W\longrightarrow{\mathbb{C}}$ at $0\in{\mathbb{C}}^{n}$ is given
by

$\zeta_{g,0}(t)=\displaystyle\prod_{S:\ \Gamma_{+}^{S}(f_{k})\not=\emptyset,\
m(S)\leq\sharp S}\ \zeta_{g,0}^{S}(t),$ (3.47)

where for each subset ${S}\subset\\{1,2,\ldots,n\\}$ such that
$\Gamma_{+}^{S}(f_{k})\not=\emptyset$ and $m(S)\leq\sharp S$ we set

$\zeta_{g,0}^{S}(t):=\prod_{i=1}^{n({{S}})}\
(1-t^{d_{i}^{{S}}})^{(-1)^{\sharp{{S}}-m(S)}K_{i}^{S}}.$ (3.48)

######  Proof.

The proof is similar to that of Theorem 3.6. Let $T_{S},i_{S}$ etc. be as in
the proof of Theorem 3.6. Then by Corollary 2.8, Lemma 2.17, Theorem 2.19 and
the decomposition $W\setminus\\{0\\}=\sqcup_{{S}\not=\emptyset}(W\cap T_{S})$
of $W\setminus\\{0\\}$ we obtain

$\zeta_{g,0}(t)=\zeta_{f_{k},0}({\mathbb{C}}_{W})(t)=\prod_{S\not=\emptyset}\
\zeta_{f_{k}\circ i_{S},0}({\mathbb{C}}_{W\cap T_{S}})(t).$ (3.49)

If $\Gamma_{+}^{S}(f_{k})=\emptyset$ we have $f_{k}\circ i_{S}=0$ and hence
$\zeta_{f_{k}\circ i_{S},0}({\mathbb{C}}_{W\cap T_{S}})(t)=1$. Then it
suffices to take the product of $\zeta_{f_{k}\circ
i_{S},0}({\mathbb{C}}_{W\cap T_{S}})(t)$ over the subsets
$S\subset\\{1,2,\ldots,n\\}$ such that $\Gamma_{+}^{S}(f_{k})\not=\emptyset$.
Moreover, for such $S$ we have

$W\cap T_{S}=\\{x\in T_{S}\ |\ f_{j}(x)=0\ (j\in I(S))\\}.$ (3.50)

Fix a subset ${S}=\\{1,2,\ldots,n\\}$ such that
$\Gamma_{+}^{S}(f_{k})\not=\emptyset$, set $l:=\sharp S>0$ and consider
$\overline{T_{S}}\simeq{\mathbb{C}}^{l}$ as the toric variety associated to
the fan $\Sigma_{0}$ in $({\mathbb{R}}^{S})^{*}\simeq{\mathbb{R}}^{l}$ formed
by the all faces of the first quadrant
$({\mathbb{R}}^{S})^{*}_{+}\subset({\mathbb{R}}^{S})^{*}$. Then we regard
$T_{S}\simeq({\mathbb{C}}^{*})^{l}$ as the open dense torus in it. Let
$\Sigma$ be a smooth subdivision of $\Sigma_{0}$ and the dual fan of
$\Gamma^{S}_{+}(f)\subset{\mathbb{R}}^{S}_{+}$ in
$({\mathbb{R}}^{S})^{*}_{+}$. Let $X_{\Sigma}$ be the smooth toric variety
associated to $\Sigma$. Then there exists a proper morphism
$\pi:X_{\Sigma}\longrightarrow\overline{T_{S}}$ of toric varieties. For a cone
$\sigma\in\Sigma$ denote by
$T_{\sigma}\simeq({\mathbb{C}}^{*})^{l-\dim\sigma}$ the $T_{S}$-orbit in
$X_{\Sigma}$ associated to $\sigma$. We have also natural affine open subsets
${\mathbb{C}}^{l}(\sigma)\simeq{\mathbb{C}}^{l}$ of $X_{\Sigma}$ associated to
$l$-dimensional cones $\sigma$ in $\Sigma$. Now apply Theorem 2.19 to the
proper morphism $\pi:X_{\Sigma}\longrightarrow\overline{T_{S}}$, the function
$f_{k}\circ i_{S}:\overline{T_{S}}\longrightarrow{\mathbb{C}}$ and the
constructible sheaf ${\mathbb{C}}_{W\cap T_{S}}\in{\bf
D}_{c}^{b}(X_{\Sigma})$. Then we obtain an equality

$\zeta_{f_{k}\circ i_{S},0}({\mathbb{C}}_{W\cap T_{S}})=\zeta_{f_{k}\circ
i_{S}}({\rm R}\pi_{*}{\mathbb{C}}_{W\cap
T_{S}})(0)=\int_{\pi^{-1}(0)}\zeta_{f_{k}\circ
i_{S}\circ\pi}({\mathbb{C}}_{W\cap T_{S}})$ (3.51)

in ${\mathbb{C}}(t)^{*}$. Hence it suffices to calculate the value of the
constructible function $\zeta_{f_{k}\circ i_{S}\circ\pi}({\mathbb{C}}_{W\cap
T_{S}})$ at each point of $\pi^{-1}(0)\subset X_{\Sigma}$. Note that
$\pi^{-1}(0)$ is the union of the $T_{S}$-orbits $T_{\tau}$ in $X_{\Sigma}$
associated to the cones $\tau\in\Sigma$ such that ${\rm
rel.int}(\tau)\subset{\rm
Int}({\mathbb{R}}^{S})^{*}_{+}\simeq({\mathbb{R}}_{>0})^{l}$. Let
$\tau\in\Sigma$ be such a cone and $\sigma\in\Sigma$ an $l$-dimensional cone
such that $\tau\prec\sigma$. Let
$w_{1},w_{2},\ldots,w_{l}\in{\mathbb{Z}}_{+}^{S}\setminus\\{0\\}$ be the
primitive vectors on the edges of the simplicial cone $\sigma$. Then there
exists an affine open subset ${\mathbb{C}}^{l}(\sigma)$ of $X_{\Sigma}$ such
that ${\mathbb{C}}^{l}(\sigma)\simeq{\mathbb{C}}^{l}_{y}$ and the functions
$f_{j}\circ i_{S}\circ\pi$ ($j\in I(S)\sqcup\\{k\\}$) have the following form
on it:

$(f_{j}\circ i_{S}\circ\pi)(y)=y_{1}^{b_{j,1}}\cdots y_{l}^{b_{j,l}}\cdot
f_{j,\sigma}(y),$ (3.52)

where we set

$b_{j,i}=\min_{v\in\Gamma_{+}^{S}(f_{j})}\langle w_{i},v\rangle\geq
0\qquad(i=1,2,\ldots,l)$ (3.53)

and $f_{j,\sigma}(y)$ is a polynomial on
${\mathbb{C}}^{l}(\sigma)\simeq{\mathbb{C}}^{l}_{y}$. Set $d:={\rm dim}\tau$
and for simplicity, assume that $w_{1},\ldots,w_{d}$ generate the simplicial
cone $\tau$. Then in the affine chart
${\mathbb{C}}^{l}(\sigma)\simeq{\mathbb{C}}^{l}_{y}$ the $T_{S}$-orbit
$T_{\tau}$ associated to $\tau$ is explicitly defined by

$T_{\tau}=\\{(y_{1},\ldots,y_{l})\in{\mathbb{C}}^{l}(\sigma)\ |\
y_{1}=\cdots=y_{d}=0,\ y_{d+1},\ldots,y_{l}\neq
0\\}\simeq({\mathbb{C}}^{*})^{l-d}.$

For $j\in I(S)\sqcup\\{k\\}$ let $h_{j}:T_{\tau}\longrightarrow{\mathbb{C}}$
be the restriction of
$f_{j,\sigma}:{\mathbb{C}}^{l}(\sigma)\longrightarrow{\mathbb{C}}$ to
$T_{\tau}\subset{\mathbb{C}}^{l}(\sigma)$. Then by the non-degeneracy of
$f=(f_{1},f_{2},\ldots,f_{k})$, the following two subvarieties of $T_{\tau}$
are non-degenerate complete intersections:

$\displaystyle V_{\tau}:=$ $\displaystyle\\{x\in T_{\tau}\ |\ h_{j}(x)=0\
(j\in I(S)\sqcup\\{k\\})\\},$ (3.54) $\displaystyle W_{\tau}:=$
$\displaystyle\\{x\in T_{\tau}\ |\ h_{j}(x)=0\ (j\in I(S))\\}.$ (3.55)

This implies that the subvariety $\cap_{j\in
I(S)\sqcup\\{k\\}}f_{j,\sigma}^{-1}(0)$ (resp. $\cap_{j\in
I(S)}f_{j,\sigma}^{-1}(0)$) of ${\mathbb{C}}^{l}(\sigma)$ is smooth and of
codimension $m(S)$ (resp. $m(S)-1=\sharp I(S)$) in ${\mathbb{C}}^{l}(\sigma)$
on a neighborhood of $T_{\tau}\subset{\mathbb{C}}^{l}(\sigma)$. Note also that
for the closure $\overline{W\cap T_{S}}$ of $W\cap T_{S}$ in
${\mathbb{C}}^{l}(\sigma)$ we have

$\overline{W\cap T_{S}}=\bigcap_{j\in I(S)}f_{j,\sigma}^{-1}(0)$ (3.56)

on a neighborhood of $T_{\tau}$. Then as in the proof of Theorem 3.6, it
suffices to consider only the $1$-dimensional cones $\tau\in\Sigma$ such that
the supporting face $\gamma(\tau)\prec\Gamma_{+}^{S}(f)$ of $\tau$ in
$\Gamma_{+}^{S}(f)$ is equal to the facet $\gamma_{i}^{S}\prec\Gamma_{+}(f)$
for some $1\leq i\leq n(S)$. In such a case, we have $d=1$,
$b_{k,1}=d_{i}^{S}>0$. But, if $m(S)>\sharp S$, then $\sharp
I(S)=m(S)-1>\sharp S-1={\rm dim}T_{\tau}$ and hence $\cap_{j\in
I(S)}f_{j,\sigma}^{-1}(0)$ does not intersect $T_{\tau}$. So we do not have to
consider the subsets $S\subset\\{1,2,\ldots,n\\}$ such that $m(S)>\sharp S$.
Moreover, for the $1$-dimensional cone $\tau\in\Sigma$ such that
$\gamma(\tau)=\gamma_{i}^{S}$, by Theorem 3.11 we have

$\chi(W_{\tau}\setminus V_{\tau})=(-1)^{\sharp S-m(S)}K_{i}^{S}.$ (3.57)

Then the assertion immediately follows. This completes the proof.

As in the proof of Lemma 3.7 we can easily prove the following result.

######  Lemma 3.15.

Assume that $f=(f_{1},f_{2},\ldots,f_{k})$ is non-degenerate at the origin
$0\in{\mathbb{C}}^{n}$ and $f_{1},f_{2},\ldots,f_{k}$ are convenient. Then the
complex subvarieties $V,W\subset{\mathbb{C}}^{n}$ are smooth or have an
isolated singular point at the origin $0\in{\mathbb{C}}^{n}$ and satisfy the
conditions ${\rm dim}V=n-k$ and ${\rm dim}W=n-k+1$.

Together with the celebrated generalization of Theorem 2.3 by Hamm [44], in
the situation of Lemma 3.15 we obtain the characteristic polynomial of the
$(n-k)$-th monodromy operator

$\Phi_{n-k,0}:H^{n-k}(F_{0};{\mathbb{C}})\overset{\sim}{\longrightarrow}H^{n-k}(F_{0};{\mathbb{C}})$
(3.58)

by Theorem 3.14.

##  4 Singularities and monodromies at infinity of polynomial maps

In this section, we introduce some basic definitions and results on
singularities and monodromies at infinity of polynomial maps
$f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$. In particular, here we
focus on Némethi-Zaharia’s theorem in [76] and Libgober-Sperber’s one in [64].
After the two fundamental papers Broughton [6] and Siersma-Tibăr [102], many
authors studied the global behavior of polynomial maps
$f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$. First of all, for a
polynomial map $f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$, it is
well-known that there exists a finite subset $B\subset{\mathbb{C}}$ such that
the restriction

${\mathbb{C}}^{n}\setminus f^{-1}(B)\longrightarrow{\mathbb{C}}\setminus B$
(4.1)

of $f$ is a $C^{\infty}$ locally trivial fibration. We denote by $B_{f}$ the
smallest subset $B\subset{\mathbb{C}}$ satisfying this condition. Let ${\rm
Sing}f\subset{\mathbb{C}}^{n}$ be the set of the critical points of
$f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$. Then by the definition
of $B_{f}$, obviously we have $f({\rm Sing}f)\subset B_{f}$. We call the
elements of $B_{f}$ the bifurcation values of $f$. The determination of the
bifurcation set $B_{f}\subset{\mathbb{C}}$ is a fundamental problem and has
been studied by a lot of mathematicians such as [6], [42], [46], [47], [76],
[77], [86], [102], [107], [109], [113], [119] and from various points of view.
For the generalizations of some of these results to more general polynomial
maps $f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}^{k}$ for $1\leq k\leq
n$, see also [12], [43], [58], [80] and [88]. The essential difficulty in
determining $B_{f}\subset{\mathbb{C}}$ lies in the fact that
$f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ is not proper and hence
we may have $f({\rm Sing}f)\not=B_{f}$ i.e. $f$ has some extra singularities
at infinity.

######  Definition 4.1.

(Broughton [6]) Let $\partial
f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}^{n}$ be the map defined by
$\partial f(x)=(\partial_{1}f(x),\ldots,\partial_{n}f(x))$
($x\in{\mathbb{C}}^{n}$). Then we say that $f$ is tame at infinity if the
restriction $(\partial f)^{-1}(B(0;\varepsilon))\longrightarrow
B(0;\varepsilon)$ of $\partial f$ to a sufficiently small ball
$B(0;\varepsilon)\subset{\mathbb{C}}^{n}$ centered at the origin
$0\in{\mathbb{C}}^{n}$ is proper.

In [6] Broughton showed that if
$f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ is tame at infinity then
we have $f({\rm Sing}f)=B_{f}$. In the tame case, the following result is also
fundamental.

######  Theorem 4.2.

(Broughton [6]) Assume that $f$ is tame at infinity. Then the generic fiber
$f^{-1}(c)$ ($c\in{\mathbb{C}}\setminus B_{f}$) has the homotopy type of the
bouquet of some $(n-1)$-spheres. In particular, we have

$H^{j}(f^{-1}(c);{\mathbb{C}})=0\quad(j\neq 0,n-1).$ (4.2)

By Siersma and Tibăr [102] this concentration result was later extended to
polynomial maps $f\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}$ with
isolated singularities with respect to some fiber-compactifying extensions of
$f$. From now on, we shall introduce the theorem of Némethi and Zaharia [76]
which gives an upper bound for $B_{f}\subset{\mathbb{C}}$ described in terms
of a Newton polyhedron of $f$.

######  Definition 4.3.

(Kouchnirenko [56]) We call the convex hull of $\\{0\\}\cup NP(f)$ in
${\mathbb{R}}^{n}$ the Newton polyhedron at infinity of $f$ and denote it by
$\Gamma_{\infty}(f)$.

For a face $\gamma\prec\Gamma_{\infty}(f)$ of $\Gamma_{\infty}(f)$ we say that
$\gamma$ is at infinity if $0\notin\gamma$.

######  Definition 4.4.

(Kouchnirenko [56]) We say that
$f(x)=\sum_{v\in{\mathbb{Z}}_{+}^{n}}a_{v}x^{v}$ ($a_{v}\in{\mathbb{C}}$) is
non-degenerate at infinity if for any face at inifinity $\gamma$ of
$\Gamma_{\infty}(f)$ the complex hypersurface $\\{x\in({\mathbb{C}}^{*})^{n}\
|\ f^{\gamma}(x)=0\\}$ of $T=({\mathbb{C}}^{*})^{n}$ is smooth and reduced,
where we set
$f^{\gamma}(x)=\sum_{v\in\gamma\cap{\mathbb{Z}}_{+}^{n}}a_{v}x^{v}$.

Kouchnirenko [56] proved that if $f$ is convenient and non-degenerate at
infinity then we have $B_{f}=f({\rm Sing}f)$. Indeed in such a case, by a
result of Broughton [6] $f$ is tame at infinity. However, in the non-
convenient case, Némethi and Zaharia [76] showed that more bifurcation values
may occur by the presence of the so-called “bad faces” of
$\Gamma_{\infty}(f)$. Here we introduce their result in the new formulation of
[109].

######  Definition 4.5.

(Takeuchi-Tibăr [112]) We say that a face $\gamma\prec\Gamma_{\infty}(f)$ is
atypical if $0\in\gamma$, $\dim\gamma\geq 1$ and the cone
$\sigma(\gamma)\subset({\mathbb{R}}^{n})^{*}$ which corresponds it in the dual
subdivision of $\Gamma_{\infty}(f)$ is not contained in the first quadrant
$({\mathbb{R}}^{n})^{*}_{+}\simeq{\mathbb{R}}^{n}_{+}$ of
$({\mathbb{R}}^{n})^{*}$.

This definition is related to that of the bad faces of $NP(f-f(0))$ in
Némethi-Zaharia [76] as follows. If a face $\Delta\prec NP(f-f(0))$ of
$NP(f-f(0))$ is bad in the sense of [76], then the convex hull $\gamma$ of
$\\{0\\}\cup\Delta$ in ${\mathbb{R}}^{n}$ is an atypical face of
$\Gamma_{\infty}(f)$. See also Remark 4.8 below.

######  Example 4.6.

Let $n=3$ and consider a non-convenient polynomial $f(x,y,z)$ on
${\mathbb{C}}^{3}$ whose Newton polyhedron at infinity $\Gamma_{\infty}(f)$ is
the convex hull of the points $(2,0,0),(2,2,0),(2,2,3)\in{\mathbb{R}}^{3}_{+}$
and the origin $0=(0,0,0)\in{\mathbb{R}}^{3}$. Then the line segment
connecting the point $(2,2,0)$ (resp. $(2,0,0)$) and the origin
$0\in{\mathbb{R}}^{3}$ is an atypical face of $\Gamma_{\infty}(f)$. However
the triangle whose vertices are the points $(2,0,0)$, $(2,2,0)$ and the origin
$0\in{\mathbb{R}}^{3}$ is not so. Note that for the line segment $\gamma$
connecting $(2,0,0)$ and the origin we have
$\dim\sigma(\gamma)\cap({\mathbb{R}}^{3})^{*}_{+}=2$.

Let $\gamma_{1},\ldots,\gamma_{m}$ be the atypical faces of
$\Gamma_{\infty}(f)$. For $1\leq i\leq m$ let $K_{i}=f^{\gamma_{i}}({\rm
Sing}f^{\gamma_{i}})\subset{\mathbb{C}}$ be the set of the critical values of
the $\gamma_{i}$-part

$f^{\gamma_{i}}:T=({\mathbb{C}}^{*})^{n}\longrightarrow{\mathbb{C}}$ (4.3)

of $f$. Let us set

$K_{f}=f({\rm Sing}f)\cup\\{f(0)\\}\cup(\cup_{i=1}^{m}K_{i}).$ (4.4)

Then Némethi-Zaharia [76] proved the following very useful result.

######  Theorem 4.7.

(Némethi-Zaharia [76, Theorem 2]) Assume that $f$ is non-degenerate at
infinity. Then we have $B_{f}\subset K_{f}$.

######  Remark 4.8.

If for an atypical face $\gamma_{i}$ of $\Gamma_{\infty}(f)$ the face
$\Delta=\gamma_{i}\cap NP(f-f(0))\prec NP(f-f(0))$ of $NP(f-f(0))$ is not bad
in the sense of Némethi-Zaharia [76], then $\dim
NP(f_{\gamma_{i}}-f(0))=\dim\Delta<\dim\gamma_{i}$, $f_{\gamma_{i}}-f(0)$ is a
positively homogeneous Laurent polynomial on $T=({\mathbb{C}}^{*})^{n}$ and
hence we have $K_{i}=\\{f(0)\\}$. Therefore the above inclusion $B_{f}\subset
K_{f}$ coincides with the one in [76].

In [76] Némethi and Zaharia also proved the equality $B_{f}=K_{f}$ for $n=2$
and conjectured its validity in higher dimensions. So the remaining important
problem is to prove the inverse inclusion $K_{f}\subset B_{f}$. Later Zaharia
[119] proved $K_{f}\setminus\\{f(0)\\}\subset B_{f}$ for $n\geq 2$ assuming
some conditions and that $f$ has isolated singularities at infinity on a fixed
smooth toric compactification of ${\mathbb{C}}^{n}$. More recently in [109],
the author relaxed the conditions in [119] with the help of some basic
properties of perverse sheaves. See also [113] for some recent progress in the
problem. Also for more general polynomial maps
$f=(f_{1},f_{2},\ldots,f_{k})\colon{\mathbb{C}}^{n}\longrightarrow{\mathbb{C}}^{k}$
($1\leq k\leq n$), one can define their bifurcation sets
$B_{f}\subset{\mathbb{C}}^{k}$ similarly. In [12] and [80], as in Theorem 4.7
the authors of them gave upper bounds for $B_{f}\subset{\mathbb{C}}^{k}$
described in terms of Newton polyhedra of $f_{j}$ ($1\leq j\leq k$). For two
polynomials
$P(x),Q(x)\in{\mathbb{C}}[x]={\mathbb{C}}[x_{1},x_{2},\ldots,x_{n}]$ on
${\mathbb{C}}^{n}$ such that $Q(x)\not=0$ we define a rational function $f(x)$
on ${\mathbb{C}}^{n}$ by

$f(x)=\frac{P(x)}{Q(x)}\qquad(x\in{\mathbb{C}}^{n}\setminus Q^{-1}(0))$ (4.5)

and consider the holomorphic map $f:{\mathbb{C}}^{n}\setminus
Q^{-1}(0)\longrightarrow{\mathbb{C}}$ associated to it. Also in this case, we
can define the bifurcation set $B_{f}\subset{\mathbb{C}}$ of $f$. Then it
would be also an important problem to determine $B_{f}$ for such rational
functions $f$. In [79] and [81] only some partial answers were given to it.

Let $C_{R}=\\{x\in{\mathbb{C}}\ |\ |x|=R\\}$ ($R\gg 0$) be a sufficiently
large circle in ${\mathbb{C}}$ such that $B_{f}\subset\\{x\in{\mathbb{C}}\ |\
|x|<R\\}$. Then by restricting the locally trivial fibration
${\mathbb{C}}^{n}\setminus f^{-1}(B_{f})\longrightarrow{\mathbb{C}}\setminus
B_{f}$ to $C_{R}$ we obtain a geometric monodromy automorphism
$\Phi_{f}^{\infty}\colon f^{-1}(R)\overset{\sim}{\longrightarrow}f^{-1}(R)$
and the linear maps

$\Phi_{j}^{\infty}\colon
H^{j}(f^{-1}(R);{\mathbb{C}})\overset{\sim}{\longrightarrow}H^{j}(f^{-1}(R);{\mathbb{C}})\
\ (j=0,1,\ldots)$ (4.6)

associated to it, where the orientation of $C_{R}$ is taken to be counter-
clockwise as usual. We call $\Phi_{j}^{\infty}$’s the (cohomological)
monodromies at infinity of $f$. Various formulas for their eigenvalues (i.e.
the semisimple parts) were obtained by Libgober-Sperber [64] and Gusein-Zade-
Luengo-Melle-Hernández [40]. Also, several results on their nilpotent parts
were obtained by García-López-Némethi [33], Dimca-Saito [22] and Matsui-
Takeuchi [73]. For example, Dimca-Saito [22] and Matsui-Takeuchi [73] obtained
upper bounds for the sizes of Jordan blocks in $\Phi_{j}^{\infty}$. For the
special case $n=2$, see also [19]. In Sections 7 and 9, we will introduce the
combinatorial expressions of the Jordan normal form of $\Phi_{n-1}^{\infty}$
obtained in Matsui-Takeuchi [71] and Stapledon [104]. The monodromies at
infinity $\Phi_{j}^{\infty}$ are important, because after a basic result [78]
of Neumann-Norbury, Dimca-Némethi [21] proved that for any $j\in{\mathbb{Z}}$
the monodromy representation

$\pi_{1}({\mathbb{C}}\setminus B_{f},c)\longrightarrow{\rm
Aut}(H^{j}(f^{-1}(c);{\mathbb{C}}))\qquad(c\in{\mathbb{C}}\setminus B_{f})$
(4.7)

is described by the matrix of $\Phi_{j}^{\infty}$ for a basis in the
decomposition of $H^{j}(f^{-1}(c);{\mathbb{C}})$ with respect to the vanishing
cycles at the points in $B_{f}$. Recall that by Theorem 4.2 if $f$ is tame at
infinity $\Phi_{n-1}^{\infty}$ is the only non-trivial monodromy at infinity
and its characteristic polynomial is calculated by the following zeta function
$\zeta_{f}^{\infty}(t)\in{\mathbb{C}}(t)^{*}$.

######  Definition 4.9.

We define the monodromy zeta function at infinity $\zeta_{f}^{\infty}(t)$ of
$f$ by

$\zeta_{f}^{\infty}(t):=\prod_{j=0}^{\infty}\det({\rm
id}-t\Phi_{j}^{\infty})^{(-1)^{j}}\quad\in{\mathbb{C}}(t)^{*}.$ (4.8)

For a subset ${S}\subset\\{1,2,\ldots,n\\}$, let
${\mathbb{R}}^{{S}}\simeq{\mathbb{R}}^{\sharp S}$ be as in Section 3 and set
$\Gamma_{\infty}^{{S}}(f):=\Gamma_{\infty}(f)\cap{\mathbb{R}}^{{S}}$. Then $f$
is convenient if and only if we have
$\dim\Gamma_{\infty}^{{S}}(f)=\sharp{{S}}$ for any
${{S}}\subset\\{1,2,\ldots,n\\}$. For each non-empty subset
${{S}}\subset\\{1,2,\ldots,n\\}$, let
$\gamma_{1}^{{S}},\gamma_{2}^{{S}},\ldots,\gamma_{n({{S}})}^{{S}}$ be the
$(\sharp{{S}}-1)$-dimensional faces at infinity of $\Gamma_{\infty}^{{S}}(f)$.
Note that for ${S}\subset\\{1,2,\ldots,n\\}$ such that ${\rm
dim}\Gamma_{\infty}^{{S}}(f)<\sharp S$ there is not such a face and hence we
set $n(S)=0$. For $1\leq i\leq n({S})$, let
$u_{i}^{{S}}\in({\mathbb{R}}^{{S}})^{*}\cap{\mathbb{Z}}^{{S}}$ be the unique
non-zero primitive vector which takes its minimum in
$\Gamma_{\infty}^{{S}}(f)$ on the whole $\gamma_{i}^{{S}}$ and set

$d_{i}^{{S}}:=-\min_{v\in\Gamma_{\infty}^{{S}}(f)}\langle
u_{i}^{{S}},v\rangle\in{\mathbb{Z}}_{>0}.$ (4.9)

We call $d_{i}^{{S}}$ the lattice distance of $\gamma_{i}^{{S}}$ from the
origin $0\in{\mathbb{R}}^{{S}}$. For each face
$\gamma_{i}^{{S}}\prec\Gamma_{\infty}^{{S}}(f)$, let
${\mathbb{L}}(\gamma_{i}^{{S}})$ be the smallest affine linear subspace of
${\mathbb{R}}^{n}$ containing $\gamma_{i}^{{S}}$ and ${\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})\in{\mathbb{Z}}_{>0}$ the normalized
$(\sharp{S}-1)$-dimensional volume of $\gamma_{i}^{{S}}$ with respect to the
lattice ${\mathbb{Z}}^{n}\cap{\mathbb{L}}(\gamma_{i}^{{S}})$.

######  Theorem 4.10.

(Libgober-Sperber [64, Theorem 1], see also [69, Theorem 3.1] for a slight
generalization) Assume that $f$ is non-degenerate at infinity. Then we have

$\zeta_{f}^{\infty}(t)=\prod_{{{S}}\neq\emptyset}\zeta^{\infty}_{f,{{S}}}(t),$
(4.10)

where for each non-empty subset ${S}\subset\\{1,2,\ldots,n\\}$ we set

$\zeta^{\infty}_{f,{{S}}}(t):=\prod_{i=1}^{n({{S}})}(1-t^{d_{i}^{{S}}})^{(-1)^{\sharp{{S}}-1}{\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})}.$ (4.11)

This theorem was first proved by Libgober-Sperber [64]. Here we introduce the
new proof given later in [69].

######  Proof.

Let
$j\colon{\mathbb{C}}\lhook\joinrel\longrightarrow{\mathbb{P}}^{1}={\mathbb{C}}\sqcup\\{\infty\\}$
be the compactification of ${\mathbb{C}}$ and take a local coordinate $h$ of
${\mathbb{P}}^{1}$ on a neighborhood of the point $\infty\in{\mathbb{P}}^{1}$
such that $\infty=\\{h=0\\}$. Then by the (generalized) Poincaré duality
isomorphisms

$H_{j}(f^{-1}(R);{\mathbb{C}})\simeq
H_{c}^{2n-2-j}(f^{-1}(R);{\mathbb{C}})\qquad(R\gg 0,j\in{\mathbb{Z}}_{+})$
(4.12)

we find that

$\zeta_{f}^{\infty}(t)=\zeta_{h,\infty}(j_{!}{\rm
R}f_{!}{\mathbb{C}}_{{\mathbb{C}}^{n}})(t)\qquad\in{\mathbb{C}}(t)^{*}.$
(4.13)

Let $T_{S},i_{S}$ etc. be as in the proof of Theorem 3.6. Then by Corollary
2.8, Lemma 2.17 and the decomposition
${\mathbb{C}}^{n}\setminus\\{0\\}=\sqcup_{{S}\not=\emptyset}T_{S}$ of
${\mathbb{C}}^{n}\setminus\\{0\\}$ we obtain

$\zeta_{h,\infty}(j_{!}{\rm
R}f_{!}{\mathbb{C}}_{{\mathbb{C}}^{n}})(t)=\prod_{S\not=\emptyset}\
\zeta_{h,\infty}(j_{!}{\rm R}(f\circ i_{S})_{!}{\mathbb{C}}_{T_{S}})(t).$
(4.14)

Note that we have ${\rm R}(f\circ i_{S})_{!}{\mathbb{C}}_{T_{S}}\simeq{\rm
R}(f|_{T_{S}})_{!}{\mathbb{C}}_{T_{S}}$. Hence it suffices to prove that for
any non-empty subset ${S}\subset\\{1,2,\ldots,n\\}$ we have

$\zeta_{h,\infty}(j_{!}{\rm
R}(f|_{T_{S}})_{!}{\mathbb{C}}_{T_{S}})(t)=\zeta^{\infty}_{f,{{S}}}(t)=\prod_{i=1}^{n({{S}})}\
(1-t^{d_{i}^{{S}}})^{(-1)^{\sharp{{S}}-1}{\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})}.$ (4.15)

Here we shall prove it only for ${S}=\\{1,2,\ldots,n\\}$. The proof for the
other cases are similar. For ${S}=\\{1,2,\ldots,n\\}$ we set

$T:=T_{S}=T_{\\{1,2,\ldots,n\\}}=({\mathbb{C}}^{*})^{n}\subset{\mathbb{C}}^{n}.$
(4.16)

Let $\Sigma$ be a smooth fan in $({\mathbb{R}}^{n})^{*}\simeq{\mathbb{R}}^{n}$
obtained by subdividing the dual subdivision of
$\Gamma^{S}_{\infty}(f)=\Gamma_{\infty}(f)$ in $({\mathbb{R}}^{n})^{*}$. Let
$X_{\Sigma}$ be the smooth toric variety associated to $\Sigma$. Recall that
$T=({\mathbb{C}}^{*})^{n}$ acts on $X_{\Sigma}$ and the $T$-orbits are
parametrized by the cones in $\Sigma$. For a cone $\sigma\in\Sigma$ denote by
$T_{\sigma}\simeq({\mathbb{C}}^{*})^{n-\dim\sigma}$ the corresponding
$T$-orbit. We have also natural affine open subsets
${\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}$ of $X_{\Sigma}$ associated to
$n$-dimensional cones $\sigma$ in $\Sigma$. Let $\sigma$ be an $n$-dimensional
cone in $\Sigma$ and $w_{1},\ldots,w_{n}\in{\mathbb{Z}}^{n}$ the primitive
vectors on the edges of $\sigma$. Then there exists an affine open subset
${\mathbb{C}}^{n}(\sigma)$ of $X_{\Sigma}$ such that
${\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}_{y}$ and the meromorphic
function $\widetilde{f}$ on $X_{\Sigma}$ obtained by extending
$f|_{T}:T\longrightarrow{\mathbb{C}}$ to $X_{\Sigma}$ has the following form
on it:

$\widetilde{f}(y)=\sum_{v\in{\mathbb{Z}}_{+}^{n}}a_{v}y_{1}^{\langle
w_{1},v\rangle}\cdots y_{n}^{\langle w_{n},v\rangle}=y_{1}^{b_{1}}\cdots
y_{n}^{b_{n}}\cdot f_{\sigma}(y),$ (4.17)

where we set $f=\sum_{v\in{\mathbb{Z}}_{+}^{n}}a_{v}x^{v}$,

$b_{i}=\min_{v\in\Gamma_{\infty}(f)}\langle w_{i},v\rangle\leq
0\qquad(i=1,2,\ldots,n)$ (4.18)

and $f_{\sigma}(y)$ is a polynomial on
${\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}_{y}$. Note that on the affine
open subset ${\mathbb{C}}^{n}(\sigma)\simeq{\mathbb{C}}^{n}$ of $X_{\Sigma}$
the hypersurface

$Z:=\overline{(f|_{T})^{-1}(0)}\quad\subset X_{\Sigma}$ (4.19)

is explicitly written as $f_{\sigma}^{-1}(0)\subset{\mathbb{C}}^{n}(\sigma)$.
So far we have extended $f|_{T}:T\longrightarrow{\mathbb{C}}$ to the
meromorphic function $\widetilde{f}$ on the smooth complete toric variety
$X_{\Sigma}$. But $\widetilde{f}$ still has points of indeterminacy. This
prevents us from using Theorem 2.19 to calculate $\zeta_{h,\infty}(j_{!}{\rm
R}(f|_{T})_{!}{\mathbb{C}}_{T})(t)\in{\mathbb{C}}(t)^{*}$. From now on, we
shall eliminate the points of indeterminacy of $\widetilde{f}$ by blowing up
$X_{\Sigma}$. Following [64], we say that a $T$-orbit $T_{\sigma}$ in
$X_{\Sigma}$ is at infinity if the supporting face
$\gamma(\sigma)\prec\Gamma_{\infty}(f)$ of $\sigma$ in $\Gamma_{\infty}(f)$ is
at infinity i.e. $0\notin\gamma(\sigma)$. Let
$\rho_{1},\rho_{2},\ldots,\rho_{m}\in\Sigma$ be the $1$-dimensional cones in
$\Sigma$ such that the $T$-orbits
$T_{i}:=T_{\rho_{i}}\simeq({\mathbb{C}}^{*})^{n-1}$ are at infinity. Then for
any $1\leq i\leq m$ the toric divisor $D_{i}:=\overline{T_{i}}$ is a smooth
hypersurface in $X_{\Sigma}$ and the poles of $\widetilde{f}$ are contained in
their union $D_{1}\cup\cdots\cup D_{m}$. Moreover by the non-degeneracy at
infinity of $f$, the hypersurface $Z=\overline{(f|_{T})^{-1}(0)}$ in
$X_{\Sigma}$ intersects $D_{I}:=\bigcap_{i\in I}D_{i}$ transversally for any
non-empty subset $I\subset\\{1,2,\ldots,m\\}$. At such intersection points,
$\widetilde{f}$ has indeterminacy. Furthermore we denote the (unique non-zero)
primitive vector in $\rho_{i}\cap{\mathbb{Z}}^{n}$ by $u_{i}$. Then the order
$a_{i}>0$ of the pole of $\widetilde{f}$ along $D_{i}$ is given by

$a_{i}=-\min_{v\in\Gamma_{\infty}(f)}\langle u_{i},v\rangle.$ (4.20)

Now, in order to eliminate the indeterminacy of the meromorphic function
$\widetilde{f}$ on $X_{\Sigma}$, we first consider the blow-up $\pi_{1}\colon
X_{\Sigma}^{(1)}\longrightarrow X_{\Sigma}$ of $X_{\Sigma}$ along the
$(n-2)$-dimensional smooth subvariety $D_{1}\cap Z$. Then the indeterminacy of
the pull-back $\widetilde{f}\circ\pi_{1}$ of $\widetilde{f}$ to
$X_{\Sigma}^{(1)}$ is improved. If $\widetilde{f}\circ\pi_{1}$ still has
points of indeterminacy on the intersection of the exceptional divisor $E_{1}$
of $\pi_{1}$ and the proper transform $Z^{(1)}$ of $Z$, we construct the blow-
up $\pi_{2}\colon X_{\Sigma}^{(2)}\longrightarrow X_{\Sigma}^{(1)}$ of
$X_{\Sigma}^{(1)}$ along $E_{1}\cap Z^{(1)}$. By repeating this procedure
$a_{1}$ times, we obtain a tower of blow-ups

$X_{\Sigma}^{(a_{1})}\underset{\pi_{a_{1}}}{\longrightarrow}\cdots\cdots\underset{\pi_{2}}{\longrightarrow}X_{\Sigma}^{(1)}\underset{\pi_{1}}{\longrightarrow}X_{\Sigma}.$
(4.21)

Then the pull-back of $\widetilde{f}$ to $X_{\Sigma}^{(a_{1})}$ has no
indeterminacy over $T_{1}$. See [69, Figures 1 and 2] and [71, Figures 1,2 and
3] for the details. Next we apply this construction to the proper transforms
of $D_{2}$ and $Z$ in $X_{\Sigma}^{(a_{1})}$. Then we obtain also a tower of
blow-ups

$X_{\Sigma}^{(a_{1})(a_{2})}\longrightarrow\cdots\cdots\longrightarrow
X_{\Sigma}^{(a_{1})(1)}\longrightarrow X_{\Sigma}^{(a_{1})}$ (4.22)

and the indeterminacy of the pull-back of $\widetilde{f}$ to
$X_{\Sigma}^{(a_{1})(a_{2})}$ is eliminated over $T_{1}\sqcup T_{2}$. By
applying the same construction to (the proper transforms of)
$D_{3},D_{4},\ldots,D_{m}$, we finally obtain a birational morphism
$\pi\colon\widetilde{X_{\Sigma}}\longrightarrow X_{\Sigma}$ such that
$g:=\widetilde{f}\circ\pi$ has no point of indeterminacy on the whole
$\widetilde{X_{\Sigma}}$. Note that the smooth compactification
$\widetilde{X_{\Sigma}}$ of $T=({\mathbb{C}}^{*})^{n}$ thus obtained is not a
toric variety any more. Nevertheless, the union of the exceptional divisors of
$\pi\colon\widetilde{X_{\Sigma}}\longrightarrow X_{\Sigma}$ and the proper
transforms of $D_{1},\ldots,D_{m}$ in $\widetilde{X_{\Sigma}}$ is normal
crossing. We thus obtain a commutative diagram of holomorphic maps

$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{f|_{T}}$$\textstyle{\widetilde{X_{\Sigma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{{\mathbb{P}}^{1},}$
(4.23)

where $\iota$ and $j$ are the inclusion maps and $g$ is proper. Then there
exists an isomorphism

$j_{!}{\rm R}(f|_{T})_{!}{\mathbb{C}}_{T}\simeq{\rm
R}g_{*}(\iota_{!}{\mathbb{C}}_{T})$ (4.24)

in ${\bf D}_{c}^{b}({\mathbb{P}}^{1})$. Now we apply Theorem 2.19 to the
proper morphism
$g\colon\widetilde{X_{\Sigma}}\longrightarrow{\mathbb{P}}^{1}$, the function
$h$ and the constructible sheaf
${\mathbb{C}}_{T}=\iota_{!}{\mathbb{C}}_{T}\in{\bf
D}_{c}^{b}(\widetilde{X_{\Sigma}})$. Then for $S=\\{1,2,\ldots,n\\}$ we obtain
an equality

$\zeta_{h,\infty}({\rm R}g_{*}{\mathbb{C}}_{T})=\zeta_{h}({\rm
R}g_{*}{\mathbb{C}}_{T})(\infty)=\int_{g^{-1}(\infty)}\zeta_{h\circ
g}({\mathbb{C}}_{T})$ (4.25)

in ${\mathbb{C}}(t)^{*}$. Note that $h\circ g$ is the meromorphic extension of
$1/(f|_{T}):T\setminus f^{-1}(0)\longrightarrow{\mathbb{C}}$ to
$\widetilde{X_{\Sigma}}$ and we have $(h\circ
g)^{-1}(0)=g^{-1}(\infty)\subset\widetilde{X_{\Sigma}}$. Then as in the proof
of Theorem 3.6, by calculating the constructible function $\zeta_{h\circ
g}({\mathbb{C}}_{T})\in{\rm CF}_{{\mathbb{C}}(t)^{*}}(g^{-1}(\infty))$ at each
point of $g^{-1}(\infty)$ with the help of Lemmas 2.9 and 2.15 and Theorem
3.3, for $S=\\{1,2,\ldots,n\\}$ we obtain the desired equality

$\int_{g^{-1}(\infty)}\zeta_{h\circ
g}({\mathbb{C}}_{T})=\prod_{i=1}^{n({{S}})}\
(1-t^{d_{i}^{{S}}})^{(-1)^{\sharp{{S}}-1}{\rm
Vol}_{{\mathbb{Z}}}(\gamma_{i}^{{S}})}.$ (4.26)

Note that for $S=\\{1,2,\ldots,n\\}$ if ${\rm dim}\Gamma_{\infty}(f)<\sharp
S=n$ then by Theorem 3.3 we have $\int_{g^{-1}(\infty)}\zeta_{h\circ
g}({\mathbb{C}}_{T})=1$ and $n(S)=0$. This completes the proof.

By this new proof of Theorem 4.10 we can easily obtain the following
globalization of Oka’s theorem. Let $f=(f_{1},f_{2},\ldots,f_{k})$, $V\subset
W\subset{\mathbb{C}}^{n}$ and $g=f_{k}|_{W}:W\longrightarrow{\mathbb{C}}$ be
as in (the last half of) Section 3. Then there exists a finite subset
$B\subset{\mathbb{C}}$ such that the restriction

$W\setminus g^{-1}(B)\longrightarrow{\mathbb{C}}\setminus B$ (4.27)

of $g$ is a locally trivial fibration. Let $C_{R}=\\{x\in{\mathbb{C}}\ |\
|x|=R\\}$ ($R\gg 0$) be a sufficiently large circle in ${\mathbb{C}}$ such
that $B\subset\\{x\in{\mathbb{C}}\ |\ |x|<R\\}$ and consider the monodromies
at infinity

$\Phi_{j}^{\infty}\colon
H^{j}(g^{-1}(R);{\mathbb{C}})\overset{\sim}{\longrightarrow}H^{j}(g^{-1}(R);{\mathbb{C}})\
\ (j=0,1,\ldots)$ (4.28)

of $g=f_{k}|_{W}:W\longrightarrow{\mathbb{C}}$ along it. Let
$\zeta^{\infty}_{g}(t)\in{\mathbb{C}}(t)^{*}$ be the monodromy zeta function
associated to them. In order to describe it in terms of the Newton polyhedra
$\Gamma_{\infty}(f_{j})$ ($1\leq j\leq k$), for simplicity here we assume that
$f_{1},f_{2},\ldots,f_{k}$ are convenient. Then, for each non-empty subset
${{S}}\subset\\{1,2,\ldots,n\\}$ we set $I(S):=\\{1,2,\ldots,k-1\\}$ and
$m(S):=k$ and consider the Minkowski sum

$\Gamma_{\infty}^{S}(f):=\displaystyle\sum_{j\in I(S)\sqcup\\{k\\}}\
\Gamma_{\infty}^{S}(f_{j})=\displaystyle\sum_{j=1}^{k}\
\Gamma_{\infty}^{S}(f_{j})\ \subset{\mathbb{R}}^{S}_{+}.$ (4.29)

Moreover we define the non-degeneracy at infinity of
$f=(f_{1},f_{2},\ldots,f_{k})$ by replacing the condition $u\in{\rm
Int}({\mathbb{R}}^{S})^{*}_{+}$ for $u\in({\mathbb{R}}^{S})^{*}$ in Definition
3.12 by the one
$u\in({\mathbb{R}}^{S})^{*}\setminus({\mathbb{R}}^{S})^{*}_{+}$. Then by
defining $n(S)$, $d_{i}^{S}$ and $K_{i}^{S}$ as in Section 3 we obtain the
following result.

######  Theorem 4.11.

(Matsui-Takeuchi [68, Theorem 5.1]) Assume that $f_{1},f_{2},\ldots,f_{k}$ are
convenient and $f=(f_{1},f_{2},\ldots,f_{k})$ is non-degenerate at infinity.
Then the monodromy zeta function $\zeta_{g}^{\infty}(t)\in{\mathbb{C}}(t)^{*}$
at infinity of $g=f_{k}|_{W}:W\longrightarrow{\mathbb{C}}$ is given by

$\zeta_{g}^{\infty}(t)=\displaystyle\prod_{S:\ k\leq\sharp S}\
\zeta_{g,S}^{\infty}(t),$ (4.30)

where for each subset ${S}\subset\\{1,2,\ldots,n\\}$ such that $k\leq\sharp S$
we set

$\zeta_{g,S}^{\infty}(t):=\prod_{i=1}^{n({{S}})}\
(1-t^{d_{i}^{{S}}})^{(-1)^{\sharp{{S}}-m(S)}K_{i}^{S}}.$ (4.31)

##  5 Combinatorial formulas for equivariant Hodge-Deligne numbers of toric
hypersurfaces

In this section, we introduce the refinements obtained in [71, Section 2] of
the results of Danilov-Khovanskii [14]. We shall freely use standard notions
of mixed Hodge structures, for which we refer to El Zein [29] and Voisin
[118]. In the sequel, let us fix an element
$\tau=(\tau_{1},\ldots,\tau_{n})\in T:=({\mathbb{C}}^{*})^{n}$ and let $g$ be
a Laurent polynomial on $({\mathbb{C}}^{*})^{n}$ such that the complex
hypersurface $Z^{*}=\\{x\in({\mathbb{C}}^{*})^{n}\ |\ g(x)=0\\}$ is non-
degenerate and invariant by the automorphism
$l_{\tau}\colon({\mathbb{C}}^{*})^{n}\underset{\tau\times}{\overset{\sim}{\longrightarrow}}({\mathbb{C}}^{*})^{n}$
induced by the multiplication by $\tau$. Set $\Delta=NP(g)$ and for simplicity
assume that ${\rm dim}\Delta=n$. Then there exists $\beta\in{\mathbb{C}}$ such
that $l_{\tau}^{*}g=g\circ l_{\tau}=\beta g$. This implies that for any vertex
$v$ of $\Delta=NP(g)$ we have
${\tau}^{v}={\tau}_{1}^{v_{1}}\cdots{\tau}_{n}^{v_{n}}=\beta$. Moreover by the
condition ${\rm dim}\Delta=n$ we see that $\tau_{1},\tau_{2},\ldots,\tau_{n}$
are roots of unity. For $p,q\geq 0$ and $k\geq 0$, let
$h^{p,q}(H_{c}^{k}(Z^{*};{\mathbb{C}}))$ be the mixed Hodge number of
$H_{c}^{k}(Z^{*};{\mathbb{C}})$ and set

$e^{p,q}(Z^{*})=\displaystyle\sum_{k}(-1)^{k}h^{p,q}(H_{c}^{k}(Z^{*};{\mathbb{C}}))$
(5.1)

as in [14]. The above automorphism of $({\mathbb{C}}^{*})^{n}$ induces a
morphism of mixed Hodge structures $l_{\tau}^{*}\colon
H_{c}^{k}(Z^{*};{\mathbb{C}})\overset{\sim}{\longrightarrow}H_{c}^{k}(Z^{*};{\mathbb{C}})$
and hence ${\mathbb{C}}$-linear automorphisms of the $(p,q)$-parts
$H_{c}^{k}(Z^{*};{\mathbb{C}})^{p,q}$ of $H_{c}^{k}(Z^{*};{\mathbb{C}})$. For
$\alpha\in{\mathbb{C}}$, let $h^{p,q}(H_{c}^{k}(Z^{*};{\mathbb{C}}))_{\alpha}$
be the dimension of the $\alpha$-eigenspace
$H_{c}^{k}(Z^{*};{\mathbb{C}})_{\alpha}^{p,q}$ of this automorphism of
$H_{c}^{k}(Z^{*};{\mathbb{C}})^{p,q}$ and set

$e^{p,q}(Z^{*})_{\alpha}=\displaystyle\sum_{k}(-1)^{k}h^{p,q}(H_{c}^{k}(Z^{*};{\mathbb{C}}))_{\alpha}.$
(5.2)

Since we have $l_{\tau}^{r}={\rm id}_{Z^{*}}$ for some $r\gg 0$, these numbers
are zero unless $\alpha$ is a root of unity. Obviously we have

$e^{p,q}(Z^{*})=\displaystyle\sum_{\alpha\in{\mathbb{C}}}e^{p,q}(Z^{*})_{\alpha},\qquad
e^{p,q}(Z^{*})_{\alpha}=e^{q,p}(Z^{*})_{\overline{\alpha}}.$ (5.3)

In this setting, along the lines of Danilov-Khovanskii [14] we can give an
algorithm for computing these numbers $e^{p,q}(Z^{*})_{\alpha}$ as follows.
First of all, as in [14, Section 3] we can easily obtain the following result.

######  Proposition 5.1.

For $p,q\geq 0$ such that $p+q>n-1$, we have

$e^{p,q}(Z^{*})_{\alpha}=\begin{cases}(-1)^{n+p+1}\binom{n}{p+1}&(\text{$\alpha=1$
and $p=q$}),\\\ &\\\ \ 0&(\text{otherwise}),\end{cases}$ (5.4)

where we used the convention $\binom{a}{b}=0$ ($0\leq a<b$) for binomial
coefficients.

For a vertex $w$ of $\Delta$, consider the translated polytope
$\Delta^{w}:=\Delta-w$ such that $0\prec\Delta^{w}$ and ${\tau}^{v}=1$ for any
vertex $v$ of $\Delta^{w}$. Then for $\alpha\in{\mathbb{C}}$ and $k\geq 0$ set

$l^{*}(k\Delta)_{\alpha}=\sharp\\{v\in{\rm
Int}(k\Delta^{w})\cap{\mathbb{Z}}^{n}\ |\
{\tau}^{v}=\alpha\\}\quad\in{\mathbb{Z}}_{+}$ (5.5)

and

$l(k\Delta)_{\alpha}=\sharp\\{v\in(k\Delta^{w})\cap{\mathbb{Z}}^{n}\ |\
{\tau}^{v}=\alpha\\}\quad\in{\mathbb{Z}}_{+}.$ (5.6)

We can easily see that these numbers $l^{*}(k\Delta)_{\alpha}$ and
$l(k\Delta)_{\alpha}$ do not depend on the choice of the vertex $w$ of
$\Delta$. Next, define two formal power series
$P_{\alpha}(\Delta;t)=\sum_{i\geq 0}\varphi_{\alpha,i}(\Delta)t^{i}$ and
$Q_{\alpha}(\Delta;t)=\sum_{i\geq 0}\psi_{\alpha,i}(\Delta)t^{i}$ by

$P_{\alpha}(\Delta;t)=(1-t)^{n+1}\left\\{\displaystyle\sum_{k\geq
0}l^{*}(k\Delta)_{\alpha}t^{k}\right\\}$ (5.7)

and

$Q_{\alpha}(\Delta;t)=(1-t)^{n+1}\left\\{\displaystyle\sum_{k\geq
0}l(k\Delta)_{\alpha}t^{k}\right\\}$ (5.8)

respectively. Then we can easily show that $P_{\alpha}(\Delta;t)$ is actually
a polynomial as in [14, Section 4.4]. Moreover as in Macdonald [65], we can
easily prove that for any $\alpha\in{\mathbb{C}}^{*}$ the function
$h_{\Delta,\alpha}(k):=l(k\Delta)_{\alpha^{-1}}$ of $k\geq 0$ is a polynomial
of degree $n$ with coefficients in ${\mathbb{Q}}$. By a straightforward
generalization of the Ehrhart reciprocity proved by [65], we obtain also an
equality

$h_{\Delta,\alpha}(-k)=(-1)^{n}l^{*}(k\Delta)_{\alpha}$ (5.9)

for $k>0$. By an elementary computation (see [14, Remark 4.6]), this implies
that we have

$\varphi_{\alpha,i}(\Delta)=\psi_{\alpha^{-1},n+1-i}(\Delta)\qquad(i\in{\mathbb{Z}}).$
(5.10)

In particular, $Q_{\alpha}(\Delta;t)=\sum_{i\geq
0}\psi_{\alpha,i}(\Delta)t^{i}$ is a polynomial for any
$\alpha\in{\mathbb{C}}^{*}$.

######  Theorem 5.2.

In the situation as above, we have

$\displaystyle\sum_{q}e^{p,q}(Z^{*})_{\alpha}=\begin{cases}(-1)^{p+n+1}\binom{n}{p+1}+(-1)^{n+1}\varphi_{\alpha,n-p}(\Delta)&(\alpha=1),\\\
&\\\ (-1)^{n+1}\varphi_{\alpha,n-p}(\Delta)&(\alpha\not=1).\end{cases}$ (5.11)

This result can be proved as in the proof for the formula in [14, Section
4.4]. With Proposition 5.1 and Theorem 5.2 at hands, we can now easily
calculate the numbers $e^{p,q}(Z^{*})_{\alpha}$ on the non-degenerate
hypersurface $Z^{*}\subset({\mathbb{C}}^{*})^{n}$ for any
$\alpha\in{\mathbb{C}}$ as in [14, Section 5.2]. Indeed for a projective toric
compactification $X$ of $({\mathbb{C}}^{*})^{n}$ such that the closure
$\overline{Z^{*}}$ of $Z^{*}$ in $X$ is smooth, the variety $\overline{Z^{*}}$
is smooth projective and hence there exists a perfect pairing

$H^{p,q}(\overline{Z^{*}};{\mathbb{C}})_{\alpha}\times
H^{n-1-p,n-1-q}(\overline{Z^{*}};{\mathbb{C}})_{\alpha^{-1}}\longrightarrow{\mathbb{C}}$
(5.12)

for any $p,q\geq 0$ and $\alpha\in{\mathbb{C}}^{*}$ (see for example [118,
Section 5.3.2]). Therefore, we obtain equalities
$e^{p,q}(\overline{Z^{*}})_{\alpha}=e^{n-1-p,n-1-q}(\overline{Z^{*}})_{\alpha^{-1}}$
which are necessary to proceed the algorithm in [14, Section 5.2]. We have
also the following analogue of [14, Proposition 5.8].

######  Proposition 5.3.

For any $\alpha\in{\mathbb{C}}$ and $p>0$ we have

$e^{p,0}(Z^{*})_{\alpha}=e^{0,p}(Z^{*})_{\overline{\alpha}}=(-1)^{n-1}\sum_{\begin{subarray}{c}\Gamma\prec\Delta\\\
{\rm dim}\Gamma=p+1\end{subarray}}l^{*}(\Gamma)_{\alpha}.$ (5.13)

The following result is an analogue of [14, Corollary 5.10]. For
$\alpha\in{\mathbb{C}}$, denote by $\Pi(\Delta)_{\alpha}$ the number of the
lattice points $v=(v_{1},\ldots,v_{n})$ on the $1$-skeleton of
$\Delta^{w}=\Delta-w$ such that ${\tau}^{v}=\alpha$, where $w$ is a vertex of
$\Delta$.

######  Proposition 5.4.

In the situation as above, for any $\alpha\in{\mathbb{C}}^{*}$ we have

$e^{0,0}(Z^{*})_{\alpha}=\begin{cases}(-1)^{n-1}\left(\Pi(\Delta)_{1}-1\right)&(\alpha=1),\\\
&\\\ (-1)^{n-1}\Pi(\Delta)_{\alpha^{-1}}&(\alpha\not=1).\end{cases}$ (5.14)

######  Proof.

By Theorem 5.2, Proposition 5.3 and the equality (5.10), the assertion can be
proved as in the proof [14, Corollary 5.10].

For a vertex $w$ of $\Delta$, we define a closed convex cone ${\rm
Con}(\Delta,w)$ by ${\rm Con}(\Delta,w)=\\{r\cdot(v-w)\ |\
r\in{\mathbb{R}}_{+},\ v\in\Delta\\}\subset{\mathbb{R}}^{n}$.

######  Definition 5.5.

Let $\Delta$ be an $n$-dimensional integral polytope in
$({\mathbb{R}}^{n},{\mathbb{Z}}^{n})$.

  1. (i)

(see [14, Section 2.3]) We say that $\Delta$ is prime if for any vertex $w$ of
$\Delta$ the cone ${\rm Con}(\Delta,w)$ is generated by a basis of
${\mathbb{R}}^{n}$.

  2. (ii)

We say that $\Delta$ is pseudo-prime if for any $1$-dimensional face
$\gamma\prec\Delta$ the number of the $2$-dimensional faces
$\gamma^{\prime}\prec\Delta$ such that $\gamma\prec\gamma^{\prime}$ is $n-1$.

By definition, prime polytopes are pseudo-prime. Moreover any face of a
pseudo-prime polytope is again pseudo-prime. From now on, we assume that
$\Delta=NP(g)$ is pseudo-prime. Let $\Sigma$ be the dual fan of $\Delta$ and
$X_{\Sigma}$ the toric variety associated to it. Then except finite points
$X_{\Sigma}$ is an orbifold and the closure $\overline{Z^{*}}$ of $Z^{*}$ in
$X_{\Sigma}$ does not intersect such points by the non-degeneracy of $g$.
Hence $\overline{Z^{*}}$ is an orbifold i.e. quasi-smooth in the sense of [14,
Proposition 2.4]. In particular, there exists a Poincaré duality isomorphism

$[H^{p,q}(\overline{Z^{*}};{\mathbb{C}})_{\alpha}]^{*}\simeq
H^{n-1-p,n-1-q}(\overline{Z^{*}};{\mathbb{C}})_{\alpha^{-1}}$ (5.15)

for any $\alpha\in{\mathbb{C}}^{*}$ (see for example [13] and [45, Corollary
8.2.22]). Then by slightly generalizing the arguments in [14] we obtain the
following analogue of [14, Section 5.5 and Theorem 5.6].

######  Proposition 5.6.

In the situation as above, for any $\alpha\in{\mathbb{C}}\setminus\\{1\\}$ and
$p,q\geq 0$, we have

$e^{p,q}(\overline{Z^{*}})_{\alpha}=\begin{cases}-\displaystyle\sum_{\Gamma\prec\Delta}(-1)^{{\rm
dim}\Gamma}\varphi_{\alpha,{\rm dim}\Gamma-p}(\Gamma)&(p+q=n-1),\\\ &\\\ \
0&(\text{otherwise}),\end{cases}$ (5.16)

and

$e^{p,q}(Z^{*})_{\alpha}=(-1)^{n+p+q}\sum_{\begin{subarray}{c}\Gamma\prec\Delta\\\
{\rm
dim}\Gamma=p+q+1\end{subarray}}\left\\{\sum_{\Gamma^{\prime}\prec\Gamma}(-1)^{{\rm
dim}\Gamma^{\prime}}\varphi_{\alpha,{\rm
dim}\Gamma^{\prime}-p}(\Gamma^{\prime})\right\\}.$ (5.17)

For $\alpha\in{\mathbb{C}}\setminus\\{1\\}$ and a face $\Gamma\prec\Delta$,
set $\widetilde{\varphi}_{\alpha}(\Gamma)=\sum_{i=0}^{{\rm
dim}\Gamma}\varphi_{\alpha,i}(\Gamma)$. Then Proposition 5.6 can be rewritten
as follows.

######  Corollary 5.7.

For any $\alpha\in{\mathbb{C}}\setminus\\{1\\}$ and $r\geq 0$, we have

$\sum_{p+q=r}e^{p,q}(Z^{*})_{\alpha}=(-1)^{n+r}\sum_{\begin{subarray}{c}\Gamma\prec\Delta\\\
{\rm
dim}\Gamma=r+1\end{subarray}}\left\\{\sum_{\Gamma^{\prime}\prec\Gamma}(-1)^{{\rm
dim}\Gamma^{\prime}}\widetilde{\varphi}_{\alpha}(\Gamma^{\prime})\right\\}.$
(5.18)

##  6 Motivic Milnor fibers and their Hodge realizations

In this section, we introduce some important results in Denef-Loeser [17]
which will be used in subsequent sections. Since their original proofs in [17]
are written in terms of Chow motives and it is not so easy to compare them
with the assertions in [18], here we rewrite them in the new style of [18]. We
shall freely use standard notions of mixed Hodge modules, for which we refer
to [45, Section 8.3], [95], [97] and [100]. First, let
$\mathrm{Var}_{{\mathbb{C}}}$ be the category of algebraic varieties defined
over ${\mathbb{C}}$. We denote by $K_{0}(\mathrm{Var}_{{\mathbb{C}}})$ the
abelian group generated by the isomorphism classes $[X]$ of the elements
$X\in\mathrm{Var}_{{\mathbb{C}}}$ of $\mathrm{Var}_{{\mathbb{C}}}$ and defined
by the relation $[X]=[X\setminus Y]+[Y]$ ($Y$ is a Zariski closed subset of
$X$). By the products of varieties $K_{0}(\mathrm{Var}_{{\mathbb{C}}})$ is a
ring. We call it the motivic Grothendieck ring of varieties. For a positive
integer $d\in{\mathbb{Z}}_{>0}$, let $\mu_{d}:=\\{t\in{\mathbb{C}}\ |\
t^{d}=1\\}\simeq{\mathbb{Z}}/{\mathbb{Z}}d$ be the multiplicative group
consisting of the $d$-roots in ${\mathbb{C}}$ and
$\zeta_{d}:=\exp(2\pi\sqrt{-1}/d)\in\mu_{d}$ its standard generator. Then they
form a projective system by the homomorphisms

$\mu_{rd}\longrightarrow\mu_{d}\qquad(t\longmapsto t^{r})$ (6.1)

and we obtain the projective limit

$\hat{\mu}:=\underset{d}{\varprojlim}\ \mu_{d}$ (6.2)

associated to it. Following Denef-Loeser [18, Section 2.4], we say that an
action of the group $\mu_{d}$ on an algebraic variety $X$ is good if any
oribit of it is contained in an affine open subset. Moreover we say that an
action of the group $\hat{\mu}$ on $X$ is good if it is induced by a good one
of $\mu_{d}$ for some $d\in{\mathbb{Z}}_{>0}$. We thus obtain the Grothendieck
ring $K^{\hat{\mu}}_{0}(\mathrm{Var}_{{\mathbb{C}}})$ of varieties over
${\mathbb{C}}$ with good $\hat{\mu}$-action. In fact, it is noting but the
inductive limit defined by the ring homomorphisms
$K_{0}^{\mu_{d}}(\mathrm{Var}_{{\mathbb{C}}})\longrightarrow
K_{0}^{\mu_{rd}}(\mathrm{Var}_{{\mathbb{C}}})$. We call the element
$\mathbb{L}:=[{\mathbb{C}}]\in K^{\hat{\mu}}_{0}(\mathrm{Var}_{{\mathbb{C}}})$
of $K^{\hat{\mu}}_{0}(\mathrm{Var}_{{\mathbb{C}}})$ defined by the affine line
${\mathbb{C}}$ endowed with the trivial $\hat{\mu}$-action the Lefschetz
motive. Then we denote by ${\mathcal{M}}_{{\mathbb{C}}}^{\hat{\mu}}$ the ring
obtained from the Grothendieck ring ${\rm K}_{0}^{\hat{\mu}}({\rm
Var}_{{\mathbb{C}}})$ by inverting the Lefschetz motive ${\mathbb{L}}\in{\rm
K}_{0}^{\hat{\mu}}({\rm Var}_{{\mathbb{C}}})$.

In what folows, let $Z$ be a (not necessarily complete) smooth algebraic
variety over ${\mathbb{C}}$ and $g\colon Z\longrightarrow{\mathbb{C}}$ a
regular function on it such that $E:=g^{-1}(0)\subset Z$ is a normal crossing
divisor. Assume also that the irreducible components $E_{1},E_{2},\dots,E_{k}$
of $E$ are smooth. We denote the multiplicity of $g$ along $E_{i}$ by
$m_{i}>0$. For each subset $I\subset\\{1,2,\dots,k\\}$ we set

$E_{I}=\bigcap_{i\in I}E_{i},\quad
E^{\circ}_{i}=E_{I}\setminus\bigcup_{i\notin I}E_{i}$ (6.3)

and $m_{I}={\rm gcd}(m_{i})_{i\in I}>0$. Then we can construct an unramified
Galois covering $\widetilde{E_{I}^{\circ}}\longrightarrow E_{I}^{\circ}$ of
$E^{\circ}_{I}$ as follows. First, for a point $p\in E_{I}^{\circ}$ we take an
affine open neighborhood $W\subset Z\setminus(\cup_{i\notin I}E_{i})$ of $p$
on which there exist regular functions $\xi_{i}$ ($i\in I$) such that
$E_{i}\cap W=\\{\xi_{i}=0\\}$ for any $i\in I$ and set

$g_{1,W}:=g\prod_{i\in I}\xi_{i}^{-m_{i}},\qquad g_{2,W}:=\prod_{i\in
I}\xi_{i}^{\frac{m_{i}}{m_{I}}}.$ (6.4)

Note that $g_{1,W}$ is a unit i.e. invertible on $W$ and $g_{2,W}\colon
W\longrightarrow{\mathbb{C}}$ is a regular function. Then on $W$ we obtain a
decomposition

$g=g_{1,W}\cdot(g_{2,W})^{m_{I}}$ (6.5)

of $g$. It is easy to see that $E_{I}^{\circ}$ is covered by such affine open
subsets $W$ of $Z\setminus(\cup_{i\notin I}E_{i})$. Then as in [18, Section
3.3] by gluing the varieties

$\widetilde{E_{I,W}^{\circ}}=\\{(t,z)\in{\mathbb{C}}^{*}\times(E_{I}^{\circ}\cap
W)\ |\ t^{m_{I}}=(g_{1,W})^{-1}(z)\\}$ (6.6)

together in the following way, we obtain the covering
$\widetilde{E_{I}^{\circ}}$ over $E_{I}^{\circ}$ of degree $m_{I}$. If
$W^{\prime}$ is another such open subset and
$g=g_{1,W^{\prime}}\cdot(g_{2,W^{\prime}})^{m_{I}}$ is the decomposition of
$g$ on it, we patch $\widetilde{E_{I,W}^{\circ}}$ and
$\widetilde{E_{I,W^{\prime}}^{\circ}}$ by the morphism
$(t,z)\longmapsto(g_{2,W^{\prime}}(z)(g_{2,W})^{-1}(z)\cdot t,z)$ defined over
$W\cap W^{\prime}$. Let $\mu_{m_{I}}=\\{t\in{\mathbb{C}}\ |\
t^{m_{I}}=1\\}\simeq{\mathbb{Z}}/{\mathbb{Z}}m_{I}$ be the cyclic group of
order $m_{I}$. Then by assigning the automorphism
$(t,z)\mapsto(\zeta_{m_{I}}\cdot t,z)$ of $\widetilde{E_{I}^{\circ}}$ to its
standard generator $\zeta_{m_{I}}=\exp(2\pi\sqrt{-1}/m_{I})\in\mu_{m_{I}}$ we
obtain a good action of $\mu_{m_{I}}$ on $\widetilde{E_{I}^{\circ}}$. We
denote the element of the motivic Grothendieck ring
${\mathcal{M}}^{\hat{\mu}}_{{\mathbb{C}}}$ with good action of $\hat{\mu}$
thus obtained by $[\widetilde{E_{I}^{\circ}}]$. Moreover we set

$\mathcal{S}_{g}=\sum_{I\not=\emptyset}(1-\mathbb{L})^{|I|-1}\cdot{[\widetilde{E_{I}^{\circ}}]}\quad\in{\mathcal{M}}^{\hat{\mu}}_{{\mathbb{C}}}$
(6.7)

and call it the motivic Milnor fiber of $g$. From now, let us explain the
Hodge realizations of the elements in
${\mathcal{M}}^{\hat{\mu}}_{{\mathbb{C}}}$ defined in Denef-Loeser [18,
Section 3.1]. In their terminology, a (mixed) Hodge structure is a vector
space $V$ over ${\mathbb{Q}}$ for which we have a direct sum decomposition
${\mathbb{C}}\otimes_{{\mathbb{Q}}}V\simeq\oplus_{p,q\in{\mathbb{Z}}}V^{p,q}$
satisfying the following two conditions:

$(i)\ V^{p,q}=\overline{V^{q,p}},\quad(ii)\ \bigoplus_{p+q=m}V^{p,q}\mbox{ is
defined over ${\mathbb{Q}}$ for any $m\in{\mathbb{Z}}$}.$ (6.8)

By defining the morphisms of Hodge structures naturally, we obtain an abelian
category $\mathrm{HS}$ of Hodge structures. Similarly, we can define an
abelian category $\mathrm{HS}^{\mathrm{mon}}$ of Hodge structures with a
quasi-unipotent automorphism (“quasi-unipotent” means that some powers of it
is unipotent as in [18, Section 3.1.3]) and denote by
$K_{0}(\mathrm{HS}^{\mathrm{mon}})$ its Grothendieck group. By the tensor
product of Hodge structures, we see that $K_{0}(\mathrm{HS}^{\mathrm{mon}})$
has a ring structure. Then we can define a ring homomorphism

$\chi_{h}\colon{\mathcal{M}}^{\hat{\mu}}_{{\mathbb{C}}}\longrightarrow
K_{0}(\mathrm{HS}^{\mathrm{mon}})$ (6.9)

in the following way. For a variety $Y$ with good action of the cyclic group
$\mu_{d}$ we set

$\chi_{h}([Y])=\sum_{j\in{\mathbb{Z}}}(-1)^{j}{[H^{j}_{c}(Y;{\mathbb{Q}})]}\qquad\in
K_{0}(\mathrm{HS}^{\mathrm{mon}}),$ (6.10)

where $H^{j}_{c}(Y;{\mathbb{Q}})$ stands for the Hodge structure endowed with
the (quasi-unipotent) automorphism induced by the morphism $Y\longrightarrow
Y$ ($y\longmapsto\zeta_{d}y$). We call the ring homomorphism $\chi_{h}$ thus
obtained the Hodge characteristic morphism. Let us define an element of
$K_{0}(\mathrm{HS}^{\mathrm{mon}})$ associated to $g$. Let

$\psi^{H}_{g}\colon{\bf D}^{b}({\rm MHM}(Z))\longrightarrow{\bf D}^{b}({\rm
MHM}(E))$ (6.11)

be the nearby cycle functor along $g$ for mixed Hodge modules. Applying it to
the constant Hodge module ${\mathbb{C}}^{H}_{Z}[n]\in{\rm MHM}(Z)$ on $Z$ we
obtain a mixed Hodge module $\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n])\in{\rm
MHM}(E)$ on $E=g^{-1}(0)$. Here, to be consistent with the notations of Denef-
Loeser [17] we denote ${\mathbb{Q}}_{Z}^{H}[n]$ by ${\mathbb{C}}^{H}_{Z}[n]$.
Note that its underlying ${\mathbb{Q}}$-perverse sheaf is

$\mathrm{rat}\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n])=^{p}\\!\psi_{g}({\mathbb{Q}}_{Z}[n])=\psi_{g}({\mathbb{Q}}_{Z}[n-1]).$
(6.12)

Let

$\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n])\simeq\bigoplus_{\lambda\in{\mathbb{C}}}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$
(6.13)

be the direct sum decomposition of $\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n])$ by
the generalized eigenspaces of its monodromy automorphism. Here
$\lambda\in{\mathbb{C}}$ ranges through roots of unity. Applying the proper
direct image functor

$(a_{E})_{!}\colon{\bf D}^{b}({\rm MHM}(E))\longrightarrow{\bf D}^{b}({\rm
MHM}({\rm pt}))$ (6.14)

for the map $a_{E}\colon E=g^{-1}(0)\longrightarrow{\rm pt}$ to the one point
space ${\rm pt}$ to $\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n])$, we obtain mixed
Hodge structures

$H^{j}(a_{E})_{!}\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n])\in{\rm MHM}({\rm
pt})\qquad(j\in{\mathbb{Z}})$ (6.15)

and quasi-unipotent automorphisms of them induced by the semisimple parts of
their monodromies. We thus obtain an element

$\chi_{c}(E;\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n])):=\sum_{j\in{\mathbb{Z}}}(-1)^{j}{[H^{j}(a_{E})_{!}\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n])]}$
(6.16)

of $K_{0}(\mathrm{HS}^{\mathrm{mon}})$.

######  Theorem 6.1.

(Denef-Loeser [17, Theorem 4.2.1]) In the Grothendieck ring
$K_{0}(\mathrm{HS}^{\mathrm{mon}})$ there exists an equality

$\chi_{c}(E;\psi^{H}_{g}({\mathbb{C}}^{H}_{Z}[n]))=(-1)^{n-1}\chi_{h}(\mathcal{S}_{g}).$
(6.17)

######  Proof.

First we recall the results explained in e.g. Dimca-Saito [23, Section 1.4].
For a root of unity $\lambda\in{\mathbb{C}}$ the direct sum
$\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])\oplus\psi^{H}_{g,\overline{\lambda}}({\mathbb{C}}^{H}_{Z}[n])$
corresponds to a ${\mathbb{Q}}$-perverse sheaf and has a weight filtration $W$
of a mixed Hodge module. We denote the filtration induced by it on
$\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$ by the same letter $W$. Then
for the logarithm

$N\colon\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])\longrightarrow\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$
(6.18)

of the unipotent part of the monodromy on
$\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$, there exists an isomorphism

$N^{k}\colon{\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])\overset{\sim}{\longrightarrow}{\rm
Gr}^{W}_{n-1-k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$ (6.19)

for any $k\geq 0$. Namely the filtration $W$ on the eigenvalue $\lambda$-part
$\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$ is the monodromy filtration
with center $n-1=\dim{Z}-1$. For $k\geq 0$ we define a submodule of ${\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$ by

$\displaystyle\mathrm{P}{\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$
$\displaystyle:=$ $\displaystyle{\rm Ker}{\left[N^{k+1}\colon{\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])\longrightarrow{\rm
Gr}^{W}_{n-1-k-2}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])\right]}.$

We call it the primitive part of ${\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$ with respect to
$N$. Then there exists an isomorphism

$\bigoplus_{j}{\rm
Gr}^{W}_{j}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])\simeq\bigoplus_{k\geq
0}\bigoplus_{i=0}^{k}{\left[N^{i}\mathrm{P}{\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])\right]}(i).$
(6.20)

Here $(i)$ stands for the Tate twist which raises weights by $2i$. We can
describe the structure of the primitive part $\mathrm{P}{\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])$ precisely as
follows. For a non-zero complex number $\alpha\in{\mathbb{C}}$ let
${\mathcal{L}}_{\alpha}$ be the rank one complex local system over
${\mathbb{C}}^{*}$ having the monodromy $\alpha$ around the origin
$0\in{\mathbb{C}}$. For the root of unity $\lambda$, we take
$d\in{\mathbb{Z}}_{>0},b\in{\mathbb{Z}}_{+}$ which are coprime each other such
that

$\lambda=\exp\left(2\pi\sqrt{-1}\frac{b}{d}\right)$ (6.21)

and consider the covering map

$\pi_{d}\colon{\mathbb{C}}^{*}\longrightarrow{\mathbb{C}}^{*}\qquad(t\longmapsto
t^{d})$ (6.22)

of degree $d$. Then there exists an isomorphism

${\rm
R}(\pi_{d})_{*}{\mathbb{C}}^{H}_{{\mathbb{C}}^{*}}\simeq\bigoplus_{\alpha:\alpha^{d}=1}{\mathcal{L}}_{\alpha}.$
(6.23)

Moreover, by the inclusion map $j\colon Z^{*}:=Z\setminus E\hookrightarrow Z$
and the restriction $g|_{Z^{*}}\colon Z^{*}\longrightarrow{\mathbb{C}}^{*}$ of
$g$ to $Z^{*}$ we define a constructible sheaf ${\cal F}_{\lambda}$ on $Z$ by

${\cal F}_{\lambda}:=j_{*}(g|_{Z^{*}})^{-1}{\mathcal{L}}_{\lambda^{-1}}$
(6.24)

and set

$J(\lambda):=\\{1\leq i\leq k\ |\ \lambda^{m_{i}}=1\ (\Longleftrightarrow
d|m_{i})\\}\ \subset\ \\{1,2,\dots,k\\}.$ (6.25)

Since the monodromy of the local system
$(g|_{Z^{*}})^{-1}{\mathcal{L}}_{\lambda^{-1}}$ on $Z^{*}$ around the divisor
$E_{i}\subset Z$ is $\lambda^{-m_{i}}\in{\mathbb{C}}^{*}$, we see that ${\cal
F}_{\lambda}$ is a rank one complex local system on a neighborhood of

$U(\lambda):=\bigsqcup_{I\subset J(\lambda)}E^{\circ}_{I}\quad\subset Z.$
(6.26)

The following result was proved by Saito [97, Section 3.3] and Denef-Loeser
[17, Lemma 4.2.4].

######  Lemma 6.2.

(Saito [97, Section 3.3] and Denef-Loeser [17, Lemma 4.2.4]) For any $k\geq 0$
there exists an isomorphism

${\rm rat}\ \mathrm{P}{\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])\simeq\bigoplus_{\begin{subarray}{c}I\subset
J(\lambda)\\\ \sharp I=k+1\end{subarray}}{\rm IC}_{E_{I}}({\cal
F}_{\lambda}|_{E^{\circ}_{I}})(-k),$ (6.27)

where ${\rm IC}_{E_{I}}({\cal F}_{\lambda}|_{E^{\circ}_{I}})$ stands for the
minimal extension of the perverse sheaf ${\cal
F}_{\lambda}|_{E^{\circ}_{I}}[n-\sharp I]$ on $E^{\circ}_{I}$ to $E_{I}$.

For a non-empty subset $I\subset J(\lambda)$ we define an open subset
$U_{I}\subset U(\lambda)$ of $E_{I}$ containing $E^{\circ}_{I}$ by

$U_{I}:=E_{I}\setminus\bigcup_{i\notin J(\lambda)}E_{i}=E_{I}\cap
U(\lambda)\subset E_{I}.$ (6.28)

Then by [45, Corollary 8.2.6] the minimal extension ${\rm IC}_{E_{I}}({\cal
F}_{\lambda}|_{E^{\circ}_{I}})$ is isomorphic to ${\cal
F}_{\lambda}|_{U_{I}}[n-\sharp I]$ on $U_{I}$. Moreover for the inclusion map
$j_{I}\colon U_{I}\hookrightarrow E_{I}$ we have an isomorphism

$(j_{I})_{!}({\cal F}_{\lambda}|_{U_{I}}[n-\sharp
I])\overset{\sim}{\longrightarrow}{\rm R}(j_{I})_{*}({\cal
F}_{\lambda}|_{U_{I}}[n-\sharp I]).$ (6.29)

Hence this is isomorphic to the minimal extension ${\rm IC}_{E_{I}}({\cal
F}_{\lambda}|_{E^{\circ}_{I}})$. We thus obtain isomorphisms

$\displaystyle\chi_{c}(E;\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n]))$
$\displaystyle=$ $\displaystyle\sum_{k\geq
0}\sum_{i=0}^{k}\chi_{c}(E;N^{i}\mathrm{P}{\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n])(i))$
$\displaystyle=$ $\displaystyle\sum_{k\geq 0}\chi_{c}(E;\mathrm{P}{\rm
Gr}^{W}_{n-1+k}\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n]))\cdot(\sum_{i=0}^{k}{\mathbb{C}}(i))$
$\displaystyle=$ $\displaystyle\sum_{k\geq 0}\sum_{\begin{subarray}{c}I\subset
J(\lambda)\\\ \sharp I=k+1\end{subarray}}\chi_{c}(U_{I};{\cal
F}_{\lambda}|_{U_{I}}[n-\sharp I])\cdot(\sum_{i=0}^{k}{\mathbb{C}}(-i)).$

Note that for the Lefschetz motive
$\mathbb{L}=[{\mathbb{C}}]\in{\mathcal{M}}_{{\mathbb{C}}}^{\hat{\mu}}$ we have
$\chi_{h}(\mathbb{L})={\mathbb{C}}(-1)$. Since for any non-empty subset
$I\subset J(\lambda)$ we have

$U_{I}=\bigsqcup_{I^{\prime}\subset J(\lambda),I^{\prime}\supset
I}E^{\circ}_{I^{\prime}},$ (6.30)

there exists an equality

$\chi_{c}(U_{I};{\cal F}_{\lambda}|_{U_{I}}[n-\sharp I])=(-1)^{n-\sharp
I}\sum_{I^{\prime}\subset J(\lambda),I^{\prime}\supset
I}\chi_{c}(E^{\circ}_{I^{\prime}};{\cal
F}_{\lambda}|_{E^{\circ}_{I^{\prime}}}).$ (6.31)

If we rewrite it replacing $I^{\prime}\subset J(\lambda)$ by $I$, we obtain an
equality

$\chi_{c}(E;\psi^{H}_{g,\lambda}({\mathbb{C}}^{H}_{Z}[n]))=\sum_{I\subset
J(\lambda)}\alpha_{I}\cdot\chi_{c}(E^{\circ}_{I};{\cal
F}_{\lambda}|_{E^{\circ}_{I}}),$ (6.32)

where we set

$\alpha_{I}=\sum_{k=0}^{\sharp I-1}(-1)^{n-k-1}\binom{\sharp
I}{k+1}\cdot\\{1+{\mathbb{C}}(-1)+\dots+{\mathbb{C}}(-k)\\}.$ (6.33)

Moreover, by the binomial theorem, we can easily show

$\alpha_{I}=(-1)^{n-1}(1-{\mathbb{C}}(-1))^{\sharp
I-1}=(-1)^{n-1}\chi_{h}((1-\mathbb{L})^{\sharp I-1}).$ (6.34)

Therefore it suffices to prove the equality

$\chi_{c}(E_{I}^{\circ};{\cal
F}_{\lambda}|_{E^{\circ}_{I}})=\chi_{h}({[\widetilde{E^{\circ}_{I}}]})_{\lambda}$
(6.35)

for any non-empty subset $I\subset J(\lambda)$. Here
$\chi_{h}([\widetilde{E^{\circ}_{I}}])_{\lambda}$ stands for the eigenvalue
$\lambda$-part of $\chi_{h}({[\widetilde{E^{\circ}_{I}}]})$. On an affine open
neighborhood $W\subset Z$ of a point of $E^{\circ}_{I}$ in $Z$ we decompose
$g$ as

$g=g_{1,W}\cdot(g_{2,W})^{m_{I}}\qquad(\mbox{$g_{1,W}$ is invertible on
$W$}).$ (6.36)

Then we can easily see that ${\cal F}_{\lambda}$ is isomorphic to
$(g_{1,\lambda})^{-1}{\mathcal{L}}_{\lambda^{-1}}$. We define a covering
$\widehat{E^{\circ}_{I}\cap W}$ of the Zariski open subset $E^{\circ}_{I}\cap
W$ of $E^{\circ}_{I}$ of degree $m_{I}$ by the Cartesian square:

$\textstyle{\widehat{E^{\circ}_{I}\cap
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{E^{\circ}_{I}\cap
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1,W}}$$\textstyle{{\mathbb{C}}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{m_{I}}}$$\textstyle{{\mathbb{C}}^{*}.}$
(6.37)

Namely we set

$\widehat{E^{\circ}_{I}\cap
W}:=\\{(t,z)\in{\mathbb{C}}^{*}\times(E^{\circ}_{I}\cap W)\ |\
t^{m_{I}}=g_{1,W}(z)\\}.$ (6.38)

Then there exist isomorphisms

$\displaystyle{\rm R}\Gamma_{c}(\widehat{E^{\circ}_{I}\cap
W};{\mathbb{C}}_{\widehat{E^{\circ}_{I}\cap W}})$ $\displaystyle\simeq$
$\displaystyle{\rm R}\Gamma_{c}(E^{\circ}_{I}\cap W;{\rm
R}p_{*}{\mathbb{C}}_{\widehat{E^{\circ}_{I}\cap W}})$ $\displaystyle\simeq$
$\displaystyle{\rm R}\Gamma_{c}(E^{\circ}_{I}\cap W;(g_{1,W})^{-1}{\rm
R}(\pi_{m_{I}})_{*}{\mathbb{C}}_{{\mathbb{C}}^{*}}).$

Since $\widehat{E^{\circ}_{I}\cap W}$ is isomorphic to the part
$\widetilde{E^{\circ}_{I}\cap W}$ of the unramified Galois covering
$\widetilde{E^{\circ}_{I}}$ over $E^{\circ}_{I}\cap W\subset E^{\circ}_{I}$ by
the map $(t,z)\mapsto(t^{-1},z)$, we see also that the eigenvalue
$\lambda$-part of ${\rm R}\Gamma_{c}(\widetilde{E^{\circ}_{I}\cap
W};{\mathbb{C}}_{\widetilde{E^{\circ}_{I}\cap W}})$ corresponds to ${\rm