#  $L^{2}$-representation of Hodge Modules

Junchao Shentu<EMAIL_ADDRESS>School of Mathematical Sciences, University
of Science and Technology of China, Hefei, 230026, China  and  Chen Zhao
<EMAIL_ADDRESS>School of Mathematical Sciences, University of Science and
Technology of China, Hefei, 230026, China

###### Abstract.

Over an arbitrary compact complex space or an arbitrary germ of complex space
$X$, we provide fine resolutions of pure Hodge modules with strict supports
$IC_{X}(\mathbb{V})$ via differential forms with locally $L^{2}$ boundary
conditions. When $\mathbb{V}=\mathbb{C}_{X_{\rm reg}}$ is the trivial
variation of Hodge structure, we give a solution to a Cheeger-Goresky-
MacPherson type conjecture: For any compact complex space $X$, there is a
complete hermitian metric $ds^{2}$ on $X_{\rm reg}$ such that there is a
canonical isomorphism

$H^{i}_{(2)}(X_{\rm reg},ds^{2})\simeq IH^{i}(X),\quad\forall i.$

Such metric $ds^{2}$ could be Kähler if $X$ is a Kähler space. As an
application, we give a differential geometrical proof of the Kähler package of
the hypercohomology of pure Hodge modules. We also prove the Kähler version of
Kashiwara’s conjecture in the absolute case.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 The construction of the distinguished metric
    3. 1.3 Strategy of the proof of the main theorem
  2. 2 Intersection Complex on Complex Spaces
  3. 3 $L^{2}$-de Rham Complex and $L^{2}$-Cohomology
    1. 3.1 $L^{2}$-cohomology
    2. 3.2 $L^{2}$-de Rham complex
  4. 4 Constructibility of $L^{2}$-de Rham Complex
    1. 4.1 $L^{2}$-Poincaré Lemma and $L^{2}$-Thom isomorphism
    2. 4.2 A formal criteria
    3. 4.3 Whitney stratified desingularization
    4. 4.4 Distinguished metric
    5. 4.5 Harmonic bundles
      1. 4.5.1 pluriharmonic bundles
      2. 4.5.2 $0$-tame harmonic bundles
  5. 5 Local Geometry of Singularities
    1. 5.1 Basic setting
    2. 5.2 Distinguished neighborhoods
    3. 5.3 Consistency of local $L^{2}$-cohomologies
  6. 6 Local Vanishings
    1. 6.1 Modify the boundary $\partial U_{\delta}$
    2. 6.2 An estimate for pluriharmonic bundle
    3. 6.3 Local vanishing
  7. 7 Proof of the Main Theorem
    1. 7.1 Proof of ${\bf IC}_{n-1}\Rightarrow{\bf VD}_{n}$
    2. 7.2 Proof of ${\bf VD}_{n}\Rightarrow{\bf IC}_{n}$
  8. 8 Applications: Kähler Package on the Intersection cohomology
    1. 8.1 Poincaré duality and Verdier duality
    2. 8.2 Lefschetz package
    3. 8.3 Pure Hodge structure

##  1\. Introduction

This paper grows out of an attempt to generalize the Topology-Geometry-
Analysis nature of the Hodge theory to compact Kähler spaces with
coefficients. The classical Hodge theory for a compact Kähler manifold $X$ can
be summarized by the graph

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\hbox{\kern
3.0pt\raise-10.30261pt\hbox{$\textstyle{\begin{subarray}{c}{\bf Geometry:}\\\
\textrm{Holomorphic de Rham complex }\Omega_{X}^{\bullet};\\\ \textrm{Pure
Hodge structures on
}H^{\ast}(X).\end{subarray}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
81.72488pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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0.0pt\hbox{\hbox{\kern
3.0pt\raise-12.16052pt\hbox{$\textstyle{\begin{subarray}{c}{\bf Analysis:}\\\
\textrm{$C^{\infty}$ de Rham complex }\mathscr{A}_{X}^{\bullet};\\\
\textrm{Hodge decomposition of differential forms;}\\\ \textrm{Harmonic
forms.}\end{subarray}}$}}}}}\ignorespaces}}}}\ignorespaces.$

This picture is generalized by P. Deligne [Delign1974] to algebraic varieties
which may admits singularities:

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0.0pt\hbox{\hbox{\kern
3.0pt\raise-11.71652pt\hbox{$\textstyle{\begin{subarray}{c}{\bf Geometry:}\\\
\textrm{Filtered de Rham Complex
}\tilde{\Omega}_{X}^{\bullet}\textrm{\cite[cite]{[\@@bibref{}{DuBois1981}{}{}]}};\\\
\textrm{Mixed Hodge structures on
}H^{\ast}(X).\end{subarray}}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern
3.0pt\raise-9.7222pt\hbox{$\textstyle{\begin{subarray}{c}{\bf Analysis:}\\\
\textrm{Differential forms; Hodge and}\\\ \textrm{weight filtrations
\cite[cite]{[\@@bibref{}{Deligne1971_2,Bernard-
Vincenzo2006}{}{}]}.}\end{subarray}}$}}}}}\ignorespaces}}}}\ignorespaces.$

In the analytic counterpart, the differential forms live either on a hyper-
covering of $X$ [Deligne1971_2, Delign1974] or a system of desingularization
diagrams of $X$ [Bernard-Vincenzo2006], instead of on $X$ itself. Moreover,
the Hodge decomposition theorem and the theory of harmonic integrals are
missing on the analytic side.

Motivated by the theory of weight in characteristic $p>0$ [BBD], the theory of
mixed Hodge structures and variation of these objects
[Cattani_Kaplan_Schmid1986, Schmid1973, Steenbrink-Zucker1985] are integrated
by M. Saito into his theory of mixed Hodge modules [MSaito1988, MSaito1990].
Saito’s theory has become an indispensable tool in the study of complex
geometry, arithmetic geometry and representation theories. Conceptually, they
establish the Hodge theory to the most generality so far:

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0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern
3.0pt\raise-9.7222pt\hbox{$\textstyle{\begin{subarray}{c}{\bf Analysis:}\\\
\textrm{Partially known
\cite[cite]{[\@@bibref{}{Zucker1979,Cattani_Kaplan_Schmid1987,Kashiwara_Kawai1987,Saper_Stern1990,Looijenga1988,Saper1992,Ohsawa1991}{}{}]}
etc.}\\\ \textrm{Mostly
missing.}\end{subarray}}$}}}}}\ignorespaces}}}}\ignorespaces,$

where RH stands for the Riemann-Hilbert correspondence. Although the theory of
mixed Hodge modules has been well established, its analytic counterpart is
much less known. The first problem that we encounter is to find a nice
analytic perspective of pure Hodge modules with strict support111Every pure
Hodge module is a (locally finite) direct sum of pure Hodge modules with
strict supports ([MSaito1988, §5.16]).:

######  Problem 1.1.

Let $X$ be a complex space and $M$ a pure Hodge module with strict support
$X$. Assume that the underlying perverse sheaf of $M$ is $IC_{X}(\mathbb{V})$
where $\mathbb{V}$ is a polarized variation of Hodge structure over some dense
Zariski open subset $X^{o}\subset X_{\rm reg}$. Find a fine resolution
$\mathscr{D}^{\bullet}$ of $IC_{X}(\mathbb{V})$ such that

  1. (1)

$\mathscr{D}^{\bullet}$ is consist of differential forms in some reasonable
sense. E.g. smooth forms with combinatory or analytic boundary conditions.

  2. (2)

When $X$ is compact, there is a Hodge decomposition theorem for elements in
$\mathscr{D}^{\bullet}$. Moreover every cohomology class of
$\mathbb{H}^{\ast}(X,IC_{X}(\mathbb{V}))$ is uniquely represented by a
harmonic form.

  3. (3)

When $X$ is compact and admits a Kähler hermitian metric, there is a natural
$(p,q)$-decomposition of $\mathscr{D}^{\bullet}$ which gives a pure Hodge
structure on $\mathbb{H}^{\ast}(X,IC_{X}(\mathbb{V}))$.

It is a general philosophy that Hodge modules are closely related to
$L^{2}$-differential forms. For one thing, the classical Hodge theory is
established via the $L^{2}$-technique. On the other hand, in Saito’s theory of
Hodge modules, the study of the cohomology of Hodge modules (e.g. the proof of
Saito’s decomposition theorem) is reduced to S. Zucker’s work [Zucker1979] on
the $L^{2}$-representation of Hodge modules over a smooth algebraic curve. In
the era when intersection cohomology was defined, J. Cheeger, M. Goresky and
R. MacPherson made a conjecture that the intersection cohomology is
represented by a certain $L^{2}$-cohomology (see Conjecture 1.3 below), so
that the intersection cohomology of a projective variety owns a package of
theorems including the Hard Lefschetz theorem and the existence of a pure
Hodge structure. These properties are proved by Beĭlinson-Bernstein-Deligne-
Gabber [BBD] and Saito [MSaito1988, MSaito1990] by alternative methods.

In recent years, the technique of mixed Hodge modules and $L^{2}$-methods both
play important roles in the development of complex and algebraic geometry. It
is interesting to search for their relations and combine them together.

According to this philosophy, a promising candidate of such a
$\mathscr{D}^{\bullet}$ is the complex of locally $L^{2}$-$\mathbb{V}$-valued
differential forms. Let $ds^{2}$ be a metric on $X^{o}$ and denote by
$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ the complex of sheaves on $X$
consisting of $\mathbb{V}$-valued measurable differential forms $\alpha$ such
that $\alpha$ and $\nabla\alpha$ are locally square integrable over $X$
(including the singular points) with respect to $ds^{2}$ and the Hodge metric
$h$ on $\mathbb{V}$. Compared to other boundary conditions (combinatory or
$L^{p}$, $p\neq 2$), this kind of complex has the following advantages.

  1. (1)

For an amount of interesting metrics $ds^{2}$,
$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ is a complex of fine sheaves
(Lemma 3.5).

  2. (2)

When $X$ is compact, $\Gamma(X,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})$
automatically admits a Hodge decomposition theorem whenever
$H^{\ast}(\Gamma(X,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}))$ are finite
dimensional ([Kashiwara_Kawai1987, Theorem A.2.2]).

  3. (3)

When $X$ is compact and $ds^{2}$ is a complete Kähler metric,
$H^{\ast}(\Gamma(X,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}))$ admits a
natural pure Hodge structure of the right weight as long as it is finite
dimensional ([Kashiwara_Kawai1987, Theorem 6.2.3] or [Zucker1979, §7]). As the
classical case, the Hodge structure is induced from the total bi-degree
decomposition of differential forms.

Now Problem 1.1 turns to a precise one, which is the main subject of this
paper.

######  Problem 1.2.

Let $X$ be a complex space and let $(\mathbb{V},h)$ underly an
$\mathbb{R}$-polarized variation of Hodge structure over some dense Zariski
open subset $X^{o}\subset X_{\rm reg}$ with quasi-unipotent local monodromies.
Is there a hermitian (resp. Kähler) metric $ds^{2}$ on $X^{o}$ such that

  1. (1)

$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ is a complex of fine sheaves
and

  2. (2)

$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ is quasi-isomorphic to
$IC_{X}(\mathbb{V})$ in $D(X)$, the derived category of sheaves of
$\mathbb{C}$-vector spaces?

Problem 1.2 has been solved in some cases.

  * •

When $ds^{2}$ is only required to be a Riemannian metric and
$\mathbb{V}=\mathbb{C}_{X_{\rm reg}}$ is the trivial variation of Hodge
structure, Problem 1.2 is solved by Cheeger [Cheeger1980]. Cheeger’s work is
the starting point of the analytical study of perverse sheaves.

  * •

When $X$ is a smooth algebraic curve and $ds^{2}$ is a hermitian metric
asymptotic of Poincaré type along the isolated points $X\backslash X^{o}$,
Problem 1.2 is solved by Zucker [Zucker1979]. Zucker’s result is generalized
by Jost-Yang-Zuo[Jost_Zuo2007] to the case when the coefficients are in a
unipotent harmonic bundle.

  * •

More generally when $X$ is a complex manifold, $D:=X\backslash X^{o}$ is a
normal crossing divisor and $ds^{2}$ is a Kähler metric asymptotic of Poincaré
type along $D$, Problem 1.2 is solved by Cattani-Kaplan-Schmid
[Cattani_Kaplan_Schmid1987] and Kashiwara-Kawai [Kashiwara_Kawai1987]
respectively.

  * •

When $X$ is Kähler and admits only isolated singularities, $X^{o}=X_{\rm reg}$
and $\mathbb{V}=\mathbb{C}_{X_{\rm reg}}$ is the trivial variation of Hodge
structure, L. Saper [Saper1985, Saper1992] constructs a family of complete
Kähler metrics $ds^{2}$ which solves Problem 1.2. Based on Saper’s work, T.
Ohsawa [Ohsawa1991, Ohsawa1993] settles this case when $ds^{2}$ is a hermitian
metric on $X$.

  * •

When $X^{o}$ is a locally symmetric variety $G/\Gamma$, $X$ is its Satake-
Baily-Borel compactification, $ds^{2}$ is the canonical complete metric and
$(\mathbb{V},h)$ is a hermitian local system associated to a representation of
$G$, Problem 1.2 is a conjecture made by Zucker [Zucker1982] and is proved by
E. Looijenga[Looijenga1988] and Saper-Stern[Saper_Stern1990] respectively.

In the case when $X^{o}=X_{\rm reg}$ and $\mathbb{V}=\mathbb{C}_{X_{\rm reg}}$
is the trivial variation of Hodge structure, Problem 1.2 is closely related to
the famous

######  Conjecture 1.3 (Cheeger-Goresky-Macpherson, [CGM1982]).

Let $X\subset\mathbb{P}^{N}$ be a projective variety and let $ds^{2}_{\rm FS}$
be the Fubini-Study metric on $X_{\rm reg}$. Then there is an isomorphism

$H^{\ast}_{(2)}(X_{\rm reg},ds^{2}_{\rm FS})\simeq IH^{\ast}(X,\mathbb{C}).$

Conjecture 1.3 is solved affirmatively by W. C. Hsiang and V. Pati
[Hsiang_Pati1985] for the case of normal surfaces, and by Ohsawa [Ohsawa1991]
for the case when $X$ admits only isolated singularities. The general case is
still unknown.

###  1.1. Main results

The aim of this paper is to give a solution to Problem 1.2 in the most general
situation. Our motivation is to relate the theory of Hodge modules and
$L^{2}$-techniques. For the purpose of applications we focus on complete
(Kähler) metrics.

Let $X$ be a complex space of pure dimension and $X^{o}\subset X_{\rm reg}$ a
dense Zariski open subset. We define a family of hermitian metrics on $X^{o}$
which is called distinguished associated to $(X,X^{o})$ (Definition 4.12).
Such kind of metric is complete and of finite volume locally at every point of
$X$ (Corollary 5.5). The distinguished metric exists in either of the
following cases (Lemma 4.13).

  1. (1)

$(X,X\backslash X^{o})$ is compactable (e.g. when $X$ is compact or
$(X,X^{o})$ is a pair of algebraic varieties).

  2. (2)

$X$ is a germ of complex space.

When $X$ is a Zariski open subset of a compact Kähler space or is a germ of a
complex space, $(X,X^{o})$ admits a Kähler distinguished metric.

The main result of this paper is

######  Theorem 1.4 (= Theorem 7.1).

Let $X$ be a complex space of pure dimension and $X^{o}\subset X_{\rm reg}$ a
dense Zariski open subset. Let $ds^{2}$ be a distinguished metric associated
to $(X,X^{o})$. Let $(\mathbb{V},h)$ be a $0$-tame harmonic bundle on
$(X,X^{o})$. Then

  1. (1)

$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ is a complex of fine sheaves.

  2. (2)

There is a canonical quasi-isomorphism

$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}\simeq IC_{X}(\mathbb{V})$

in $D(X)$.

Moreover if $X$ is compact, there is a natural isomorphism

$H^{k}_{(2)}(X^{o},\mathbb{V};ds^{2},h)\simeq
IH^{k}(X,\mathbb{V}):=\mathbb{H}^{k}(X,IC_{X}(\mathbb{V}))$

for every $k$. In particular, $H^{k}_{(2)}(X^{o},\mathbb{V};ds^{2},h)$ is
finite dimensional for every $k$ when $X$ is compact.

A $0$-tame harmonic bundle on $(X,X^{o})$ (Definition 4.20) is a pluriharmonic
bundle on $X^{o}$ whose metric satisfies a certain constrain of the asymptotic
behavior near $X\backslash X^{o}$. The class of $0$-tame harmonic bundles
contains (Remark 4.22) the underlying hermitian local system of an
$\mathbb{R}$-polarized variation of Hodge structure with quasi-unipotent local
monodromies and semisimple local systems (do not necessarily have quasi-
unipotent local monodromies) endowed with the (canonical) Corlette-Jost-Zuo
metric ([Jost_Zuo1997],[Mochizuki20072]). Hence Theorem 1.4 gives an
$L^{2}$-representation of (the $\mathscr{D}$-module part of) twisted
$\mathscr{D}$-modules, which are introduced by C. Sabbah [Sabbah2005] as
generalizations of Hodge modules. In particular, Theorem 1.4 gives a solution
to Problem 1.2.

By taking $\mathbb{V}=\mathbb{C}_{X_{\rm reg}}$ we solve Conjecture 1.3 for
the distinguished metric.

######  Corollary 1.5.

Let $X$ be a compact complex space, then there exists a complete hermitian
metric $ds^{2}$ on $X_{\rm reg}$ such that there is a natural isomorphism

$H^{\ast}_{(2)}(X_{\rm reg},ds^{2})\simeq IH^{\ast}(X,\mathbb{C}).$

When $X$ admits a Kähler hermitian metric, $ds^{2}$ could be Kähler.

######  Remark 1.6.

  1. (1)

Compared to the Fubini-Study metric in Conjecture 1.3, the distinguished
metric is complete and it provides good Hodge theoretic properties (c.f.
[Zucker1979, §7]). However, the distinguished metric is not canonical.

  2. (2)

When $X$ admits only isolated singularities, the distinguished metric is
different from Saper’s distinguished metric [Saper1992] since the
distinguished metric in this paper admits bounded potential functions locally.
This property ensures a MacPherson’s conjecture type result for the
$L^{2}$-representation of Hodge modules (Theorem 8.6).

  3. (3)

When $X$ is a compact manifold and $D:=X\backslash X^{o}$ is a normal crossing
divisor, the distinguished metric is not asymptotic of Poincaré type along
$D$. The technique of this paper does not work on Poincaré type metric
directly. We will deal with this natural metric in another paper.

The $L^{2}$-representation of the intersection complex allows us, as it is
hoped, to prove the Kähler package of the intersection cohomology with
coefficients in a variation of Hodge structure or the hypercohomology of a
semisimple local system more generally. This package of theorems includes

  1. (1)

The Poincaré duality (Theorem 8.1, 8.2) and its interpretation by integration.

  2. (2)

Hard Lefschetz theorem, Lefschetz decomposition (Theorem 8.3). This is a
generalization of C. Simpson’s Hard Lefschetz theorem for semisimple local
systems to semisimple perverse sheaves, which is the Kähler version of a
special case of Kashiwara’s conjecture [Kashiwara1998].

Precisely we show that (Corollary 8.4) for a semisimple perverse sheaf $P$ on
a compact Kähler space $X$, the Hard Lefschetz theorem holds for the
hypercohomology $\mathbb{H}^{\ast}(X,P)$. The proofs use the $L^{2}$-harmonic
representatives of cohomology classes followed by standard arguments. It is
important to have $ds^{2}$ being complete since one needs the Kähler identity
in the sense of unbounded operators ([Zucker1979, (7.5)]).

  3. (3)

The existence of the pure Hodge structure on $\mathbb{H}^{\ast}(X,M)$ via the
$(p,q)$-decomposition of $L^{2}$-forms ([Kashiwara_Kawai1987, §6],
[Zucker1979, §7]) where $M$ is a pure Hodge module. We also make a conjecture
that this $L^{2}$-Hodge structure is compatible with Saito’s (Conjecture 8.5).
Evidences for this conjecture will be provided in §8.3.

###  1.2. The construction of the distinguished metric

One of the main difficulties for the $L^{2}$-representation over a general
complex analytic singularity, compared to isolated singularities, is the
problem of equisingularity. The first step towards the $L^{2}$-representation
of the intersection complex is to find a stratification along which
$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ is constructible. In order to
achieve this, we require the metrics $ds^{2}$ and $h$ to be quasi-isotrivial
along each stratum. In [Saper_Stern1990, Looijenga1988, Kashiwara_Kawai1987,
Cattani_Kaplan_Schmid1987], the singularity loci is naturally stratified so
that $ds^{2}$ and $h$ are quasi-isotrivial along each stratum in a rather nice
way. More precisely, for every stratum $S$ and every point $x\in S$, there is
a neighborhood $x\in U_{x}$ in $X$, a contractible precompact neighborhood
$x\in U_{S,x}$ in $S$, an analytic subspace $C_{x}$ which intersects
transversally with $S$ at the single point $x$ and a $C^{\infty}$
diffeomorphic fibration

(1.1) $\displaystyle\alpha_{x}:U_{S,x}\times C_{x}\simeq U_{x}$

such that

(1.2) $\displaystyle\alpha_{x}^{\ast}ds^{2}\sim
ds^{2}_{S}+ds^{2}|_{C_{x}},\quad\alpha_{x}^{\ast}h\sim q^{\ast}(h|_{C_{x}})$

for some hermitian metric $ds^{2}_{S}$ on $S$. Here $q:U_{S,x}\times C_{x}\to
C_{x}$ is the projection. This ensures that the cohomology sheaves of
$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ are locally trivial along the
stratum $S$.

However, due to the possible appearance of Whitney’s four plane
$x(x+y)(x-y)(x-zy)=0$ ([Goresky2012, p.g. 470]), it is too much to hope
$\alpha_{x}$ to be $C^{\infty}$ for a general analytic singularity. For a
hermitian metric $ds^{2}_{0}$ on $X$, i.e. locally $ds^{2}_{0}$ is quasi-
isometric to the euclidean metric via a local embedding
$X\subset\mathbb{C}^{N}$, T. Mostowski [Mostowski1985] introduces the
Lipschitz stratification so that $ds^{2}_{0}$ is locally quasi-isotrivial
along each stratum. In this case, (1.1) is a bi-Lipschitz homeomorphism.

For other interesting metrics, a bi-Lipschitz fiberation is not enough to
ensure the quasi-isotriviality (1.2). One way to solve this difficulty is to
use the resolution of singularity. As the case of Whitney’s four plane, one
may resolve the singularity into a normal crossing divisor to get a
$C^{\infty}$ fiberation. Although this fiberation may not desend to a
stratified fiberation on $X$, this causes no serious problem for this paper
since the $L^{2}$-cohomology is unchanged under bimeromorphic transformations.

Therefore the problem turns to find a stratification which admits an
equisingular desingularization. In this paper, for every pair $(X,X^{o})$ we
introduce a Whitney stratification $0\subset X_{0}\subset\dots\subset X_{n}=X$
and a desingularization $\pi:\widetilde{X}\to X$ so that the preimage of every
$X_{i}$ is a simple normal crossing divisor. Moreover, along each stratum the
fibers $\pi^{-1}(x)$ are smoothly stratified isotrivial. Such a pair
$(\\{X_{i}\\},\pi:\widetilde{X}\to X)$ is called a Whitney stratified
desingularization (Definition 4.6). It exists when $X$ is compact or is a germ
of complex space (Theorem 4.7). For this construction, along each stratum the
$C^{\infty}$ fiberation (1.1) exists on the smooth model $\widetilde{X}$
(Proposition 4.9), rather than on $X$. A general phenomenon is that every
motivic hermitian metric (a metric defined by holomorphic functions) is quasi-
isotrivial along the $C^{\infty}$ fibration associated to some Whitney
stratified desingularization, as long as its quasi-isometric local expression
is independent of the choice of $C^{\infty}$ complex coordinates. This is
useful to produce many interesting constructible $L^{2}$-de Rham complexes
(Proposition 4.5).

The distinguished metric could be described as follows (Definition 4.12). Let
$X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. A hermitian metric on $X^{o}$ is called distinguished associated to
$(X,X^{o})$ if for every point $x\in X$ there is a neighborhood $U$ and a
Whitney stratified desingularization $\pi:\widetilde{U}\to U$ of $(U,X^{o}\cap
U)$ such that $ds^{2}$ is quasi-isometric to

$\displaystyle\sum_{i=1}^{r}\frac{dz_{i}d\overline{z}_{i}}{|z_{i}|^{2}\log^{2}|z_{1}\cdots
z_{r}|^{2}\log^{2}(-\log|z_{1}\cdots
z_{r}|^{2})}+\frac{\sum_{i=1}^{n}dz_{i}d\overline{z}_{i}}{-\log|z_{1}\cdots
z_{r}|^{2}\log^{2}(-\log|z_{1}\cdots z_{r}|^{2})}+\pi^{\ast}\omega_{0}.$

Here $\omega_{0}$ is a hermitian metric on $X$ and $(z_{1},\dots,z_{n})$ is a
holomorphic coordinate on $\widetilde{U}$ such that $\pi^{-1}(U\backslash
X^{o})=\\{z_{1}\cdots z_{r}=0\\}$. The distinguished metric witnesses the
stratification $\\{X_{i}\\}$, which is necessary to construct an
$L^{2}$-representation of the intersection complex.

###  1.3. Strategy of the proof of the main theorem

One of the consequences of Theorem 1.4 is the Verdier duality

$\displaystyle\mathbb{D}(\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}[\dim
X])\simeq\mathscr{D}^{\bullet}_{X,\mathbb{V}^{\ast};ds^{2},h^{\ast}}[\dim
X]\in D(X).$

Here $\mathbb{D}$ is the Verdier dual operator on $D(X)$. Locally at every
point $x\in X$ this implies the local duality

######  Theorem 1.7 (Locally Duality, Theorem 8.1).

Notations as in Theorem 1.4. Let $x\in X$ be a point. Then for every $0\leq
k\leq 2\dim X$ there is a duality

$\displaystyle\varinjlim_{x\in U}\mathbb{H}^{2\dim
X-k}(U,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})\simeq\varinjlim_{x\in
U}\left(\mathbb{H}^{k}_{\rm
cpt}(U,\mathscr{D}^{\bullet}_{X,\mathbb{V}^{\ast};ds^{2},h^{\ast}})\right)^{\ast},$

where $U$ ranges over all open neighborhoods of $x$. Moreover both sides of
the isomorphism are finite dimensional.

Denote the statements

  * •

${\bf IC}_{n}$: Theorem 1.4 holds when $\dim X\leq n$.

  * •

${\bf VD}_{n}$: Theorem 1.7 holds for $\dim X\leq n$ and $k\leq\dim X$, under
the additional assumptions that

$\varinjlim_{x\in U}\left(\mathbb{H}^{k}_{\rm
cpt}(U,\mathscr{D}^{\bullet}_{X,\mathbb{V}^{\ast};ds^{2},h^{\ast}})\right)^{\ast}$

is finite dimensional for every $k\leq\dim X$ and

$\varinjlim_{x\in U}\left(\mathbb{H}^{\dim X}_{\rm
cpt}(U,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})\right)^{\ast}$

is finite dimensional.

The proof of Theorem 1.4 follows from a bi-induction procedure:

Initial Step:

${\bf IC}_{0}$. This case is trivial.

Bi-induction Step:

${\bf IC}_{n-1}\Rightarrow{\bf VD}_{n}$, ${\bf VD}_{n}\Rightarrow{\bf
IC}_{n}$.

Notice that in the induction procedure, non-irreducible complex space appears
even $X$ is irreducible. Therefore in Theorem 1.4 we allow the space to have
many components.

Now let us explain the ideas in the proof of the bi-induction steps.

${\bf IC}_{n-1}\Rightarrow{\bf VD}_{n}$ follows from an abstract
$L^{2}$-Poincaré duality (Proposition 3.1). The main difficulty is to show
that $H^{\dim X}_{(2),\rm max}(U,\mathbb{V};ds^{2},h)$ is finite dimensional
for a certain neighborhood $U$ of $x$. Our strategy is to use an
$L^{2}$-Mayer-Vietoris sequence to cut this cohomology into two parts: one is
$\mathbb{H}^{\dim X}_{\rm
cpt}(U,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2}_{\pi},h})$ which is finite
dimensional by the assumption. The other is the $L^{2}$-cohomology over the
boundary $\partial U$. Since $\partial U$ is a compact stratified space which
intersects with $X$ on the strata of positive dimensions, ${\bf IC}_{n-1}$
ensures that the $L^{2}$-cohomology over $\partial U$ is finite dimensional.

${\bf VD}_{n}\Rightarrow{\bf IC}_{n}$ is divided into six steps.

  1. (1)

Reduce the problem ${\bf IC}_{n}$ to the case of a germ of complex space $0\in
X\subset\mathbb{C}^{N}$ (Corollary 2.4). This is a consequence of the Goresky-
MacPherson-Deligne criterion (Proposition 2.2).

  2. (2)

Show that the distinguished metric and the harmonic metric are
bimeromorphically stratified (Definition 4.4) along the fibering
stratification of $X\subset\mathbb{C}^{N}$ with respect to some Whitney
stratified desingularization of $X$. This implies that the cohomology sheaves
of $\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ are weakly constructible.

  3. (3)

According to Step (2), the $L^{2}$-Poincaré lemma (Lemma 4.1) and the
$L^{2}$-Thom isomorphism (Lemma 4.2) reduce the problem (via Proposition 4.5)
to two types of vanishings: Notations as in Theorem 1.4. Assume that $0\in
Z:=X$ is a germ of complex space. Then there are vanishings

(1.3) $\displaystyle\varinjlim_{0\in
U}\mathbb{H}^{k}(U,\mathscr{D}^{\bullet}_{Z,\mathbb{V};ds^{2},h})=0,\quad\forall
k\geq\dim Z$

and

(1.4) $\displaystyle\varprojlim_{0\in U}\mathbb{H}^{k}_{\rm
cpt}(U,\mathscr{D}^{\bullet}_{Z,\mathbb{V};ds^{2},h})=0,\quad\forall k\leq\dim
Z.$

Here $U$ ranges over the open neighborhoods of $0$.

  4. (4)

Assuming ${\bf VD}_{n}$, it suffices to prove the vanishing (1.4) for both
$\mathbb{V}$ and $\mathbb{V}^{\ast}$. Since $(\mathbb{V}^{\ast},h^{\ast})$ is
also a $0$-tame harmonic bundle, it is enough to show (1.4) for an arbitrary
$(\mathbb{V},h)$. This is the only place where we assume ${\bf VD}_{n}$.

  5. (5)

(1.4) is proved via calculating the $L^{2}$-cohomologies of
$U_{\delta}:=\\{z\in Z|\|z\|<\delta\\}$, $0<\delta\ll 1$. The key feature of
$U_{\delta}$ is that its $L^{2}$-cohomologies are independent of $\delta>0$.
I.e. the ’extension by zero’ morphism

(1.5) $\displaystyle\mathbb{H}^{k}_{\rm
cpt}(U_{\delta^{\prime}},\mathscr{D}^{\bullet}_{Z,\mathbb{V};ds^{2},h})\to\mathbb{H}^{k}_{\rm
cpt}(U_{\delta},\mathscr{D}^{\bullet}_{Z,\mathbb{V};ds^{2},h})$

is an isomorphism for every $0<\delta^{\prime}<\delta\ll 1$ (Proposition 5.6).
Hence (1.4) is equivalent to the vanishing

(1.6) $\displaystyle\mathbb{H}^{k}_{\rm
cpt}(U_{\delta},\mathscr{D}^{\bullet}_{Z,\mathbb{V};ds^{2},h})=0,\quad\forall
k\leq\dim Z.$

We prove this vanishing by solving the equation $\nabla\beta=-\alpha$ for
$\nabla\alpha=0$ in the space of $L^{2}$ differential forms whose support is
compact in $U_{\delta}$. In order to do this, we introduce another cofinal
system of neighborhoods $V_{\delta}$, $0<\delta\ll 1$. The main feature of
$V_{\delta}$ is the existence of a fiberation

(1.7) $\displaystyle V_{\delta}\cap Z^{o}\simeq(0,\delta)\times L$

for some real submanifold $L\subset Z^{o}$, such that the distinguished metric
satisfies the estimate (Proposition 5.4)

(1.8)
$\displaystyle\frac{d\rho^{2}}{\rho^{2}\log^{2}\rho\log^{2}(-\log\rho)}+\frac{ds^{2}_{L}}{\log^{2}\rho\log^{2}(-\log\rho)}\lesssim
ds^{2}\lesssim\frac{d\rho^{2}}{\rho^{2}\log^{2}\rho\log^{2}(-\log\rho)}+ds^{2}_{L}.$

Here $ds^{2}_{L}$ is a Riemannian metric on $L$ which can be extended smoothly
to the boundary.

To get rid of the complicated boundary condition ([Hormander1965, Proposition
2.1.1]) caused by the incompleteness on $\partial U_{\delta}$, we make a
modification on $ds^{2}$ along $\partial U_{\delta}$, denoted by
$ds^{2}_{\delta}$, so that $ds^{2}_{\delta}$ is complete and
$\|-\|_{ds^{2},h}\lesssim\|-\|_{ds^{2}_{\delta},w_{\delta}h}$ for some weight
function $w_{\delta}$. If ${\rm supp}\alpha$ is compact in $U_{\delta}$, the
modification on $\partial U_{\delta}$ does not affect the analytical behavior
of $\alpha$. This is one of the reasons why we choose to prove (1.4) instead
of (1.3).

Due to the completeness of $ds^{2}_{\delta}$, the equation
$\nabla\beta=-\alpha$ is solvable in the Hilbert space of $L^{2}$ differential
forms with respect to $ds^{2}_{\delta}$ and $w_{\delta}h$ if one shows the
estimate ([Ohsawa2018, §2.1.2 Lemma 2.1])

(1.9) $\displaystyle|(\alpha,u)|_{ds^{2}_{\delta},w_{\delta}h}\leq
C\|\nabla^{\ast}_{\delta}u\|_{ds^{2}_{\delta},w_{\delta}h},\quad\forall u\in
A^{k}_{\rm cpt}(U_{\delta}\cap Z^{o},\mathbb{V}),\forall k\leq\dim Z.$

Here $\nabla^{\ast}_{\delta}$ is the Hilbert adjoint of $\nabla$ with respect
to $(ds^{2}_{\delta},w_{\delta}h)$. Thanks to (1.5), we may assume that ${\rm
supp}\alpha$ lies in some $V_{\delta^{\prime\prime}}\subset U_{\delta}$. Let
$\alpha=\alpha_{1}+d\rho\wedge\alpha_{0}$ be the decomposition according to
(1.7). There is a simple observation that the integration of $\alpha_{0}$
along $(0,\delta^{\prime\prime})$ in (1.7) gives a formal solution to the
equation $\nabla\beta=-\alpha$. However this solution fails to be square
integrable with respect to $(ds^{2}_{\delta},w_{\delta}h)$. Our strategy is to
deform $(ds^{2}_{\delta},w_{\delta}h)$ to
$(ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h)$, $\epsilon\in(0,1)$ so that

$\|\alpha\|_{ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h}\lesssim\|\alpha\|_{ds^{2}_{\delta},w_{\delta}h}.$

The asymptotic behavior (1.8) and the $0$-tameness of the harmonic metric $h$
ensure that

$\beta\in L_{(2)}^{k-1}(U_{\delta}\cap
Z^{o},\mathbb{V};ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h)$

for every $\epsilon\in(0,1)$. Hence there is an estimate

(1.10)
$\displaystyle|(\alpha,u)|_{ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h}\leq
C_{\epsilon}\|\nabla^{\ast}_{\delta,\epsilon}u\|_{ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h},\quad\forall
u\in A^{k}_{\rm cpt}(U_{\delta}\cap Z^{o},\mathbb{V}),\forall k\leq\dim Z.$

Here $\nabla^{\ast}_{\delta,\epsilon}$ is the Hilbert adjoint of $\nabla$ with
respect to $(ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h)$ and

$C_{\epsilon}\leq\|\beta\|_{ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h}.$

Although $\|\beta\|_{ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h}$ may tend
to infinity when $\epsilon\to 0$, a twisted Donelly-Fefferman type estimate
(Theorem 6.2) for $(ds^{2}_{\delta,\epsilon},w_{\delta,\epsilon}h)$ shows that
the constants $C_{\epsilon}$ could be chosen in a way independent of
$\epsilon\in(0,1)$. Hence (1.9) is the limit of (1.10). This solves the
equation $\nabla\beta=-\alpha$.

To get a solution of $\nabla\beta=-\alpha$ with compact supports, one may
shrink $U_{\delta}$ first and use the zero extension of the solution. This is
valid due to the introducing of the weight $w_{\delta}$ (Lemma 6.1). As a
consequence of the local vanishings there is a quasi-isomorphism
$IC_{X}(\mathbb{V})\simeq_{\rm
qis}\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$.

  6. (6)

Make the quasi-isomorphism $IC_{X}(\mathbb{V})\simeq_{\rm
qis}\mathscr{D}_{X,\mathbb{V};ds^{2},h}$ canonical. This can be done by
Proposition 2.2.

######  Remark 1.8.

Recall that in [Saper1992, Cattani_Kaplan_Schmid1987, Kashiwara_Kawai1987],
the purity theorem which comes from the Hodge theory of singularities plays
crucial roles in the proof of (1.3). Our proof uses a different strategy. The
vanishings come from the $L^{2}$-estimate for the $\nabla$-operator. It is
interesting to formulate a purity theorem for an arbitrary complex analytic
singularity.

######  Remark 1.9.

The choice of the deformation $(ds^{2}_{\delta,\epsilon},w_{\delta})$ is
deeply inspired by Ohsawa’s work [Ohsawa1993] on isolated singularities. To
perform the trick by Ohsawa to non-isolated singularities, one faces the
difficulty that the intersection $(X\backslash X^{o})\cap\partial U_{\delta}$
is always non-empty when $X\backslash X^{o}$ is non-isolated. The metric
$ds^{2}_{\delta}$ on $(X\backslash X^{o})\cap\partial U_{\delta}$ is a
complicated mixture of $ds^{2}$ and the modifier $ds^{2}_{\delta}-ds^{2}$.
Hence it is difficult to study the local $L^{2}$-cohomology at $(X\backslash
X^{o})\cap\partial U_{\delta}$. To handle this analytical difficulty we choose
to calculate (1.4) instead of (1.3) and introduce the trick of two
neighborhoods $U_{\delta}$, $V_{\delta}$. So we may assume that ${\rm
supp}\alpha$ is far away from $(X\backslash X^{o})\cap\partial U_{\delta}$ in
the equation $\nabla\beta=-\alpha$.

The paper is organized as follows.

Section 2 and 3 are the review of some basic knowledge of the intersection
complex, the $L^{2}$-cohomology and the $L^{2}$-de Rham complex, in order to
fix notations. Some preliminary results are proved.

In Section 4 we establish the basic theory of Whitney stratified
desingularization of a pair of complex spaces. Distinguished metrics and the
notion of $0$-tame harmonic bundles are introduced. We also prove that they
are stratified along the bimeromorphic fibering neighborhood associated to a
given Whitney stratified desingularization. The weakly constructibility of the
$L^{2}$-de Rham complex is then a consequence.

Section 5 is devoted to the geometry and the asymptotic behavior of the
distinguished metric at an arbitrary complex singularity. The Whitney
stratified desingularization is the basic tool. We fix the basic setting and
constructions in §5.1 and §5.2 which will be used in the rest of the paper.

We prove the key vanishing theorem of this paper in Section 6.

Everything is combined together in Section 7 to give a bi-induction proof of
the main theorem.

Section 8 contains some applications of the main theorem which have been
mentioned in the introduction.

Acknowledgment: The deepest thank goes to professor Jun Li for bringing this
problem to the authors’ interest and his continual help and discussions during
their postdoctoral period and beyond. We would like to thank professor Takeo
Ohsawa and professor Adam Parusiński for answering our questions patiently. We
also wish to thank Zhenqian Li and Guozhen Wang for helpful conversations.
Finally, we would like to thank professor Sen Hu, professor Kefeng Liu,
professor Mao Sheng, professor Xiaotao Sun and professor Kang Zuo for their
interest in this work and their continual encouragements.

Notations:

  * •

All complex spaces are assume to be reduced, separated, paracompact, countable
at infinity and of pure dimension.

  * •

$h^{i}(\mathscr{A}^{\bullet})$ stands for the $i$-th cohomology sheaf of the
complex of sheaves $\mathscr{A}^{\bullet}$.

  * •

Let $\alpha,\beta$ be functions (or metrics, $(1,1)$-forms, etc.) on a complex
space. Denote by $\alpha\lesssim\beta$ if $C\beta-\alpha\geq 0$ for some
constant $C>0$. Denote $\alpha\sim\beta$ if both $\alpha\lesssim\beta$ and
$\beta\lesssim\alpha$ are satisfied.

  * •

Let $M$ be a complex manifold and $\mathbb{V}$ a local system,
$A^{k}(M,\mathbb{V})$ (resp. $A^{k}_{\rm cpt}(M,\mathbb{V})$) stands for the
space of $C^{\infty}$ $\mathbb{V}$-valued $k$-forms (resp. with compact
supports) on $M$. Denote by $\mathscr{A}^{k}_{M}(\mathbb{V})$ the sheaf of
$\mathbb{V}$-valued $C^{\infty}$ $k$-forms on $M$.

##  2\. Intersection Complex on Complex Spaces

Due to the nature of the $L^{2}$-cohomology, we need to consider constructible
sheaves that may not be locally finitely generated. In this section we review
the basic knowledge of such sheaves and prove some technical results for later
usage.

Let $X$ be a complex space of pure dimension $n$ and $X^{o}\subset X_{\rm
reg}$ a dense Zariski open subset. Throughout this paper, a stratification of
the pair $(X,X^{o})$ is a topological stratification

$\displaystyle\emptyset=X_{-1}\subset X_{0}\subset\cdots\subset X_{n}=X$

in the sense of [Borel1984, §I-1.1], in addition that $X_{n-1}=X\backslash
X^{o}$ and $X_{i}$ is a Zariski closed analytic subset for every
$i=0,\dots,n$. Roughly speaking, $X_{i}\backslash X_{i-1}$ is an
$i$-dimensional complex submanifold of $X$ for every $i$ and the links along
each $X_{i}\backslash X_{i-1}$ are topologically isotrivial. A typical example
of a stratification is the Whitney stratification (c.f. [Mather2012]).

A connected component of $\coprod_{i}X_{i}\backslash X_{i-1}$ is called a
stratum of $\\{X_{i}\\}$.

######  Definition 2.1.

Let $X$ be a complex space and $A$ a sheaf of $\mathbb{C}$-vector spaces on
$X$. $A$ is called a weakly local system on $X$ if for every point $x\in X$
there is a neighborhood $x\in U$ such that $A|_{U}\simeq\mathbb{C}_{U}^{I}$,
where $I$ is an (possibly infinite) index set. In this case we say that $A$ is
a local system if the $I$s are finite sets.

$A$ is called (weakly) constructible if there is a stratification
$\\{X_{i}\\}$ such that $A|_{X_{i}\backslash X_{i-1}}$ is a (weakly) local
system for every $i$. In this situation we also say that $A$ is (weakly)
constructible with respect to $\\{X_{i}\\}$.

The category of weakly constructible sheaves on $X$ is an abelian category.

Let $\mathbb{V}$ be a local system on $X^{o}$. Denote $U_{i}:=X\backslash
X_{i}$ and denote by $j_{i}:U_{i}\to U_{i-1}$ ($U_{-1}:=X$) the inclusion. The
intersection complex $IC_{X}(\mathbb{V})$ is defined, in the derived category
$D(X)$ of sheaves of $\mathbb{C}$-vector spaces on $X$, by

(2.1) $\displaystyle
IC_{X}(\mathbb{V}):=R^{<n}j_{0\ast}R^{<n-1}j_{1\ast}\cdots
R^{<1}j_{n-1\ast}\mathbb{V}.$

The right handside is independent of the stratification $\\{X_{i}\\}$ up to
quasi-isomorphisms and (2.1) gives a well defined bounded complex in $D(X)$.

In order to make the intersection complex a perverse sheaf ([BBD]), one
usually uses a Tate twist $IC_{X}(\mathbb{V})[n]$ instead of
$IC_{X}(\mathbb{V})$. In this paper the difference between twisting causes no
problem.

The following proposition is a mild generalization of Goresky-MacPherson-
Deligne criterion for intersection complexes to weakly constructible
complexes. It also shows that in the set of isomorphisms ${\rm
Iso}_{D(X)}(\mathscr{A}^{\bullet},IC_{X})$, there is a canonical one,
represented by a natural zigzag of quasi-isomorphisms between complexes, as
long as a quasi-isomorphism
$\iota_{0}:\mathbb{V}\to\mathscr{A}^{\bullet}|_{X^{o}}$ is fixed. A canonical
isomorphism is necessary if we want to compare the Hodge structures on both
sides of Theorem 1.4 (Conjecture 8.5).

######  Proposition 2.2 (Goresky-MacPherson-Deligne criterion).

Let $X$ be a complex space of pure dimension $n$ and $X^{o}\subset X_{\rm
reg}$ a dense Zariski open subset. Let $\\{X_{i}\\}$ be a stratification of
$(X,X^{o})$ and $\mathbb{V}$ a local system on $X^{o}$. A bounded complex of
sheaves of $\mathbb{C}$-vector spaces $\mathscr{A}^{\bullet}$ is isomorphic to
the intersection complex $IC_{X}(\mathbb{V})$ in $D(X)$ if and only if the
following conditions hold.

  1. (1)

$\mathscr{A}^{\bullet}|_{X^{o}}$ is quasi-isomorphic to $\mathbb{V}$.

  2. (2)

For every $j$, $h^{j}(\mathscr{A}^{\bullet})$ is weakly constructible with
respect to $\\{X_{i}\\}$.

  3. (3)

For each stratum $S$ of $\\{X_{i}\\}$ and every point $x\in S$, denote by
$i_{x}:\\{x\\}\to X$ the inclusion. Then

(2.2) $\displaystyle h^{k}(i_{x}^{\ast}\mathscr{A}^{\bullet})=0,\quad\forall
k\geq n-\dim S$

and

(2.3) $\displaystyle h^{k}(i_{x}^{!}\mathscr{A}^{\bullet})=0,\quad\forall
k\leq n+\dim S.$

Moreover, given a quasi-isomorphism of complexes
$\iota_{0}:\mathbb{V}\to\mathscr{A}^{\bullet}|_{X^{o}}$, there is a canonical
zigzag of quasi-isomorphisms between complexes (rather than morphisms in the
derived category)

$\displaystyle\mathscr{A}^{\bullet}\leftarrow\tau^{<n}\mathscr{A}^{\bullet}\to\tau^{<n}Rj_{0\ast}\left(\mathscr{A}^{\bullet}|_{U_{0}}\right)\leftarrow\tau^{<n}Rj_{0\ast}\left(\tau^{<n-1}\mathscr{A}^{\bullet}|_{U_{0}}\right)\to\tau^{<n}Rj_{0\ast}\tau^{<n-1}Rj_{1\ast}\left(\mathscr{A}^{\bullet}|_{U_{1}}\right)\leftarrow$
$\displaystyle\cdots\leftarrow\tau^{<n}Rj_{0\ast}\cdots\tau^{<{2}}Rj_{n-2\ast}\left(\tau^{<{1}}\mathscr{A}^{\bullet}|_{U_{n-2}}\right)\to\tau^{<n}Rj_{0\ast}\cdots\tau^{<1}Rj_{n-1\ast}\left(\mathscr{A}^{\bullet}|_{X^{o}}\right)\leftarrow
IC_{X}(\mathbb{V})$

###### Proof.

The first part is well known (c.f. [Borel1984, chap. 5] or [Ryoshi-Kiyoshi-
Toshiyuki2008, Prop. 8.2.11]). To prove the second part, it suffices to show
that the natural diagram

is a diagram of quasi-isomorphisms for every $d$. Consider the distinguished
triangle

$\displaystyle\tau^{<n-d}\mathscr{A}^{\bullet}|_{U_{d-1}}\to\mathscr{A}^{\bullet}|_{U_{d-1}}\to\tau^{\geq
n-d}\mathscr{A}^{\bullet}|_{U_{d-1}}\to\tau^{<n-d}\mathscr{A}^{\bullet}|_{U_{d-1}}[1].$

By (2.2), $\tau^{\geq n-d}\mathscr{A}^{\bullet}|_{U_{d-1}}=0$. Hence the
canonical map

$\tau^{<n-d}\mathscr{A}^{\bullet}|_{U_{d-1}}\to\mathscr{A}^{\bullet}|_{U_{d-1}}$

is a quasi-isomorphism. Denote by $i:U_{d-1}\backslash U_{d}\to U_{d-1}$ and
$i_{x}:\\{x\\}\to U_{d-1}\backslash U_{d}$ the closed immersions. Since
$U_{d-1}\backslash U_{d}$ is smooth with real dimension $2d$, we have
$i_{x}^{!}\simeq i_{x}^{\ast}[-2d]$. Hence by (2.3), we see that

$\displaystyle h^{k}((i_{x}i)^{!}\mathscr{A}^{\bullet}|_{U_{d-1}})\simeq
i^{\ast}_{x}h^{k-2d}(i^{!}\mathscr{A}^{\bullet}|_{U_{d-1}})=0,\quad\forall
k\leq n+d.$

Equivalently $\tau^{\leq n-d}i_{!}i^{!}\mathscr{A}^{\bullet}|_{U_{d-1}}=0$.

This, together with the distinguished triangle

$\displaystyle
i_{!}i^{!}\mathscr{A}^{\bullet}|_{U_{d-1}}\to\mathscr{A}^{\bullet}|_{U_{d-1}}\to
Rj_{d\ast}\left(\mathscr{A}^{\bullet}|_{U_{d}}\right)\to
i_{!}i^{!}\mathscr{A}^{\bullet}|_{U_{d-1}}[1]$

shows that the canonical map

$\tau^{<n-d}\mathscr{A}^{\bullet}|_{U_{d-1}}\to\tau^{<n-d}Rj_{d\ast}\left(\mathscr{A}^{\bullet}|_{U_{d}}\right)$

is a quasi-isomorphism. ∎

######  Remark 2.3.

Condition (3) in Proposition 2.2 is equivalent to the following statement. For
every $x\in X_{i}\backslash X_{i-1}$,

$\displaystyle\varinjlim_{x\in
U_{x}}\mathbb{H}^{k}(U_{x},\mathscr{A}^{\bullet})=0,\quad\forall k\geq n-i$

and

$\displaystyle\varprojlim_{x\in U_{x}}\mathbb{H}^{k}_{\rm
cpt}(U_{x},\mathscr{A}^{\bullet})=0,\quad\forall k\leq n+i.$

######  Corollary 2.4.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. Let $\mathbb{V}$ be a local system on $X^{o}$ and $X=\bigcup_{i\in
I}U_{i}$ an open covering. Then a complex of sheaves of $\mathbb{C}$-vector
spaces $\mathscr{A}^{\bullet}$ is quasi-isomorphic to $IC_{X}(\mathbb{V})$ if
and only if $\mathscr{A}^{\bullet}|_{X^{o}}\simeq\mathbb{V}$ and
$\mathscr{A}^{\bullet}|_{U_{i}}$ is quasi-isomorphic to
$IC_{U_{i}}(\mathbb{V}|_{U_{i}\cap X^{o}})$ for every $i\in I$.

###### Proof.

Notice that a sheaf $A$ is weakly constructible if $A|_{U_{i}}$ is weakly
constructible for every $i\in I$ ([Dimca2004, Proposition 4.1.13]). Now the
corollary follows from the Goresky-MacPherson-Deligne Criterion (Proposition
2.2). ∎

##  3\. $L^{2}$-de Rham Complex and $L^{2}$-Cohomology

###  3.1. $L^{2}$-cohomology

Let $(M,ds^{2}_{M})$ be an orientable Riemannian stratified space with
boundary and $\mathbb{V}$ a complex local system on $M$. For this notion we
always assume that $ds^{2}_{M}$ can be extended smoothly to the boundary.
Denote by
$(\mathcal{V}:=\mathbb{V}\otimes_{\mathbb{C}}\mathscr{O}_{M},\nabla)$ the
associated flat bundle. Let $h$ be a hermitian metric on $\mathcal{V}$ which
is not necessarily compatible with $\nabla$. Such a pair $(\mathbb{V},h)$ is
called a hermitian local system. Denote by
$L_{(2)}^{k}(M,\mathbb{V};ds^{2}_{M},h)$ the space of $\mathbb{V}$-valued
measurable $k$-forms which are square integrable with respect to
$(ds^{2}_{M},h)$. Let

$D^{k}_{\rm max}(M,\mathbb{V};ds^{2}_{M},h):=\left\\{\alpha\in
L_{(2)}^{k}(M,\mathbb{V};ds^{2}_{M},h)\big{|}\nabla\alpha\in
L_{(2)}^{k+1}(M,\mathbb{V};ds^{2}_{M},h)\right\\}$

and

$D^{\bullet}_{\rm max}(M,\mathbb{V};ds^{2}_{M},h):=D^{0}_{\rm
max}(M,\mathbb{V};ds^{2}_{M},h)\stackrel{{\scriptstyle\nabla}}{{\to}}D^{1}_{\rm
max}(M,\mathbb{V};ds^{2}_{M},h)\stackrel{{\scriptstyle\nabla}}{{\to}}\cdots\stackrel{{\scriptstyle\nabla}}{{\to}}D^{\dim_{\mathbb{R}}M}_{\rm
max}(M,\mathbb{V};ds^{2}_{M},h).$

Here $\nabla$ are operators in the sense of distribution.

The $L^{2}$-cohomology of $(M,\mathbb{V};ds^{2}_{M},h)$ with the maximal ideal
boundary condition is defined by

$H^{\ast}_{(2),\rm
max}(M,\mathbb{V};ds^{2}_{M},h):=H^{\ast}\left(D^{\bullet}_{\rm
max}(M,\mathbb{V};ds^{2}_{M},h)\right).$

Subscripts such as $ds^{2}_{M}$ and $h$ would be omitted when they are clear
from the context. $\mathbb{V}$ would be omitted when
$\mathbb{V}\simeq\mathbb{C}$ is the trivial local system.

There is another ideal boundary condition for the operator $\nabla$. Let

$D^{k}_{\rm min}(M,\mathbb{V};ds^{2}_{M},h):=\left\\{\alpha\in D^{k}_{\rm
max}(M,\mathbb{V};ds^{2}_{M},h)\big{|}\exists\alpha_{n}\in A^{k}_{\rm
cpt}(M,\mathbb{V}),\alpha_{n}\to\alpha,\nabla\alpha_{n}\to\nabla\alpha\right\\}$

and

$D_{\rm min}^{\bullet}(M,\mathbb{V};ds^{2}_{M},h):=D^{0}_{\rm
min}(M,\mathbb{V};ds^{2}_{M},h)\stackrel{{\scriptstyle\nabla}}{{\to}}D^{1}_{\rm
min}(M,\mathbb{V};ds^{2}_{M},h)\stackrel{{\scriptstyle\nabla}}{{\to}}\cdots\stackrel{{\scriptstyle\nabla}}{{\to}}D^{\dim_{\mathbb{R}}M}_{\rm
min}(M,\mathbb{V};ds^{2}_{M},h).$

The $L^{2}$-cohomology of $(M,ds^{2}_{M})$ with minimal ideal boundary
condition is defined by

$H^{\ast}_{(2),\rm
min}(M,\mathbb{V};ds^{2}_{M},h):=H^{\ast}\left(D^{\bullet}_{\rm
min}(M,\mathbb{V};ds^{2}_{M},h)\right).$

To distinguish the ideal boundary conditions we denote by $\nabla_{\rm max}$
and $\nabla_{\rm min}$ the unbounded operators with domains ${\rm
Dom}\nabla_{\rm max}:=\bigoplus_{i}D^{i}_{\rm max}(M,\mathbb{V};ds^{2}_{M},h)$
and ${\rm Dom}\nabla_{\rm min}:=\bigoplus_{i}D^{i}_{\rm
min}(M,\mathbb{V};ds^{2}_{M},h)$ respectively. These two ideal boundary
conditions coincide when $M$ is a manifold without boundary and $ds^{2}_{M}$
is complete ([Zucker1979, Theorem 7.2]). Hence there is a canonical
isomorphism

$\displaystyle H^{k}_{(2),\rm min}(M,\mathbb{V};ds^{2}_{M},h)\simeq
H^{k}_{(2),\rm max}(M,\mathbb{V};ds^{2}_{M},h)$

whenever $(M,ds^{2}_{M})$ is a complete Riemannian manifold (without
boundary). In this case we denote the $L^{2}$-cohomology by
$H^{\ast}_{(2)}(-)$ for short.

Generally these two kinds of $L^{2}$-cohomologies are related by the
$L^{2}$-Poincaré duality.

######  Proposition 3.1.

Let $(M,ds^{2}_{M})$ be an orientable Riemannian manifold. Assume that either
of the following statements holds.

  1. (1)

$H^{k}_{(2),\rm min}(M,\mathbb{V}^{\ast};ds^{2}_{M},h^{\ast})$ and
$H^{k+1}_{(2),\rm min}(M,\mathbb{V}^{\ast};ds^{2}_{M},h^{\ast})$ are finite
dimensional.

  2. (2)

$H^{k}_{(2),\rm min}(M,\mathbb{V}^{\ast};ds^{2}_{M},h^{\ast})$ and
$H^{\dim_{\mathbb{R}}M-k}_{(2),\rm max}(M,\mathbb{V};ds^{2}_{M},h)$ are finite
dimensional.

Then there is a well defined perfect pairing

$H^{\dim_{\mathbb{R}}M-k}_{(2),\rm max}(M,\mathbb{V};ds^{2}_{M},h)\times
H^{k}_{(2),\rm
min}(M,\mathbb{V}^{\ast};ds^{2}_{M},h^{\ast})\to\mathbb{C},\quad([\alpha],[\beta])\mapsto\int_{M}\alpha\wedge\beta.$

Here $(\mathbb{V}^{\ast},h^{\ast})$ is the dual hermitian local system.

###### Proof.

Denote $L^{k}_{(2)}:=L^{k}_{(2)}(M,\mathbb{V};ds^{2}_{M},h)$ and $L^{\ast
k}_{(2)}:=L^{k}_{(2)}(M,\mathbb{V}^{\ast};ds^{2}_{M},h^{\ast})$ for short.
Consider the diagram

(3.3)

and its Hodge star dual

(3.6) $\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.14038pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L^{n-k+1}_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.99036pt\raise 11.27083pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.21529pt\hbox{$\scriptstyle{(\nabla_{\rm max})^{\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 38.14038pt\raise
4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
38.14038pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{L^{n-k}_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
59.07112pt\raise 11.27083pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.21529pt\hbox{$\scriptstyle{(\nabla_{\rm max})^{\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.2656pt\raise
4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.7126pt\raise-10.12776pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9611pt\hbox{$\scriptstyle{\nabla_{\rm max}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
14.1404pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.2656pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{L^{n-k-1}_{(2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
62.79337pt\raise-10.12776pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9611pt\hbox{$\scriptstyle{\nabla_{\rm max}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
59.2656pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,\quad
n=\dim_{\mathbb{R}}M.$

Under either of the assumptions all the eight operators in (3.3) and (3.6)
have closed ranges. Therefore there are Hodge decompositions
([Kashiwara_Kawai1987, Theorem A.2.2])

$\displaystyle L^{\ast k}_{(2)}=\mathscr{H}^{\ast k}_{\rm
min}\oplus\nabla_{\rm min}(L^{\ast k-1}_{(2)})\oplus(\nabla_{\rm
min})^{\ast}(L^{\ast k+1}_{(2)})$

and

$\displaystyle L^{n-k}_{(2)}=\mathscr{H}^{n-k}_{\rm max}\oplus\nabla_{\rm
max}(L^{n-k-1}_{(2)})\oplus(\nabla_{\rm max})^{\ast}(L^{n-k+1}_{(2)}),$

where

$\mathscr{H}^{\ast k}_{\rm min}={\rm Ker}\nabla_{\rm min}\cap{\rm
Ker}(\nabla_{\rm min})^{\ast}\cap L^{\ast k}_{(2)}$

and

$\mathscr{H}^{n-k}_{\rm max}={\rm Ker}\nabla_{\rm max}\cap{\rm
Ker}(\nabla_{\rm max})^{\ast}\cap L^{n-k}_{(2)}.$

Hence there are isomorphisms

$\displaystyle H^{k}_{(2),\rm
min}(M,\mathbb{V}^{\ast};ds^{2}_{M},h^{\ast})\simeq\mathscr{H}^{\ast k}_{\rm
min}$

and

$\displaystyle H^{n-k}_{(2),\rm
max}(M,\mathbb{V};ds^{2}_{M},h)\simeq\mathscr{H}^{n-k}_{\rm max}.$

Since the Hodge star operator gives an isomorphisms between $\mathscr{H}^{\ast
k}_{\rm min}$ and $\mathscr{H}^{n-k}_{\rm max}$, the pairing

$\mathscr{H}^{n-k}_{\rm max}\times\mathscr{H}^{\ast k}_{\rm
min}\to\mathbb{C},\quad(\alpha,\beta)\mapsto\int_{M}\alpha\wedge\beta=\pm\int_{M}\langle\alpha,\ast\beta\rangle{\rm
vol}$

is perfect. ∎

The $L^{2}$-cohomology with minimal ideal boundary condition admits the
Kunneth-type formula, as is observed by Zucker [Zucker1982].

######  Proposition 3.2.

Let $(M,ds^{2}_{M})$ and $(N,ds^{2}_{N})$ be Riemannian manifolds with
boundaries. Let $(\mathbb{V}_{1},h_{1})$ and $(\mathbb{V}_{2},h_{2})$ be
hermitian local systems on $M$ and $N$ respectively. Let
$ds^{2}_{X}:=ds^{2}_{M}+ds^{2}_{N}$ be the product metric on $X:=M\times N$
and let $(\mathbb{V},h):=(\mathbb{V}_{1}\boxtimes\mathbb{V}_{2},h_{1}h_{2})$
be the product hermitian local system on $X$. Assume that $H^{\ast}_{(2),\rm
min}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})$ are finite dimensional. Then there
are canonical isomorphisms

(3.7) $\displaystyle H^{k}_{(2),\rm
min}(X,\mathbb{V};ds^{2}_{X},h)\simeq\bigoplus_{i+j=k}H^{i}_{(2),\rm
min}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\otimes H^{j}_{(2),\rm
min}(N,\mathbb{V}_{2};ds^{2}_{N},h_{2}).$

###### Proof.

The proof is originated by [Zucker1982, Theorem 2.29] with a mild
modification. For the convenience of readers we give a sketched proof.

Since $\bigoplus_{i+j=k}A_{\rm cpt}^{i}(M,\mathbb{V}_{1})\otimes A_{\rm
cpt}^{j}(N,\mathbb{V}_{2})$ is dense in $A_{\rm cpt}^{k}(X,\mathbb{V})$ under
the graph norm $\|-\|+\|\nabla-\|$, by taking completion under the graph norm
we obtain

$\displaystyle D^{k}_{\rm
min}(X,\mathbb{V};ds^{2}_{X},h)\simeq\bigoplus_{i+j=k}D^{i}_{\rm
min}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\widehat{\otimes}D^{j}_{\rm
min}(N,\mathbb{V}_{2};ds^{2}_{N},h_{2}).$

Denote by $\nabla_{1}$ the flat connection associated to $\mathbb{V}_{1}$.
Since $H^{\ast}_{(2),\rm min}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})$ is finite
dimensional, ${\rm Im}\nabla_{1,\rm min}$ is closed and there is a bounded
operator

$\displaystyle H_{M}:D_{\rm
min}^{\bullet}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\to D_{\rm
min}^{\bullet-1}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})$

such that

(3.8) $\displaystyle{\rm Id}=P_{M}+\nabla_{1,\rm min}H_{M}+H_{M}\nabla_{1,\rm
min}$

on $D^{\bullet}_{\rm min}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})$. Here

$P_{M}:L_{(2)}^{\bullet}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\to\mathscr{H}^{\bullet}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1}):={\rm
Ker}(\nabla_{1,\rm min})\cap{\rm Ker}(\nabla_{1,\rm min})^{\ast}$

is the projection via the abstract Hodge decomposition

$\displaystyle
L_{(2)}^{\bullet}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})=\mathscr{H}^{\bullet}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\oplus{\rm
Im}\nabla_{1,\rm min}\oplus{\rm Im}(\nabla_{1,\rm min})^{\ast}.$

$(\nabla_{1,\rm min})^{\ast}$ is the Hilbert adjoint of $\nabla_{1,\rm min}$
and $H_{M}=(\nabla_{1,\rm min})^{\ast}G_{M}$ where $G_{M}$ is the abstract
Green operator ([Kashiwara_Kawai1987, Theorem A.2.2]). This shows that the
bounded operator $P_{M}$ gives a homotopy equivalence from $D^{\bullet}_{\rm
min}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})$ to the null complex
$\mathscr{H}^{\bullet}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})$.

By boundedness, (3.8) naturally extends to the homotopy to identity formula on

$D^{\bullet}_{\rm
min}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\widehat{\otimes}D^{\bullet}_{\rm
min}(N,\mathbb{V}_{2};ds^{2}_{N},h_{2})$

and implies that it is homotopic equivalent to

(3.9)
$\displaystyle\mathscr{H}^{\bullet}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\widehat{\otimes}D^{\bullet}_{\rm
min}(N,\mathbb{V}_{2};ds^{2}_{N},h_{2})=\mathscr{H}^{\bullet}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\otimes
D^{\bullet}_{\rm min}(N,\mathbb{V}_{2};ds^{2}_{N},h_{2}).$

The reason for the last equality is the assumption that

$H^{\ast}_{(2),\rm
min}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})\simeq\mathscr{H}^{\ast}(M,\mathbb{V}_{1};ds^{2}_{M},h_{1})$

is finite dimensional. Now the proposition is proved because the cohomologies
of (3.9) are the right hand side of (3.7). ∎

Another feature of the minimal boundary condition is that the extension of a
form by zero is well defined. Let $U\subset M$ be an open subset, then the map

$\displaystyle A^{\bullet}_{\rm cpt}(U,\mathbb{V})\to A^{\bullet}_{\rm
cpt}(M,\mathbb{V}),\quad\alpha\mapsto\widetilde{\alpha}=\begin{cases}\alpha,&x\in
U\\\ 0,&x\in M\backslash U\end{cases}$

is bounded under the graph norm $\|-\|+\|\nabla-\|$. Hence it induces a map

$\displaystyle D^{\bullet}_{\rm min}(U,\mathbb{V};ds^{2}_{M},h)\to
D^{\bullet}_{\rm
min}(M,\mathbb{V};ds^{2}_{M},h),\quad\alpha\mapsto\widetilde{\alpha}.$

We will simply call this map extension by zero when the spaces are clear in
the context.

###  3.2. $L^{2}$-de Rham complex

Let $X$ be a real or complex analytic space and $X^{o}\subset X_{\rm reg}$ an
open subset of the loci of regular points $X_{\rm reg}$ so that $X^{o}$ is
dense in $X$. Let $ds^{2}$ be a Riemannian metric on $X^{o}$ and
$(\mathbb{V},h)$ a hermitian local system on $X^{o}$. Denote by
$(\mathcal{V}:=\mathbb{V}\otimes_{\mathbb{C}}\mathscr{O}_{M},\nabla)$ the
associated flat bundle. Denote by $\mathscr{L}^{k}_{X,\mathbb{V};ds^{2},h}$
the sheaf of $\mathbb{V}$-valued measurable $k$-forms on $X$ which are locally
square integrable with respect to the metrics $ds^{2}$ and $h$, i.e. for every
open subset $U\subset X$, $\mathscr{L}^{k}_{X,\mathbb{V};ds^{2},h}(U)$ is the
space of measurable $\mathbb{V}$-valued $k$-forms $\alpha$ on $U\cap X^{o}$
such that locally at every point $x\in U$ there is a neighborhood
$V_{x}\subset U$ of $x$ so that

$\int_{V_{x}\cap X^{o}}|\alpha|^{2}_{ds^{2},h}{\rm vol}_{ds^{2}}<\infty.$

Denote

$\mathscr{D}^{k}_{X,\mathbb{V};ds^{2},h}:=\left\\{\alpha\in\mathscr{L}^{k}_{X,\mathbb{V};ds^{2},h}\big{|}\nabla\alpha\in\mathscr{L}^{k+1}_{X,\mathbb{V};ds^{2},h}\right\\}.$

The $L^{2}$-de Rham complex of $(X,\mathbb{V};ds^{2},h)$ is defined by

$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}:=\mathscr{D}^{0}_{X,\mathbb{V};ds^{2},h}\stackrel{{\scriptstyle\nabla}}{{\to}}\mathscr{D}^{1}_{X,\mathbb{V};ds^{2},h}\stackrel{{\scriptstyle\nabla}}{{\to}}\cdots\stackrel{{\scriptstyle\nabla}}{{\to}}\mathscr{D}^{\dim_{\mathbb{R}}X}_{X,\mathbb{V};ds^{2},h},$

where $\nabla$ are operators in the sense of distribution.

######  Definition 3.3.

Let $X$ be a complex space and $ds^{2}$ a hermitian metric on $X_{\rm reg}$.
We say that $ds^{2}$ is a hermitian metric on $X$ if for every $x\in X$ there
is a neighborhood $U$ and a holomorphic closed immersion $U\subset V$ into a
complex manifold such that $ds^{2}|_{U}\sim ds^{2}_{V}|_{U}$ for some
hermitian metric $ds^{2}_{V}$ on $V$. If moreover the associated $(1,1)$-form
associated to $ds^{2}$ is $d$-closed on $X^{o}$, we say that $ds^{2}$ is a
Kähler hermitian metric on $X$

######  Remark 3.4.

To distinguish hermitian metrics on $X$ and hermitian metrics on the open set
$X^{o}$, we will always emphasize the space on which the metric lies.

Many interesting metrics, including the distinguished metric introduced in
this paper, are locally bounded from below by a hermitian one. In this case
the $L^{2}$-de Rham complexes are complexes of fine sheaves.

######  Lemma 3.5.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. Let $ds^{2}$ be a hermitian metric on $X^{o}$ and $(\mathbb{V},h)$ a
hermitian local system on $X^{o}$. Suppose that for every point $x\in
X\backslash X^{o}$ there is a neighborhood $x\in U_{x}$ and a hermitian metric
$ds^{2}_{0}$ on $U_{x}$ such that $ds^{2}_{0}|_{X^{o}\cap U_{x}}\lesssim
ds^{2}|_{X^{o}\cap U_{x}}$. Then

  1. (1)

for every $k$, $\mathscr{D}^{k}_{X,\mathbb{V};ds^{2},h}$ is a fine sheaf.

  2. (2)

For every $C^{\infty}$ form $\alpha$ on $X$, $|\alpha|_{ds^{2}}$ is locally
$L^{\infty}$ bounded.

###### Proof.

It suffices to show that for every $W\subset\overline{W}\subset V\subset X$
where $W$ and $V$ are small open subsets, there is a continuous function $f$
on $V$ such that

  * •

${\rm supp}(f)\subset\overline{W}$,

  * •

$f$ is $C^{\infty}$ on $V\cap X^{o}$

  * •

$df$ has bounded fiberwise norm with respect to the metric $ds^{2}$.

We may assume that there is a closed embedding $V\subset M$ where $M$ is a
complex manifold. Let $W^{\prime}\subset\overline{W^{\prime}}\subset M$ where
$W^{\prime}$ is an open subset such that $W^{\prime}\cap V=W$. Let
$ds^{2}_{M}$ be a hermitian metric on $M$ so that $ds^{2}_{0}|_{V\cap
X^{o}}\sim ds^{2}_{M}|_{V\cap X^{o}}$. Let $g$ be a smooth function on $M$
whose support lies in $\overline{W^{\prime}}$. Denote $f=g|_{V}$, then ${\rm
supp}(f)\subset\overline{W}$ and $f$ is $C^{\infty}$ on $V\cap X^{o}$. Since
$V\cap X^{o}\subset M$ is a submanifold, one has the orthogonal decomposition

$\displaystyle T_{M,x}=T_{V\cap X^{o},x}\oplus T_{V\cap
X^{o},x}^{\bot},\quad\forall x\in V\cap X^{o}.$

Consequently, the restriction map $T^{\ast}_{M,x}\to T^{\ast}_{V\cap X^{o},x}$
is distance-decreasing for every $x\in V\cap X^{o}$. Therefore
$|df|_{ds^{2}}\lesssim|df|_{ds^{2}_{0}}\lesssim|dg|_{ds^{2}_{M}}<\infty$. This
proves the first claim of the lemma. For the second claim, assume that
$\alpha|_{V}=\alpha^{\prime}|_{V}$ for some smooth form $\alpha^{\prime}$ on
$M$, then
$|\alpha|_{ds^{2}}\lesssim|\alpha|_{ds^{2}_{0}}\lesssim|\alpha^{\prime}|_{ds^{2}_{M}}<\infty$.
∎

When $X$ is compact,

$\Gamma\left(X,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}\right)=D_{\rm
max}^{\bullet}(X,\mathbb{V};ds^{2},h).$

Therefore we have

######  Corollary 3.6.

Notations as in Lemma 3.5. Assume that $X$ is compact, then there is a natural
isomorphism

$H^{\ast}_{(2),\rm
max}(X^{o},\mathbb{V};ds^{2},h)\simeq\mathbb{H}^{\ast}(X,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}).$

##  4\. Constructibility of $L^{2}$-de Rham Complex

###  4.1. $L^{2}$-Poincaré Lemma and $L^{2}$-Thom isomorphism

This subsection is a preparation for the proof of the main result (Theorem
4.5) of the next subsection. In particular we prove an $L^{2}$-Poincaré lemma
(Lemma 4.1) and an $L^{2}$-Thom isomorphism for trivial line bundles (Lemma
4.2).

Let $(X,\mathbb{V};ds^{2},h)$ be a Riemannian stratified space with a
hermitian local system on the union of open strata $X^{o}:=X\backslash
X_{\dim_{\mathbb{R}}X-2}$ . Denote by $q:(0,1)\times X\to X$ the projection.
Let $ds^{2}_{(0,1)\times X}:=d\rho^{2}+ds^{2}$ and let $q^{\ast}h$ be the
pullback metric on $q^{\ast}\mathbb{V}$. The pullback operator $q^{\ast}$
sends a locally $L^{2}$-form to a locally $L^{2}$-form. Hence it gives a well
defined map

(4.1) $\displaystyle
q^{\ast}:\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}(X)\to\mathscr{D}^{\bullet}_{(0,1)\times
X,q^{\ast}\mathbb{V};ds^{2}_{(0,1)\times X},q^{\ast}h}((0,1)\times X).$

######  Lemma 4.1.

Notations as above, the canonical morphism (4.1) is a homotopy equivalence.

###### Proof.

For simplicity we denote

$A^{\bullet}_{X,(2)}:=A^{\bullet}(X^{o},\mathbb{V})\cap\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$

and

$A^{\bullet}_{(0,1)\times X,(2)}:=A^{\bullet}((0,1)\times
X^{o},\mathbb{V})\cap\mathscr{D}^{\bullet}_{(0,1)\times
X,q^{\ast}\mathbb{V};ds^{2}_{(0,1)\times X},q^{\ast}h}((0,1)\times X).$

Using the decomposition

$\displaystyle\alpha=\alpha_{0}+d\rho\wedge\alpha_{1},\quad\iota_{\frac{\partial}{\partial\rho}}\alpha_{0}=0,~{}~{}\iota_{\frac{\partial}{\partial\rho}}\alpha_{1}=0,$

we define

$\displaystyle Q(\alpha)=2\int_{\frac{1}{4}}^{{\frac{3}{4}}}\alpha_{0}(t,x)dt$

and

$\displaystyle
H(\alpha)=2\int_{\frac{1}{4}}^{{\frac{3}{4}}}\int_{t}^{\rho}\alpha_{1}(\rho^{\prime},x)d\rho^{\prime}dt.$

It follows from the definition that $Qq^{\ast}={\rm Id}$. We are going to show
that $q^{\ast}Q$ is homotopic to the identity. For this we need to show the
following two claims.

A:

$\nabla H+H\nabla={\rm Id}-q^{\ast}Q$. Recall that

$\nabla=d\rho\wedge\frac{\partial}{\partial\rho}+\nabla_{\mathbb{V}}$

where $\nabla_{\mathbb{V}}$ is the flat connection associated to $\mathbb{V}$
and $\nabla=q^{\ast}\nabla_{\mathbb{V}}$. Claim A follows from the equalities
(4.2) and (4.3) below:

(4.2) $\displaystyle\nabla H(\alpha)$
$\displaystyle=2\int_{\frac{1}{4}}^{{\frac{3}{4}}}\left(d\rho\wedge\frac{\partial}{\partial\rho}\int_{t}^{\rho}\alpha_{1}(\rho^{\prime},x)d\rho^{\prime}+\int_{t}^{\rho}\nabla_{\mathbb{V}}\alpha_{1}(\rho^{\prime},x)d\rho^{\prime}\right)dt$
$\displaystyle=d\rho\wedge\alpha_{1}(\rho,x)+2\int_{\frac{1}{4}}^{{\frac{3}{4}}}\int_{t}^{\rho}\nabla_{\mathbb{V}}\alpha_{1}(\rho^{\prime},x)d\rho^{\prime}dt.$
(4.3) $\displaystyle H\nabla(\alpha)$
$\displaystyle=2\int_{\frac{1}{4}}^{{\frac{3}{4}}}\int_{t}^{\rho}\left(\frac{\partial}{\partial\rho^{\prime}}\alpha_{0}(\rho^{\prime},x)-\nabla_{\mathbb{V}}\alpha_{1}(\rho^{\prime},x)\right)d\rho^{\prime}dt$
$\displaystyle=\alpha_{0}(\rho,x)-2\int_{\frac{1}{4}}^{\frac{3}{4}}\alpha_{0}(t,x)dt-2\int_{\frac{1}{4}}^{{\frac{3}{4}}}\int_{t}^{\rho}\nabla_{\mathbb{V}}\alpha_{1}(\rho^{\prime},x)d\rho^{\prime}dt.$

B:

$Q$, $H$ send locally $L^{2}$ forms to locally $L^{2}$ forms. This follows
from the following estimates (4.4) and (4.5).

(4.4)
$\displaystyle\int_{M}\left|2\int_{\frac{1}{4}}^{\frac{3}{4}}\alpha_{0}(t,x)dt\right|^{2}{\rm
vol}_{M}\leq
2\int_{M}\int_{\frac{1}{4}}^{\frac{3}{4}}|\alpha_{0}(t,x)|^{2}dt{\rm
vol}_{M}\leq 2\|\alpha\|^{2}_{[\frac{1}{4},\frac{3}{4}]\times M}.$ (4.5)
$\displaystyle\int_{[a,b]\times
M}\left|2\int_{\frac{1}{4}}^{\frac{3}{4}}\int_{t}^{\rho}\alpha_{1}(\rho^{\prime},x)d\rho^{\prime}dt\right|^{2}{\rm
vol}_{[a,b]\times M}$ $\displaystyle\leq$ $\displaystyle 2\int_{[a,b]\times
M}\int_{\frac{1}{4}}^{\frac{3}{4}}\int_{a}^{b}\left|\alpha_{1}(\rho^{\prime},x)\right|^{2}d\rho^{\prime}dt{\rm
vol}_{[a,b]\times M}$ $\displaystyle=$
$\displaystyle\int_{M}\int_{a}^{b}\left(\int_{a}^{b}\left|\alpha_{1}(\rho^{\prime},x)\right|^{2}d\rho^{\prime}\right)d\rho{\rm
vol}_{M}$ $\displaystyle\leq$
$\displaystyle\int_{M}\left(\int_{a}^{b}|\alpha_{1}(\rho,x)|^{2}d\rho\right){\rm
vol}_{M}$ $\displaystyle\leq$
$\displaystyle\int_{M}\left(\int_{a}^{b}|\alpha(\rho,x)|^{2}d\rho\right){\rm
vol}_{M}$ $\displaystyle=$ $\displaystyle\|\alpha\|^{2}_{[a,b]\times M}.$

Here $M\subset X$ is an arbitrary compact subset and $0<a<\frac{1}{4}$,
$\frac{3}{4}<b<1$. As a consequence, $Q:A^{\bullet}_{(0,1)\times X,(2)}\to
A^{\bullet}_{X,(2)}$ is well defined. Also $H:A^{\bullet}_{(0,1)\times
X,(2)}\to A^{\bullet-1}_{(0,1)\times X,(2)}$ is well defined and gives a
homotopy of $q^{\ast}Q$ to the identity. On the other hand, $Qq^{\ast}={\rm
Id}$. Hence $q^{\ast}$ is a homotopy equivalence. Since both $Q$ and $H$ are
continuous operators (claim B), $\nabla H+H\nabla={\rm Id}-q^{\ast}Q$ extends
to a homotopy from

$q^{\ast}Q:\mathscr{D}^{\bullet}_{(0,1)\times
X,q^{\ast}\mathbb{V};ds^{2}_{(0,1)\times X},q^{\ast}h}((0,1)\times
X)\to\mathscr{D}^{\bullet}_{(0,1)\times
X,q^{\ast}\mathbb{V};ds^{2}_{(0,1)\times X},q^{\ast}h}((0,1)\times X)$

to the identity. This proves the lemma. ∎

For compact supported smooth forms, there is a pushforward map via integral
along the fibers of $q$. For each locally $L^{2}$ integrable
$q^{\ast}\mathbb{V}$-valued smooth form $\alpha$ on $(0,1)\times X^{o}$ such
that ${\rm supp}\alpha$ is precompact, there is a decomposition

$\displaystyle\alpha=\alpha_{0}+d\rho\wedge\alpha_{1},\quad\iota_{\frac{\partial}{\partial\rho}}\alpha_{0}=0,~{}~{}\iota_{\frac{\partial}{\partial\rho}}\alpha_{1}=0.$

Define

$q_{\ast}(\alpha)=\int_{0}^{1}dt\wedge\alpha_{1}.$

Because

$\displaystyle\int_{M}\left|\int_{0}^{1}dt\wedge\alpha_{1}\right|^{2}{\rm
vol}_{M}\leq\int_{M}\int_{0}^{1}|\alpha_{1}|^{2}dt{\rm
vol}_{M}\lesssim\|\alpha\|^{2}_{(0,1)\times M}$

where $M\subset X$ is any precompact subset, $q_{\ast}$ is a continuous
operator. On the other hand

$\displaystyle q_{\ast}\nabla(\alpha)$
$\displaystyle=q_{\ast}(\nabla_{\mathbb{V}}\alpha_{0}+dt\wedge(\frac{\partial\alpha_{0}}{\partial
t}-\nabla_{\mathbb{V}}\alpha_{1}))$
$\displaystyle=\int^{1}_{0}(\frac{\partial\alpha_{0}}{\partial
t}-\nabla_{\mathbb{V}}\alpha_{1})dt$
$\displaystyle=-\int^{1}_{0}dt\wedge\nabla_{\mathbb{V}}\alpha_{1}\quad\left(\int^{1}_{0}dt\wedge\frac{\partial\alpha_{0}}{\partial
t}=\alpha_{0}(1,x)-\alpha_{0}(0,x)=0\right)$
$\displaystyle=\nabla_{\mathbb{V}}\int^{1}_{0}dt\wedge\alpha_{1}=\nabla
q_{\ast}(\alpha).$

Therefore $q_{\ast}$ extends uniquely to a continuous morphism

(4.6) $\displaystyle q_{\ast}:\Gamma_{\rm cpt}\left((0,1)\times
X,\mathscr{D}^{\bullet}_{(0,1)\times X,q^{\ast}\mathbb{V};ds^{2}_{(0,1)\times
X},q^{\ast}h}\right)\to\Gamma_{\rm
cpt}\left(X,\mathscr{D}^{\bullet-1}_{X,\mathbb{V};ds^{2},h}\right).$

######  Lemma 4.2.

Notations as above, the pushforward map (4.6) is a homotopy equivalence.

###### Proof.

For simplicity we denote

$A^{\bullet}_{X,{\rm cpt},(2)}:=\Gamma_{\rm
cpt}\left(X,\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}\right)\cap
A^{\bullet}(X^{o},\mathbb{V})$

and $A^{\bullet}_{(0,1)\times X,{\rm cpt},(2)}$ similarly.

Fix a compact supported smooth 1-form $e(\rho)=f(\rho)d\rho$ on $(0,1)$ where
$f\geq 0$ with its total integral 1. Define

$\displaystyle e_{\ast}:A^{\bullet}_{X,{\rm cpt},(2)}\to
A^{\bullet}_{(0,1)\times X,{\rm cpt},(2)},\quad
e_{\ast}(\alpha)=e\wedge\alpha,$

and

$\displaystyle
H(\alpha)=\int^{\rho}_{0}\alpha_{1}dt-\int^{\rho}_{0}e\int^{1}_{0}\alpha_{1}dt.$

It follows directly from the definition that $H\alpha$ has compact support and
$q_{\ast}e_{\ast}={\rm Id}$. We are going to show that $e_{\ast}q_{\ast}$ is
homotopic to the identity. For this we need to show the following two claims.

A:

$\nabla H+H\nabla={\rm Id}-e_{\ast}q_{\ast}$. This follows from the equalities
(4.7) and (4.8) below:

(4.7) $\displaystyle\nabla H(\alpha)$
$\displaystyle=\nabla\left(\int^{\rho}_{0}\alpha_{1}dt-\int^{\rho}_{0}e\int^{1}_{0}\alpha_{1}dt\right)$
$\displaystyle=d\rho\wedge\alpha_{1}+\int_{0}^{\rho}\nabla_{\mathbb{V}}\alpha_{1}dt-e\wedge\int_{0}^{1}\alpha_{1}dt-\int_{0}^{\rho}e\int_{0}^{1}\nabla_{\mathbb{V}}\alpha_{1}dt.$
(4.8) $\displaystyle H\nabla(\alpha)$
$\displaystyle=\int_{0}^{\rho}\left(\frac{\partial}{\partial
t}\alpha_{0}-\nabla_{\mathbb{V}}\alpha_{1}\right)dt-\int_{0}^{\rho}e\int_{0}^{1}\left(\frac{\partial}{\partial
t}\alpha_{0}-\nabla_{\mathbb{V}}\alpha_{1}\right)dt$
$\displaystyle=\alpha_{0}-\int_{0}^{\rho}\nabla_{\mathbb{V}}\alpha_{1}dt+\int_{0}^{\rho}e\int_{0}^{1}\nabla_{\mathbb{V}}\alpha_{1}dt.$

B:

$H$ sends compact supported $L^{2}$ forms to compact supported $L^{2}$ forms.
This follows from the estimate

(4.9) $\displaystyle\int_{(0,1)\times
X}\left|\int^{t}_{0}d\rho\wedge\alpha_{1}-\int^{t}_{0}e\int^{1}_{0}d\rho\wedge\alpha_{1}\right|^{2}{\rm
vol}_{(0,1)\times X}$ $\displaystyle\leq$ $\displaystyle 2\int_{(0,1)\times
X}\left|\int^{t}_{0}d\rho\wedge\alpha_{1}\right|^{2}{\rm vol}_{(0,1)\times
X}+2\int_{(0,1)\times
X}\left|\int^{t}_{0}e\int^{1}_{0}d\rho\wedge\alpha_{1}\right|^{2}{\rm
vol}_{(0,1)\times X}$ $\displaystyle\leq$ $\displaystyle 2\int_{(0,1)\times
X}\int^{1}_{0}\left|\alpha_{1}\right|^{2}d\rho{\rm vol}_{(0,1)\times
X}+2\int_{(0,1)\times
X}\left|\int^{1}_{0}e\right|^{2}\int^{1}_{0}\left|\alpha_{1}\right|^{2}d\rho{\rm
vol}_{(0,1)\times X}$ $\displaystyle=$ $\displaystyle
4\int_{X}\int_{0}^{1}|dt\wedge\alpha_{1}|^{2}dt{\rm vol}_{X}$
$\displaystyle\leq$ $\displaystyle 4\|\alpha\|^{2}_{(0,1)\times X}<\infty.$

As a consequence,

$H:A^{\bullet}_{(0,1)\times X,{\rm cpt},(2)}\to A^{\bullet-1}_{(0,1)\times
X,{\rm cpt},(2)}$

is well defined and gives a homotopy from $e_{\ast}q_{\ast}$ to the identity.
Since both $e_{\ast}$ and $H$ are continuous operators (claim B), $\nabla
H+H\nabla={\rm Id}-e_{\ast}q_{\ast}$ extends to a homotopy from

$e_{\ast}q_{\ast}:\Gamma_{\rm cpt}\left((0,1)\times
X,\mathscr{D}^{\bullet}_{(0,1)\times X,q^{\ast}\mathbb{V};ds^{2}_{(0,1)\times
X},q^{\ast}h}\right)\to\Gamma_{\rm cpt}\left((0,1)\times
X,\mathscr{D}^{\bullet}_{(0,1)\times X,q^{\ast}\mathbb{V};ds^{2}_{(0,1)\times
X},q^{\ast}h}\right)$

to the identity. This proves the lemma. ∎

###  4.2. A formal criteria

In this subsection we give a sufficient condition for the metrics $ds^{2}$ and
$h$ to let the cohomology sheaves of
$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ be weakly constructible.

######  Definition 4.3.

Let $X$ be a complex space of pure dimension $n$ and $X^{o}\subset X_{\rm
reg}$ a dense Zariski open subset. A stratification $\\{X_{i}\\}$ of
$(X,X^{o})$ is called a bimeromorphic fibering stratification if for every
stratum $S$ and every point $x\in S$, there is an analytic subspace
$C_{x}\subset X$ passing through $x$, a neighborhood $x\in U_{x}$ in $X$, a
neighborhood $x\in U_{S,x}$ in $S$ and a proper bimeromorphic morphism
$\pi_{x}:\widetilde{U_{x}}\to U_{x}$ such that

  1. (1)

$\pi_{x}$ is biholomorphic over $U_{x}\cap X^{o}$.

  2. (2)

There is a neighborhood $\pi^{-1}_{x}(U_{S,x})\subset
V\subset\widetilde{U_{x}}$ and a $C^{\infty}$ diffeomorphism

$\alpha_{x}:V\simeq(V\cap\pi^{-1}_{x}C_{x})\times U_{S,x}$

such that

     * •

$\alpha_{x}(y)=(y,x)$ for any $y\in V\cap\pi^{-1}_{x}C_{x}$,

     * •

$p_{2}\circ\alpha_{x}|_{\pi^{-1}_{x}U_{S,x}}=\pi_{x}|_{\pi^{-1}_{x}U_{S,x}}$
where $p_{2}:(V\cap\pi^{-1}_{x}C_{x})\times U_{S,x}\to U_{S,x}$ is the
projection.

     * •

$\alpha_{x}(V\cap\pi^{-1}_{x}X^{o})=(V\cap\pi^{-1}_{x}(C_{x}\cap X^{o}))\times
U_{S,x}$.

The six tuple $(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$ is called a
bimeromorphic fibering neighborhood of $x\in S$.

In this paper we mainly use bimeromorphic fibering stratifications and
bimeromorphic fibering neighborhoods produced by a Whitney stratified
desingularization (Definition 4.10).

######  Definition 4.4.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset with a hermitian metric $ds^{2}$ on $X^{o}$. Let $(\mathbb{V},h)$ be a
hermitian local system on $X^{o}$. Let $\\{X_{i}\\}$ be a bimeromorphic
fibering stratification of $(X,X^{o})$. We say that $(ds^{2},h)$ is
bimeromorphically stratified along $\\{X_{i}\\}$ if for every stratum $S$ and
every point $x\in S$, there is a bimeromorphic fibering neighborhood
$(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$ of $x\in S$ such that the
following statements hold.

  1. (1)

$\pi^{\ast}_{x}ds^{2}|_{V}\sim\pi^{\ast}_{x}ds^{2}|_{\pi^{-1}_{x}C_{x}\cap
V}+ds^{2}_{S}$ via $\alpha_{x}$ for some hermitian metric $ds^{2}_{S}$ on $S$.

  2. (2)

There is an isomorphism of the local systems222This trivialization exists when
$U_{S_{x}}$ is contractible.

$q^{\ast}(\mathbb{V}|_{\pi^{-1}_{x}(C_{x}\cap X^{o})\cap
V})\simeq\mathbb{V}|_{V\cap\pi^{-1}_{x}X^{o}}$

where $q:=p_{1}\circ\alpha_{x}:V\simeq(\pi^{-1}_{x}C_{x}\cap V)\times
U_{S,x}\to(\pi^{-1}_{x}C_{x}\cap V)$ is the projection. Moreover

$h|_{V\cap\pi^{-1}_{x}X^{o}}\sim q^{\ast}(h|_{\pi^{-1}_{x}(C_{x}\cap
X^{o})\cap V}).$

In this case we say that $(ds^{2},h)$ is stratified along
$(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$.

######  Proposition 4.5.

Let $X$ be a complex space of pure dimension $n$ and $X^{o}\subset X_{\rm
reg}$ a dense Zariski open subset with a hermitian metric $ds^{2}$ on $X^{o}$.
Let $(\mathbb{V},h)$ be a hermitian local system on $X^{o}$. Assume that
$(ds^{2},h)$ is bimeromorphically stratified along a bimeromorphic fibering
stratification $\\{X_{i}\\}$ of $(X,X^{o})$, then the cohomology sheaves of
$\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ are weakly constructible with
respect to $\\{X_{i}\\}$.

Assume moreover that $\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ is a
complex of fine sheaves. Let $(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$ be a
bimeromorphic fibering neighborhood of $x\in S$ for some stratum $S$ along
which $(ds^{2},h)$ is bimeromorphically stratified. For every $k$ there are
canonical isomorphisms

(4.10) $\displaystyle
H^{k}\left(i^{\ast}_{x}\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}\right)\simeq\varinjlim_{x\in
U}H^{k}\left(\Gamma(U,\mathscr{D}^{\bullet}_{C_{x},\mathbb{V}|_{C_{x}};ds^{2}|_{C_{x}},h|_{C_{x}}})\right)$

and

(4.11) $\displaystyle
H^{k}\left(i^{!}_{x}\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}\right)\simeq\varinjlim_{x\in
U}H^{k-2\dim S}\left(\Gamma_{\rm
cpt}(U,\mathscr{D}^{\bullet}_{C_{x},\mathbb{V}|_{C_{x}};ds^{2}|_{C_{x}},h|_{C_{x}}})\right)^{\ast}.$

Here $i_{x}:\\{x\\}\to X$ is the inclusion and $U$ ranges over the open
neighborhoods of $x$ in $C_{x}$.

###### Proof.

To show that
$h^{k}(\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})|_{X_{d}\backslash
X_{d-1}}$ is a weakly local system for every $k$ and every $d$, it is
equivalent to show that: for every point $x\in X_{d}\backslash X_{d-1}$ and
every precompact neighborhood $x\in W\subset X_{d}\backslash X_{d-1}$ such
that $W$ is diffeomorphic to $(-1,1)^{2d}$, the natural maps

$\displaystyle\Gamma(\overline{W_{\epsilon}},h^{k}(\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}))\to
H^{k}(\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})_{x},\quad\forall\epsilon\in(0,1)$

are isomorphisms. Here we assume that $x=0\in(-1,1)^{2d}$ and denote
$W_{\epsilon}=(-\epsilon,\epsilon)^{2d}$. Equivalently, we shall show that

(4.12) $\displaystyle\varinjlim_{\overline{W_{\epsilon}}\subset
W^{\prime}}H^{k}(\Gamma(W^{\prime},\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}))\to
H^{k}(\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})_{x},\quad\forall\epsilon\in(0,1)$

are isomorphisms where $W^{\prime}$ ranges over open neighborhoods of
$\overline{W_{\epsilon}}$ in $X$.

Let $(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x}:\widetilde{U_{x}}\to U_{x})$ be
a bimeromorphic fibering neighborhood along which $(ds^{2},h)$ is stratified,
i.e. (1)-(2) of Definition 4.4 hold. Without loss of generality we assume that
$\overline{W}\subset U_{S,x}$. Since $\pi_{x}$ is a proper map and
$\pi_{x}={\rm Id}$ over $U_{x}\cap X^{o}$, we obtain that

$\displaystyle H^{k}(\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})_{x}$
$\displaystyle\simeq\varinjlim_{x\in W^{\prime}\subset
U_{x}}H^{k}\left(\Gamma(W^{\prime},\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})\right)$
$\displaystyle\simeq\varinjlim_{\pi^{-1}\\{x\\}\subset\widetilde{W}^{\prime}\subset\widetilde{U_{x}}}H^{k}\left(\Gamma(\widetilde{W}^{\prime},\mathscr{D}^{\bullet}_{\widetilde{U_{x}},\mathbb{V};ds^{2},h})\right)$
$\displaystyle\simeq\varinjlim_{\begin{subarray}{c}\epsilon\to 0\\\ x\in
W_{x}\subset
C_{x}\end{subarray}}H^{k}\left(\Gamma(\alpha^{-1}_{x}(\pi^{-1}_{x}W_{x}\times
W_{\epsilon}),\mathscr{D}^{\bullet}_{\widetilde{U_{x}},\mathbb{V};ds^{2},h})\right)$

and

$\displaystyle\varinjlim_{\overline{W_{\epsilon}}\subset
W^{\prime}}H^{k}(\Gamma(W^{\prime},\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}))$
$\displaystyle\simeq\varinjlim_{\pi^{-1}(\overline{W_{\epsilon}})\subset\widetilde{W}^{\prime}\subset\widetilde{U_{x}}}H^{k}\left(\Gamma(\widetilde{W}^{\prime},\mathscr{D}^{\bullet}_{\widetilde{U_{x}},\mathbb{V};ds^{2},h})\right)$
$\displaystyle\simeq\varinjlim_{\begin{subarray}{c}\epsilon<\epsilon^{\prime}<1\\\
x\in W_{x}\subset
C_{x}\end{subarray}}H^{k}\left(\Gamma(\alpha^{-1}_{x}(\pi^{-1}_{x}W_{x}\times
W_{\epsilon^{\prime}}),\mathscr{D}^{\bullet}_{\widetilde{U_{x}},\mathbb{V};ds^{2},h})\right).$

As a consequence, (4.12) is isomorphic to

$\displaystyle\varinjlim_{\begin{subarray}{c}\epsilon<\epsilon^{\prime}<1\\\
x\in W_{x}\subset
C_{x}\end{subarray}}H^{k}\left(\Gamma(\alpha^{-1}_{x}(\pi^{-1}_{x}W_{x}\times
W_{\epsilon^{\prime}}),\mathscr{D}^{\bullet}_{\widetilde{U_{x}},\mathbb{V};ds^{2},h})\right)\to\varinjlim_{\begin{subarray}{c}\epsilon^{\prime}\to
0\\\ x\in W_{x}\subset
C_{x}\end{subarray}}H^{k}\left(\Gamma(\alpha^{-1}_{x}(\pi^{-1}_{x}W_{x}\times
W_{\epsilon^{\prime}}),\mathscr{D}^{\bullet}_{\widetilde{U_{x}},\mathbb{V};ds^{2},h})\right)$

for every $\epsilon\in(0,1)$, where $W_{x}$ runs over open neighborhoods of
$x\in C_{x}$. By Lemma 4.1, both sides are isomorphic to

$\displaystyle\varinjlim_{x\in W_{x}\subset
C_{x}}H^{k}\left(\Gamma(W_{x},\mathscr{D}^{\bullet}_{C_{x},\mathbb{V}|_{C_{x}};ds^{2}|_{C_{x}},h|_{C_{x}}})\right).$

This proves that (4.12) is an isomorphism. This argument also deduces (4.10).
If moreover $\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}$ is a complex of
fine sheaves, (4.11) follows from the calculation

$\displaystyle
h^{k}\left(i^{!}_{x}\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h}\right)$
$\displaystyle\simeq\varinjlim_{x\in W^{\prime}}\mathbb{H}^{k}_{\rm
cpt}(W^{\prime},\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})^{\ast}$
$\displaystyle\simeq\varinjlim_{x\in W^{\prime}}H^{k}\left(\Gamma_{\rm
cpt}(W^{\prime},\mathscr{D}^{\bullet}_{X,\mathbb{V};ds^{2},h})\right)^{\ast}\quad(\textrm{fineness})$
$\displaystyle\simeq\varinjlim_{\pi^{-1}\\{x\\}\subset\widetilde{W}^{\prime}\subset\widetilde{U_{x}}}H^{k}\left(\Gamma_{\rm
cpt}(\widetilde{W}^{\prime},\mathscr{D}^{\bullet}_{\widetilde{U_{x}},\mathbb{V};ds^{2},h})\right)^{\ast}$
$\displaystyle\simeq\varinjlim_{\begin{subarray}{c}\epsilon>0\\\ x\in
W_{x}\subset C_{x}\end{subarray}}H^{k}\left(\Gamma_{\rm
cpt}(\alpha_{x}^{-1}(\pi_{x}^{-1}W_{x}\times
W_{\epsilon}),\mathscr{D}^{\bullet}_{\widetilde{U_{x}},\mathbb{V};ds^{2},h})\right)^{\ast}$
$\displaystyle\simeq\varinjlim_{x\in W_{x}\subset C_{x}}H^{k-2\dim
S}\left(\Gamma_{\rm
cpt}(W_{x},\mathscr{D}^{\bullet}_{C_{x},\mathbb{V}|_{C_{x}};ds^{2}|_{C_{x}},h|_{C_{x}}})\right)^{\ast}\quad\textrm{(Lemma
\ref{lem_Thom_locL2})}.$

This proves Proposition 4.5. ∎

###  4.3. Whitney stratified desingularization

A desingularization of a complex space provides local coordinates to study the
asymptotic behavior of a given metric and to calculate its local
$L^{2}$-cohomologies (see [Hsiang_Pati1985, Saper1992]). However it is
difficult to study the $L^{2}$-representation of the intersection complex via
a general desingularization because it does not witnesses the stratification.
The aim of this subsection is to combine the desingularization and the
stratification of a complex space.

######  Definition 4.6.

Let $X$ be a complex space of dimension $n$ and $X^{o}\subset X_{\rm reg}$ a
dense Zariski open subset. A Whitney stratified desingularization of
$(X,X^{o})$ consists of the following data.

  1. (1)

A projective morphism $\pi:\widetilde{X}\to X$ such that

    1. (a)

$\tilde{X}$ is smooth,

    2. (b)

$\pi|_{\pi^{-1}X^{o}}:\pi^{-1}X^{o}\to X^{o}$ is biholomorphic.

  2. (2)

A Whitney stratification $\\{X_{i}\\}_{0\leq i\leq n}$ of $(X,X^{o})$ such
that

    1. (a)

For every $i<n$, $\pi^{-1}X_{i}$ is a simple normal crossing divisor of
$\widetilde{X}$.

    2. (b)

Denote by $(\pi^{-1}X_{n-1})_{\rm red}=:E=\cup_{i\in I}E_{i}$ the irreducible
decomposition of the (reduced) exceptional divisor of $\pi$. For every stratum
$S$ of $\\{X_{i}\\}$ and every stratum $E_{J}:=\bigcap_{j\in J}E_{j}$,
$\emptyset\neq J\subset I$, either $S\cap\pi(E_{J})=\emptyset$ or
$\pi:\pi^{-1}S\cap E_{J}\to S$ is a submersion.

######  Theorem 4.7.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. Let $Y\subset X$ be a compact analytic subspace. Then there is a
neighborhood $U$ of $Y$ and a Whitney stratified desingularization of
$(U,U^{o}=X^{o}\cap U)$. In particular, Whitney stratified desingularization
exists in the following cases.

  * •

$X$ is a compact complex space.

  * •

$\overline{X}$ is a compact complex space and
$\overline{X}^{o}\subset\overline{X}_{\rm reg}$ is a dense Zariski open
subset. $X\subset\overline{X}$ is a Zariski open subset and
$X^{o}=\overline{X}^{o}\cap X$. For example $(X,X^{o})$ is a pair of complex
algebraic varieties.

  * •

$X$ is a germ of complex space.

###### Proof.

The proof is done by induction on the depth. Let $n=\dim X$.

Initial step: Let $U$ be some neighborhood of $Y$ so that there is a
projective map $\pi_{1}:U^{1}\to U$ satisfying the following statements.

  * •

$U^{1}$ is smooth, $\pi_{1}:\pi_{1}^{-1}U^{o}\to U^{o}$ is biholomorphic.

  * •

$E^{(1)}:=\pi_{1}^{-1}(U\backslash U^{o})$ is a simple normal crossing
divisor.

Take an arbitrary Whitney stratification $\\{U_{i}^{(1)}\\}$ of $(U,U^{o})$
(c.f. [Verdier1976, Theorem 2.2]). Condition (2) in Definition 4.6 holds for
$\pi_{1}:\pi_{1}^{-1}(U^{o})\to U^{o}$.

Induction step: Assume that there is a map $\pi_{k}:U^{k}\to U$ and a Whitney
stratification $\\{U_{i}^{(k)}\\}$ such that condition (1) holds for $\pi_{k}$
and condition (2) holds over $U\backslash U^{(k)}_{n-k}$.

After a possible shrinking of $U$, we may take a projective map
$\rho:U^{k+1}\to U^{k}$ (an embedding resolution of
$\pi^{-1}_{k}(U^{(k)}_{n-k})\subset U^{k}$) so that

  * •

$U^{k+1}$ is smooth and $\rho:\rho^{-1}\pi_{k}^{-1}(U\backslash
U^{(k)}_{n-k})\to\pi_{k}^{-1}(U\backslash U^{(k)}_{n-k})$ is biholomorphic.

  * •

Denote $\pi_{k+1}=\rho\pi_{k}$, then
$E^{\prime}:=\pi^{-1}_{k+1}(U^{(k)}_{n-k})$ and $\pi^{-1}_{k+1}(U\backslash
U^{o})$ are simple normal crossing divisors.

Denote by $E^{\prime}=\bigcup_{j\in I^{\prime}}E^{\prime}_{j}$ the irreducible
decomposition and denote by $E^{\prime}_{J}=\bigcup_{j\in J}E^{\prime}_{j}$
for every $\emptyset\neq J\subset I^{\prime}$. By generic smoothness (Bertini-
Sard theorem) there is a smooth dense Zariski open subset $S\subset
U_{n-k}^{(k)}\backslash U_{n-k-1}^{(k)}$ so that the following claims hold.

  * •

For every stratum $E^{\prime}_{J}$, $\emptyset\neq J\subset I^{\prime}$,
either $S\cap\pi_{k+1}(E^{\prime}_{J})=\emptyset$ or
$\pi_{k+1}:\pi^{-1}_{k+1}(S)\cap E^{\prime}_{J}\to S$ is a submersion.

  * •

By [Verdier1976, Theorem 2.2], after removing a nowhere dense Zariski closed
subspace of $S$, we assume that $(S,S^{\prime})$ satisfies Verdier’s condition
(w) for every stratum $S^{\prime}$ of $\\{U_{i}^{(k)}\\}$ such that
$S\subset\overline{S^{\prime}}$. In particular, such $(S,S^{\prime})$
satisfies Whitney’s conditions (a) and (b).

By [Verdier1976, Theorem 2.2], there is a Whitney stratification
$\\{U^{(k+1)}_{i}\\}$ of $(U,U^{o})$ so that

  * •

$\\{U^{(k+1)}_{i}\\}$ refines $\\{U^{(k)}_{i}\\}$,

  * •

For every $i\geq n-k$, $U^{(k+1)}_{i}=U^{(k)}_{i}$.

  * •

$S=U^{(k+1)}_{n-k}\backslash U_{n-k-1}^{(k+1)}$.

By the constructions, $(\\{U^{(k+1)}_{i}\\},\pi_{k+1}:U^{k+1}\to U)$ satisfies
condition (1) in Definition 4.6 and condition (2) holds over $U\backslash
U^{(k+1)}_{n-k-1}$. Since $\dim(U\backslash
U^{(k+1)}_{n-k-1})<\dim(U\backslash U^{(k)}_{n-k})$, this procedure stops
after finite steps. We prove the theorem. ∎

######  Lemma 4.8.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. Let $(\\{X_{i}\\},\pi:\widetilde{X}\to X)$ be a Whitney stratified
desingularization of $(X,X^{o})$. Let $Z\subset X$ be an analytic subspace
such that $Z\cap X^{o}\neq\emptyset$ and $Z$ intersects transversally with
every stratum of $\\{X_{i}\\}$. Then $(\\{X_{i}\cap
Z\\},\pi|_{\pi^{-1}Z}:\pi^{-1}Z\to Z)$ is a Whitney stratified
desingularization of $(Z,X^{o}\cap Z)$.

###### Proof.

Since $Z$ intersects transversally with every stratum of $\\{X_{i}\\}$,
$\\{X_{i}\cap Z\\}$ is a Whitney stratification of $Z$. It remains to show
that $\pi^{-1}Z$ is smooth and $\pi^{-1}(X_{i}\cap Z)\subset\pi^{-1}Z$ is a
simple normal crossing divisor for every $i$ such that $X_{i}\cap
Z\neq\emptyset$. Denote by $E=\cup E_{i}$ the underlying reduced subspace of
the exceptional divisor of $\pi$. By (2-b) in Definition 4.6, the (schematic)
intersection $E_{J}\cap\pi^{-1}Z$ is either empty or smooth for every stratum
$E_{J}$ of $E$. Recall the following fact: for a point $x\in Y$ of a complex
space and $D\subset Y$ a Cartier divisor passing through $x$, if $D$ is smooth
at $x$, then $Y$ is smooth at $x$. As a consequence $\pi^{-1}Z$ is smooth and
$\pi^{-1}(X_{i}\cap Z)\subset\pi^{-1}Z$ is either empty or a simple normal
crossing divisor of $\pi^{-1}Z$. Condition (2-b) follows directly from the
construction. ∎

######  Proposition 4.9.

Let $X$ be complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. Let $(\\{X_{i}\\},\pi:\widetilde{X}\to X)$ be a Whitney stratified
desingularization of $(X,X^{o})$. Then $\\{X_{i}\\}$ is a bimeromorphic
fibering stratification of $(X,X^{o})$.

To be precise, let $S$ be a stratum of $\\{X_{i}\\}$ and $x\in S$. Let
$C_{x}\subset X$ be a $\dim X-\dim S$ dimensional germ of analytic subspace
passing through $x$ and intersecting transversally with every stratum of
$\\{X_{i}\\}$ ($C_{x}$ could be the cut off of $X$ by $\dim S$ hypersurfaces
in general positions). There is a neighborhood $V$ of $\pi^{-1}\\{x\\}$ in
$\widetilde{X}$, a neighborhood $x\in U_{S,x}$ in $S$ and a $C^{\infty}$
diffeomorphism

(4.13) $\displaystyle\alpha_{x}:V\simeq(V\cap\pi^{-1}C_{x})\times U_{S,x}$

such that

  1. (1)

$\alpha_{x}(y)=(y,x)$ for any $y\in V\cap\pi^{-1}C_{x}$,

  2. (2)

$p_{2}\circ\alpha_{x}|_{\pi^{-1}(U_{S,x})}=\pi|_{\pi^{-1}(U_{S,x})}$ where
$p_{2}:(V\cap\pi^{-1}C_{x})\times U_{S,x}\to U_{S,x}$ is the projection.

  3. (3)

$\alpha_{x}(V\cap\pi^{-1}X^{o})=(V\cap\pi^{-1}(C_{x}\cap X^{o}))\times
U_{S,x}$.

  4. (4)

For every stratum $E_{J}$ ($\emptyset\neq J\subset I$) of the exceptional
divisor $E=\bigcup_{i\in I}E_{i}$ of $\pi$,

$\alpha_{x}(V\cap E_{J})=(V\cap\pi^{-1}C_{x}\cap E_{J})\times U_{S,x}.$

###### Proof.

Assume that $S\subset X_{r}\backslash X_{r-1}$, i.e. $\dim S=r$. By the
definition of Whitney stratified desingularization, $\pi^{-1}S$ is a simple
normal crossing divisor of the complex manifold $Y:=\pi^{-1}(X_{r-1}\cup
S)\subset\widetilde{X}$. Let $E=\bigcup_{i\in I}E_{i}$ be the reduced
exceptional divisor of $\pi$. Let $I_{S}\subset I$ be the subset of indexes
$i\in I$ so that $E_{i}\cap Y\subset\pi^{-1}S$ and $I^{\prime}_{S}\subset I$
is the subset of indexes $i\in I$ so that $E_{i}\cap\pi^{-1}S\neq\emptyset$.
Then for every $J\subset I_{S}$, $\pi:E_{J}\to S$ is a proper submersion. Let
$U_{S,x}\subset S$ be an open neighborhood of $x$ so that its holomorphic
tangent bundle admits a frame $w_{1},\dots,w_{r}$. To get the data
$(V,\alpha_{x})$ claimed in the theorem, it suffices to construct $C^{\infty}$
complex vector fields $v_{1},\dots,v_{r}$ on a neighborhood of
$\pi^{-1}\\{x\\}\subset Y$ such that the following statements hold.

  1. (1)

For every $i=1,\dots,r$ and every $J\subset I_{S}$, $v_{i}|_{E_{J}}\in
T_{E_{J}}$ and lifts $w_{i}$ through $\pi$, i.e. for every $x\in E_{J}$,
$\pi_{\ast}(v_{i}(x))=w_{i}(\pi(x))$.

  2. (2)

The restrictions of $v_{1},\dots,v_{r}$ on $V\cap\pi^{-1}C_{x}$ form a
$C^{\infty}$ frame of the holomorphic normal bundle of
$V\cap\pi^{-1}C_{x}\subset\widetilde{X}$.

  3. (3)

For every $i=1,\dots,r$ and every $J\subset I$, $v_{i}|_{E_{J}}\in T_{E_{J}}$.

Then the vector fields $v_{i}$ and $\overline{v_{i}}$ give the one-parameter
groups $h_{i},\overline{h}_{i}:(-\epsilon,\epsilon)\times W\to\widetilde{X}$
where $\epsilon>0$ is a constant independent of $i$ and
$\pi^{-1}\\{x\\}\subset W\subset\pi^{-1}C_{x}$ is an open subset. By the
conditions (1)-(3) of the vector fields $v_{1},\dots,v_{r}$, the map

$\alpha_{x}^{-1}:(-\epsilon,\epsilon)^{2r}\times W\to\widetilde{X},$
$\alpha_{x}^{-1}(t_{1},\dots,t_{2r},x)=h_{1}(t_{1},\overline{h}_{1}(t_{2},h_{2}(t_{3},\overline{h}_{2}(\dots
h_{r}(t_{2r-1},\overline{h}_{r}(t_{2r},x))\dots))))$

is well defined and its inverse is the fibering $\alpha_{x}$ that we need for
the theorem.

It remains to construct the vector fields $v_{1},\dots,v_{r}$. This can be
done by three steps.

Step 1:

Notice that $\pi^{-1}S\to S$ is a stratified submersion, i.e. every stratum is
mapped submersively onto $S$. One may lift $w_{1},\dots,w_{r}$ to $C^{\infty}$
vector fields $w^{\prime}_{1},\dots,w^{\prime}_{r}$ on $\pi^{-1}S$ so that
$w^{\prime}_{i}|_{E_{J}}\in T_{E_{J}}$, $\forall J\subset I_{S}$. Actually one
can lift $w_{1},\dots,w_{r}$ to the minimal strata of $\pi^{-1}S$ and extend
them to strata of higher dimensions step by step.

Step 2:

Extend $w^{\prime}_{1},\dots,w^{\prime}_{r}$ to $C^{\infty}$ vector fields
$w^{\prime\prime}_{1},\dots,w^{\prime\prime}_{r}$ on $\bigcup_{i\in
I^{\prime}_{S}}E_{i}$ so that $w^{\prime\prime}_{i}|_{E_{J}}\in T_{E_{J}}$ for
every $J\subset I^{\prime}_{S}$.

Step 3:

Extend $w^{\prime\prime}_{1},\dots,w^{\prime\prime}_{r}$ to $C^{\infty}$
vector fields $v_{1},\dots,v_{r}$ on $\widetilde{X}$. By the previous two
steps, conditions (1) and (3) hold. Since the vector fields are $C^{\infty}$,
condition (2) automatically holds when $V$ is sufficiently small.

Notice that $E$ admits only simple normal crossing singularities. The
$C^{\infty}$ extensions of the vector fields are possible in each step. ∎

######  Definition 4.10.

Notations as in Proposition 4.9, $(U_{x},U_{S,x},C_{x},V,\pi,\alpha_{x})$ is
called a bimeromorphic fibering neighborhood of $x\in S$ with respect to the
Whitney stratified desingularization $(\\{X_{i}\\},\pi:\widetilde{X}\to X)$.

The following proposition shows that the Whitney stratified desingularization
owns good metrical properties. That is, if $(\\{X_{i}\\},\pi:\widetilde{X}\to
X)$ is a Whitney stratified desingularization of $X\subset\mathbb{C}^{N}$, the
euclidean metric is bimeromorphically stratified along every stratum of
$\\{X_{i}\\}$ in a nice way.

######  Proposition 4.11.

Let $X$ be a germ of complex space and $X^{o}\subset X_{\rm reg}$ a dense
Zariski open subset. Let $ds^{2}_{0}$ be a hermitian metric on $X$ and let
$(\\{X_{i}\\},\pi:\widetilde{X}\to X)$ be a Whitney stratified
desingularization of $(X,X^{o})$. Then $ds^{2}_{0}$ is stratified along any
bimeromorphic fibering neighborhood with respect to
$(\\{X_{i}\\},\pi:\widetilde{X}\to X)$.

Precisely, let $x\in S$ be a point in a stratum $S$ of $\\{X_{i}\\}$ and let
$(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$ be a bimeromorphic fibering
neighborhood of $x\in S$ with respect to $(\\{X_{i}\\},\pi:\widetilde{X}\to
X)$. Then $ds^{2}_{0}$ is stratified along
$(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$.

###### Proof.

Let $v_{1},\cdots,v_{2d}$ be the defining vector fields of the diffeomorphism
$\alpha_{x}$, as constructed in the proof of Proposition 4.9. Denote by
$\omega_{0}$ the $(1,1)$-form associated to $ds^{2}_{0}$. We claim that there
is a neighborhood $U\subset X$ of $x$ such that

  1. (1)

For every $i=1,\dots,2d$, $\|v_{i}\|_{\omega_{0}}\sim 1$ over $U^{o}:=U\cap
X^{o}$. For simplicity we identify $U^{o}$ and $\pi^{-1}U^{o}$.

  2. (2)

There is a constant $\epsilon>0$ such that
$\angle_{\omega_{0}}(v_{i},v_{j})>\epsilon$ over $U^{o}$ for every $1\leq
i\neq j\leq 2d$.

  3. (3)

There is a constant $\epsilon>0$ such that
$\angle_{\omega_{0}}(v_{i},w)>\epsilon$ over $U^{o}$ for every $i=1,\dots,2d$
and every $w\in T_{C_{x}\cap X^{o}}$.

  4. (4)

Denote by

$q:V\simeq V\cap\pi^{-1}C_{x}\times U_{S,x}\to V\cap\pi^{-1}C_{x}$

and

$p:V\simeq V\cap\pi^{-1}C_{x}\times U_{S,x}\to U_{S,x}$

the projections. By abuse of notations we identify $X^{o}$ and
$\pi^{-1}X^{o}$. Let $\widetilde{C}_{s}:=p^{-1}\\{s\\}$ for $s\in U_{S,x}$.
There is a constant $C>0$ such that

$C^{-1}\omega_{0}|_{\widetilde{C}_{s}}\leq\left(q^{\ast}(\omega_{0}|_{\widetilde{C}_{x}})\right)|_{\widetilde{C}_{s}}\leq
C\omega_{0}|_{\widetilde{C}_{s}}$

for every $s\in U\cap U_{S,x}$.

Assuming these claims, for every $\xi=\sum_{i=1}^{d}f_{i}v_{i}+q^{\ast}w$
where $w\in T_{C_{x}\cap X^{o}}$ we have

$\displaystyle\|\xi\|_{\omega_{0}}^{2}$
$\displaystyle=\sum_{i=1}^{d}\|f_{i}\|^{2}\|v_{i}\|^{2}_{\omega_{0}}+\sum_{i=1}^{d}2{\rm
Re}(f_{i}v_{i},q^{\ast}w)+\sum_{i\neq j}2{\rm
Re}(f_{i}v_{i},f_{j}v_{j})+\|q^{\ast}w\|^{2}_{\omega_{0}}$ $\displaystyle\leq
3\sum_{i=1}^{d}\|f_{i}\|^{2}\|v_{i}\|^{2}_{\omega_{0}}+3\|q^{\ast}w\|^{2}_{\omega_{0}}$
$\displaystyle\leq
3\sum_{i=1}^{d}\|f_{i}\|^{2}\|v_{i}\|^{2}_{\omega_{0}}+3C\|w\|^{2}_{\omega_{0}|_{C_{x}}}$

and

$\displaystyle\|\xi\|_{\omega_{0}}^{2}$
$\displaystyle=\sum_{i=1}^{d}\|f_{i}\|^{2}\|v_{i}\|^{2}_{\omega_{0}}+\sum_{i=1}^{d}2{\rm
Re}(f_{i}v_{i},q^{\ast}w)+\sum_{i\neq j}2{\rm
Re}(f_{i}v_{i},f_{j}v_{j})+\|q^{\ast}w\|^{2}_{\omega_{0}}$
$\displaystyle\geq\sum_{i=1}^{d}\|f_{i}\|^{2}\|v_{i}\|^{2}_{\omega_{0}}+\|q^{\ast}w\|^{2}_{\omega_{0}}-\sum_{i=1}^{d}2\epsilon\|f_{i}v_{i}\|_{\omega_{0}}\|q^{\ast}w\|_{\omega_{0}}-\sum_{i\neq
j}2\epsilon\|f_{i}v_{i}\|_{\omega_{0}}\|f_{j}v_{j}\|_{\omega_{0}}$
$\displaystyle\geq\sum_{i=1}^{d}(1-\epsilon)\|f_{i}\|^{2}\|v_{i}\|^{2}_{\omega_{0}}+(1-\epsilon)C^{-1}\|w\|^{2}_{\omega_{0}|_{C_{x}}}.$

As a consequence,

$\omega_{0}\sim ds^{2}_{U_{S,x}}+q^{\ast}\omega_{0}|_{\widetilde{C}_{x}}$

is stratified along $(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$. Here
$ds^{2}_{U_{S,x}}$ is some hermitian metric of $U_{S,x}$.

Now we prove claims (1)-(4). By the construction,
$\pi_{\ast}(v_{1})(x),\dots,\pi_{\ast}(v_{d})(x)\in T_{S,x}$ is a basis, hence
we obtain claims (1) and (2) by continuity. Since $T_{C_{x},x}\cap T_{S,x}=0$,
claim (3) follows by continuity. Now we prove claim (4). If it is false, then
there is a sequence of points $\\{x_{n}\\}\in
V\cap\widetilde{C}_{s_{n}}\cap\pi^{-1}X^{o}$ and vectors $v_{n}\in
T_{V,x_{n}}$ such that

  1. (1)

$\\{s_{n}\\}$ converges to $x$ and $\\{x_{n}\\}$ converges to
$x^{\prime\prime}\in\pi^{-1}\\{x\\}$,

  2. (2)

For every $n$, $\|v_{n}\|_{\pi^{\ast}\omega_{0}}=1$.

  3. (3)

For every $n$, either

(4.14) $\displaystyle\|v_{n}\|_{\pi^{\ast}\omega_{0}}\geq
n\|\rho_{s_{n}\ast}v_{n}\|_{\pi^{\ast}\omega_{0}}$

or

(4.15)
$\displaystyle\|v_{n}\|_{\pi^{\ast}\omega_{0}}\leq\frac{1}{n}\|\rho_{s_{n}\ast}v_{n}\|_{\pi^{\ast}\omega_{0}}.$

Here $\rho_{s}:\widetilde{C}_{s}\subset V\stackrel{{\scriptstyle
q}}{{\to}}\widetilde{C}_{x}$ is the diffeomorphism induced by $\alpha_{x}$.

Without loss of generality we assume that (4.14) holds for all $n$. Let
$\gamma:[0,1)\to V\cap\pi^{-1}C_{x}$ be a curve such that

  1. (1)

$\gamma$ is continuous and $\gamma|_{(0,1)}$ is $C^{\infty}$,

  2. (2)

$\gamma(0)=x^{\prime\prime}$ and $q(x_{n})\in\gamma((0,1))$ for every $n$.

Let $Y=q^{-1}{\rm Im}\gamma$, then $Y\cap\pi^{-1}X^{o}$ is an immersed
submanifold with its boundary
$Y\backslash\pi^{-1}X^{o}=q^{-1}\gamma^{-1}\\{0\\}\subset\pi^{-1}S$. Then
$\rho_{s\ast}:T_{V}|_{Y\cap\widetilde{C}_{s}}\to
T_{V}|_{Y\cap\widetilde{C}_{x}}$ is a $C^{\infty}$ map smoothly depending on
$s\in U_{S,x}$. By the constructions, $\pi$ sends $Y$ homeomorphically onto
its image $\pi(Y)$ and $\pi(Y)\cap X^{o}$ is an immersed submanifold of
$X^{o}$ whose boundary is $\pi(Y)\backslash X^{o}$. Hence
$\rho_{s}:\widetilde{C}_{s}\to\widetilde{C}_{x}$ induces a continuous map
$\tau_{s}:\pi(Y\cap\widetilde{C}_{s})\to\pi(Y\cap\widetilde{C}_{x})$ which
continuously depends on $s$. Note that $\alpha_{x}$ does not desend to $X$
since $v_{1},\dots,v_{d}$ are not necessarily liftings of vector fields on
$X$. This is the purpose that we introduce $Y$.

Denote $E_{s}:={\rm Im}\left(\pi_{\ast}T_{V}|_{Y\cap\widetilde{C}_{s}}\to
T_{X}|_{\pi(Y\cap\widetilde{C}_{s})}\right)$, then $E_{s}\subset
T_{X}|_{\pi(Y\cap\widetilde{C}_{s})}$ is a closed subset. Note that
$E_{s,\pi(y)}=T_{X,\pi(y)}\simeq T_{V,y}$ for every $y\in Y\cap\pi^{-1}X^{o}$
and $E_{s,\pi(y)}\simeq T_{V,y}/{\rm Ker}\pi_{\ast,y}$ for $y\in
Y\backslash\pi^{-1}X^{o}$. Since ${\rm Ker}\pi_{\ast,z}=T_{\pi^{-1}\\{x\\},z}$
is the linear span of the tangent cone of $\pi^{-1}\\{x\\}$ at every
$z\in\pi^{-1}\\{s\\}$, $\rho_{s\ast}$ sends ${\rm Ker}\pi_{\ast,y}$ into ${\rm
Ker}\pi_{\ast,x}$ for every (unique) $y\in Y\cap\pi^{-1}\\{s\\}$. Therefore
$\rho_{s\ast}$ induces a continuous map $\tau_{s\ast}:E_{s}\to E_{x}$ which
continuously depends on $s$.

By assumption, $\|\pi_{\ast}(v_{n})\|_{\omega_{0}}=1$. So there is a
subsequence $\pi_{\ast}(v_{n_{k}})\in E_{s_{n_{k}}}$ which converges to some
$v\in E_{x}$ with $\|v\|_{\omega_{0}}=1$. By taking the limit of (4.14)

$\|\pi_{\ast}(v_{n_{k}})\|_{\omega_{0}}\geq
n\|\tau_{s_{n_{k}}\ast}\pi_{\ast}(v_{n_{k}})\|_{\omega_{0}},$

we obtain $1\geq+\infty$. Neither will (4.15) happen by the same argument.
This proves claim (4) and we finish the proof of the proposition. ∎

###  4.4. Distinguished metric

Let $X$ be a complex space of pure dimension $n$ and $X^{o}\subset X_{\rm
reg}$ a dense Zariski open subset. Let $\widetilde{X}\to X$ be a good
desingularization of $X\backslash X^{o}$ in the sense that

  * •

$\widetilde{X}$ is a complex manifold, $\pi$ is a projective holomorphic map
and $\pi|_{\pi^{-1}X^{o}}:\pi^{-1}X^{o}\to X^{o}$ is biholomorphic.

  * •

The (reduced) exceptional loci $E:=\pi^{-1}(X\backslash X^{o})$ is a simple
normal crossing divisor of $\widetilde{X}$ with finite components.

Let $E=\bigcup_{i\in I}E_{i}$ be the irreducible decomposition. Since $\pi$ is
projective, there are constants $a_{i}\in\mathbb{Z}_{>0}$, $i\in I$ such that
$\mathscr{O}_{\widetilde{X}}(-\sum_{i\in I}a_{i}E_{i})$ is $\pi$-positive,
i.e. locally there is a hermitian metric $h$ on
$\mathscr{O}_{\widetilde{X}}(-\sum_{i\in I}a_{i}E_{i})$ such that
$\sqrt{-1}\Theta_{h}(\mathscr{O}_{\widetilde{X}}(-\sum_{i\in
I}a_{i}E_{i}))+\pi^{\ast}\omega_{0}$ is positive definite on $\widetilde{X}$
for some hermitian metric $\omega_{0}$ on $X$.

For each $i\in I$, let $h_{i}$ be a hermitian metric on
$\mathscr{O}_{\widetilde{X}}(E_{i})$ such that $h^{\ast}=\prod_{i\in
I}h_{i}^{\otimes a_{i}}$. Let
$s_{i}\in\Gamma(\widetilde{X},\mathscr{O}_{\widetilde{X}}(E_{i}))$ be the
defining section of $E_{i}$ and denote $|s_{i}|:=|s_{i}|_{h_{i}}$ for short.

By perturbing the vector $(a_{i})_{i\in I}$ slightly in general directions and
multiplying the common denominators we get integers
$a_{ij}\in\mathbb{Z}_{>0}$, $i\in I$, $j=1,\dots,m$ so that

  1. (1)

After a possible shrinking of $X$, $(\mathscr{O}_{\widetilde{X}}(-\sum_{i\in
I}a_{ij}E_{i}),\prod_{i\in I}h^{a_{ij}\ast}_{i})$ is a hermitian line bundle
with $\pi$-positive curvature for every $j=1,\dots,m$.

  2. (2)

${\rm rank}(a_{ij})_{i\in I,j\in J}=m$ for any $J\subset\\{1,\dots,m\\}$ such
that $|J|\leq\min(m,|I|)$.

The distinguished potential is defined by

$\displaystyle\varphi_{\pi}:=\sum_{j=1}^{m}\frac{1}{\log\left(-\log\prod_{i\in
I}|s_{i}|^{2a_{ij}}\right)}.$

Denote $\psi_{j}=-\log\prod_{i\in I}|s_{i}|^{2a_{ij}}$, then

$\displaystyle\partial\bar{\partial}\varphi_{\pi}=\sum_{j=1}^{m}\frac{2+\log\psi_{j}}{\psi_{j}^{2}\log^{3}\psi_{j}}\partial\psi_{j}\wedge\bar{\partial}\psi_{j}+\sum_{j=1}^{m}\frac{-\partial\bar{\partial}\psi_{j}}{\psi_{j}\log^{2}\psi_{j}}.$

Since $-\sqrt{-1}\partial\bar{\partial}\psi_{j}$ is $\pi$-positive for each
$j=1,\dots,m$, locally
$\sqrt{-1}\partial\bar{\partial}\varphi_{\pi}+\omega_{0}$ is positive definite
for some hermitian metric $\omega_{0}$ on $X$.

######  Definition 4.12.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. A hermitian metric $ds^{2}$ on $X^{o}$ is called distinguished
associated to $(X,X^{o})$ if for every point $x\in X$ there is a neighborhood
$U$ and a Whitney stratified desingularization $\pi:\widetilde{U}\to U$ of
$(U,X^{o}\cap U)$ (Definition 4.6) such that

$ds^{2}|_{U}\sim\sqrt{-1}\partial\bar{\partial}\varphi_{\pi}+\omega_{0}.$

Here $\omega_{0}$ is a hermitian metric on $U$. If
$ds^{2}\sim\sqrt{-1}\partial\bar{\partial}\varphi_{\pi}+\omega_{0}$ for some
global Whitney stratified desingularization $\pi:\widetilde{X}\to X$ and some
hermitian metric $\omega_{0}$ on $X$, we say $ds^{2}$ is a distinguished
metric associated to $\pi$.

Every distinguished metric is locally complete and locally of finite volume.
In particular they are complete and of finite volume when $X$ is compact. The
proof of this fact will be postponed to Corollary 5.5.

######  Lemma 4.13.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. Let $Y\subset X$ be a compact analytic subspace. Then there is a
distinguished metric on a neighborhood of $Y$. If moreover $X$ admits a Kähler
hermitian metric, then there is a Kähler distinguished metric on a
neighborhood of $Y$. In particular,

  * •

if $X$ is a germ of complex space, then after a possible shrinking of $X$
there is a Kähler distinguished metric associated to $(X,X^{o})$;

  * •

if $X$ is a compact (Kähler) complex space, then there is a (Kähler)
distinguished metric associated to $(X,X^{o})$;

  * •

if $(X,X^{o})$ is a pair of (quasi-projective) algebraic varieties, then there
is a (Kähler) distinguished metric associated to $(X,X^{o})$.

###### Proof.

By Theorem 4.7, there is an open neighborhood $U$ of $Y$ and a Whitney
stratified desingularization $\pi:\widetilde{U}\to U$ of $(U,X^{o}\cap U)$.
Let $\omega_{0}$ be a (Kähler) hermitian metric on $X$. Then
$\sqrt{-1}\partial\bar{\partial}\varphi_{\pi}+K\omega_{0}$, $K\gg 0$ is a
positive $(1,1)$-form which gives a (Kähler) distinguished metric. ∎

######  Lemma 4.14.

Let $(\\{X_{i}\\},\pi:\widetilde{X}\to X)$ be a Whitney stratified
desingularization of $(X,X^{o})$ and $ds^{2}_{\pi}$ a distinguished metric
associated to $\pi$. Let $\omega_{\widetilde{X}}$ be a hermitian metric on
$\widetilde{X}$ and $\omega_{0}$ a hermitian metric on $X$. Let $S$ be a
stratum of $\\{X_{i}\\}$ and $x\in\widetilde{X}$ such that $\pi(x)\in S$. Let
$(z_{1},\dots,z_{n})$ be a $C^{\infty}$ complex coordinate of $\widetilde{X}$
near $x$ such that $x=(0,\dots,0)$ and the exceptional divisor $E$ of $\pi$ is
defined by $\\{z_{1}\cdots z_{r}=0\\}$. Then there is a quasi-isometry

$\displaystyle ds^{2}_{\pi}$
$\displaystyle\sim\sum_{i=1}^{r}\frac{\sqrt{-1}dz_{i}\wedge
d\overline{z}_{i}}{|z_{i}|^{2}\log^{2}|z_{1}\cdots
z_{r}|^{2}\log^{2}(-\log|z_{1}\cdots z_{r}|^{2})}$
$\displaystyle+\frac{\omega_{\widetilde{X}}}{-\log|z_{1}\cdots
z_{r}|^{2}\log^{2}(-\log|z_{1}\cdots z_{r}|^{2})}+\pi^{\ast}\omega_{0}.$

###### Proof.

We assume that $\\{s_{i}=0\\}=\\{z_{i}=0\\}$ for $i=1,\dots,r$. Hence for
every $i=1,\dots,r$,

$\displaystyle|s_{i}|=f_{i}z_{i}+g_{i}\bar{z_{i}}$

for smooth functions $f_{i}$ and $g_{i}$. On the other hand
$|s_{i}|\simeq\|w_{i}\|$ for some holomorphic function $w_{i}$ which vanishes
on $\\{s_{i}=0\\}$ of order 1. Hence for every $i=1,\dots,r$,

$\displaystyle z_{i}=\tilde{f}_{i}w_{i}+\tilde{g}_{i}\bar{w_{i}}$

for smooth functions $\tilde{f}_{i}$ and $\tilde{g}_{i}$. As a consequence
$|s_{i}|/\|z_{i}\|$ is an invertible $C^{\infty}$ function for every
$i=1,\dots,r$. Denote $\psi_{j}=-\log\prod_{i=1}^{r}|s_{i}|^{2a_{ij}}$. Then

(4.16)
$\displaystyle-\sqrt{-1}\partial\bar{\partial}\psi_{j}+K\pi^{\ast}\omega_{0}\sim\omega_{\widetilde{X}},\quad
K\gg 0$

and $\psi_{j}=\lambda_{j}-\log\prod_{i=1}^{r}\|z_{i}\|^{2a_{ij}}$ for some
invertible $C^{\infty}$ function $\lambda_{j}$. In particular

$\displaystyle\psi_{j}\sim u:=-\log\prod_{i=1}^{r}\|z_{i}\|^{2}.$

Hence we have

(4.17) $\displaystyle\partial\bar{\partial}\varphi_{\pi}$
$\displaystyle=\sum_{j=1}^{m}\frac{2+\log(\psi_{j})}{\psi_{j}^{2}\log^{3}(\psi_{j})}\partial\psi_{j}\wedge\bar{\partial}\psi_{j}+\sum_{j=1}^{m}\frac{-\partial\bar{\partial}\psi_{j}}{\psi_{j}\log^{2}(\psi_{j})}$
$\displaystyle\sim\frac{1}{u^{2}\log^{2}u}\sum_{j=1}^{m}\partial\psi_{j}\wedge\bar{\partial}\psi_{j}+\frac{1}{u\log^{2}u}\sum_{j=1}^{m}-\partial\bar{\partial}\psi_{j}.$

Denote $u_{j}:=-\log\prod_{i=1}^{r}\|z_{i}\|^{2a_{ij}}$, then
$\psi_{j}=\lambda_{j}+u_{j}$. So

(4.18) $\displaystyle\sqrt{-1}\partial\psi_{j}\wedge\bar{\partial}\psi_{j}$
$\displaystyle=\sqrt{-1}\partial(\lambda_{j}+u_{j})\wedge\bar{\partial}(\lambda_{j}+u_{j})$
$\displaystyle=2\sqrt{-1}\partial
u_{j}\wedge\bar{\partial}u_{j}+2\sqrt{-1}\partial\lambda_{j}\wedge\bar{\partial}\lambda_{j}-\sqrt{-1}\partial(\lambda_{j}-u_{j})\wedge\bar{\partial}(\lambda_{j}-u_{j})$
$\displaystyle\lesssim\sqrt{-1}\partial
u_{j}\wedge\bar{\partial}u_{j}+\sqrt{-1}\partial\lambda_{j}\wedge\bar{\partial}\lambda_{j}.$

By the same calculation we know that

(4.19) $\displaystyle\sqrt{-1}\partial
u_{j}\wedge\bar{\partial}u_{j}\lesssim\sqrt{-1}\partial\psi_{j}\wedge\bar{\partial}\psi_{j}+\sqrt{-1}\partial\lambda_{j}\wedge\bar{\partial}\lambda_{j}.$

Combining (4.16), (4.17), (4.18) and (4.19) we obtain that

$\displaystyle ds^{2}_{\pi}$
$\displaystyle\sim\frac{\sqrt{-1}}{u^{2}\log^{2}u}\sum_{\alpha=1}^{m}\partial
u_{\alpha}\wedge\bar{\partial}u_{\alpha}+\frac{\omega_{\widetilde{X}}}{u\log^{2}u}+\pi^{\ast}\omega_{0}$
$\displaystyle=\frac{\sqrt{-1}}{u^{2}\log^{2}u}\sum_{1\leq i,j\leq
r}\left(\sum_{\alpha=1}^{m}a_{\alpha i}a_{\alpha j}\right)\frac{dz_{i}\wedge
d\bar{z}_{j}}{z_{i}\bar{z}_{j}}+\frac{\omega_{\widetilde{X}}}{u\log^{2}u}+\pi^{\ast}\omega_{0}$
$\displaystyle\sim\frac{\sqrt{-1}}{u^{2}\log^{2}u}\sum_{i=1}^{r}\frac{dz_{i}\wedge
d\bar{z}_{i}}{|z_{i}|^{2}}+\frac{\omega_{\widetilde{X}}}{u\log^{2}u}+\pi^{\ast}\omega_{0}.$

∎

######  Remark 4.15.

As a corollary, the distinguished metric only depends on the choice of the
desingularization $\pi:\widetilde{X}\to X$. Its local quasi-isometric class is
independent of the choice of the sections $\\{s_{i}\\}$, the hermitian metrics
$\\{h_{i}\\}$ and the constants $\\{a_{ij}\\}$.

######  Proposition 4.16.

Let $X$ be a germ of complex space and $X^{o}\subset X_{\rm reg}$ a dense
Zariski open subset. Let $(\\{X_{i}\\},\pi:\widetilde{X}\to X)$ be a Whitney
stratified desingularization of $(X,X^{o})$ and $ds^{2}_{\pi}$ the associated
distinguished metric. Then $ds^{2}_{\pi}$ is stratified along any
bimeromorphic fibering neighborhood with respect to
$(\\{X_{i}\\},\pi:\widetilde{X}\to X)$.

More precisely, let $x\in S$ be a point in a stratum $S$ of $\\{X_{i}\\}$ and
let $(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$ be a bimeromorphic fibering
neighborhood of $x\in S$ with respect to the Whitney stratified
desingularization (Definition 4.10). Then $ds^{2}_{\pi}$ is stratified along
$(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$.

Moreover, $ds^{2}_{\pi}|_{C_{x}}$ is a distinguished metric associated to the
Whitney stratified desingularization $(\\{C_{x}\cap
X_{i}\\},\pi:\pi^{-1}C_{x}\to C_{x})$ (c.f. Lemma 4.8).

###### Proof.

Let $x\in S$ for some stratum $S$ of dimension $d$ and let
$(U_{x},U_{S,x},C_{x},V,\pi,\alpha_{x})$ be a bimeromorphic fibering
neighborhood of $x\in S$ with respect to the Whitney stratified
desingularization. Denote by $E$ the exceptional divisor of $\pi$. Let
$y\in\pi^{-1}\\{x\\}$ and let $(z^{\prime}_{1},\dots,z^{\prime}_{n-d})$ be a
holomorphic coordinate of $x^{\prime}\in\pi^{-1}C_{x}$ so that
$E\cap\pi^{-1}C_{x}:=\\{z^{\prime}_{1}\cdots z^{\prime}_{r}=0\\}$ for some
$r\leq n-d$. Let $(z^{\prime}_{n-d+1},\dots,z^{\prime}_{n})$ be a holomorphic
coordinate of $x\in U_{S,x}$, then the diffeomorphism
$\alpha_{x}:V\simeq(V\cap\pi^{-1}C_{x})\times U_{S,x}$ (4.13) gives a
$C^{\infty}$ complex coordinate $(z_{1},\dots,z_{n})$ of $x^{\prime}\in V$
where $z_{i}=z^{\prime}_{i}\alpha_{x}$. Under this coordinate we have
$E=\\{z_{1}\cdots z_{r}=0\\}$.

By Lemma 4.14 there is a quasi-isometry

$\displaystyle ds^{2}_{\pi}$ $\displaystyle\sim\Phi+\pi^{\ast}\omega_{0}$

where

$\Phi=\sum_{i=1}^{r}\frac{\sqrt{-1}dz_{i}\wedge
d\overline{z}_{i}}{|z_{i}|^{2}\log^{2}|z_{1}\cdots
z_{r}|^{2}\log^{2}(-\log|z_{1}\cdots
z_{r}|^{2})}+\frac{\sum_{i=1}^{n}dz_{i}\wedge
d\overline{z}_{i}}{-\log|z_{1}\cdots z_{r}|^{2}\log^{2}(-\log|z_{1}\cdots
z_{r}|^{2})}.$

The quasi-isometric class of $\Phi$ does not depend on the choice of the
coordinate $(z_{n-d+1},\dots,z_{n})$. Hence by Proposition 4.11,
$ds^{2}_{\pi}$ is stratified along $(U_{x},U_{S,x},C_{x},V,\pi,\alpha_{x})$.

By Lemma 4.8, $(\\{C_{x}\cap
X_{i}\\},\pi_{x}:=\pi|_{\pi^{-1}C_{x}}:\pi^{-1}C_{x}\to C_{x})$ is a Whitney
stratified desingularization of $(C_{x},C_{x}\cap X^{o})$. Hence
$\varphi_{\pi}|_{C_{x}}=\varphi_{\pi_{x}}$. Therefore

$ds^{2}_{\pi}|_{C_{x}}=\sqrt{-1}\partial\bar{\partial}\varphi_{\pi_{x}}+\pi^{\ast}_{x}(\omega_{0}|_{C_{x}})$

is a distinguished metric associated to $(\\{C_{x}\cap X_{i}\\},\pi_{x})$. ∎

###  4.5. Harmonic bundles

####  4.5.1. pluriharmonic bundles

Let $(M,ds^{2})$ be a Kähler manifold and $(\mathcal{V},\nabla)$ a holomorphic
vector bundle with a flat connection on $M$. Let $h$ be a hermitian metric on
$\mathcal{V}$ which is not necessarily compatible with $\nabla$. Denote by
$\mathbb{V}:={\rm Ker}\nabla$ the complex local system associated to
$(\mathcal{V},\nabla)$. $(\mathbb{V},h)$ is called a hermitian local system.

Let $[-,-]$ be the graded Lie bracket of operators on
$\mathscr{A}^{\bullet}_{M}(\mathbb{V})$, i.e.

$[T_{1},T_{2}]=T_{1}T_{2}-(-1)^{\deg T_{1}\deg T_{2}}T_{2}T_{1}.$

Let $\nabla=\nabla^{1,0}+\nabla^{0,1}$ be the bi-degree decomposition. There
are unique operators $\delta^{\prime}_{h}$ and $\delta^{\prime\prime}_{h}$ so
that $\nabla^{1,0}+\delta^{\prime\prime}_{h}$ and
$\nabla^{0,1}+\delta^{\prime}_{h}$ are connections compatible with $h$. Denote
$\nabla^{c}_{h}=\delta^{\prime\prime}_{h}-\delta^{\prime}_{h}$. Then we have
the Kähler identities ([Simpson1992, Page 15])

(4.20)
$\displaystyle[\Lambda_{\omega},\nabla]=\sqrt{-1}(\nabla^{c}_{h})^{\ast},\quad[\Lambda_{\omega},\nabla^{c}_{h}]=-\sqrt{-1}\nabla^{\ast}_{h}.$

Here $\omega$ is the Kähler form associated to $ds^{2}$ and $()^{\ast}$ is the
formal adjoint with respect to the metric $h$. Denote

$\displaystyle\Theta_{h}(\nabla):=[\nabla,\nabla^{c}_{h}]=\nabla\nabla^{c}_{h}+\nabla^{c}_{h}\nabla$

and

$\displaystyle\Delta_{\nabla}=\nabla\nabla^{\ast}_{h}+\nabla^{\ast}_{h}\nabla,\quad\Delta_{\nabla^{c}_{h}}={\nabla^{c}_{h}}{\nabla^{c}_{h}}^{\ast}+{\nabla^{c}_{h}}^{\ast}{\nabla^{c}_{h}}.$

We have

######  Lemma 4.17 (Kodaira-Bochner formula for $\nabla$).

$\displaystyle\Delta_{\nabla}=\Delta_{\nabla^{c}_{h}}+[\Lambda_{\omega},\sqrt{-1}\Theta_{h}(\nabla)].$

###### Proof.

$\displaystyle\Delta_{\nabla}-\Delta_{\nabla^{c}_{h}}$
$\displaystyle=\sqrt{-1}\nabla[\Lambda_{\omega},\nabla^{c}_{h}]+\sqrt{-1}[\Lambda_{\omega},\nabla^{c}_{h}]\nabla+\sqrt{-1}\nabla^{c}_{h}[\Lambda_{\omega},\nabla]+\sqrt{-1}[\Lambda_{\omega},\nabla]\nabla^{c}_{h}\quad(\ref{align_Kahler_id2})$
$\displaystyle=\sqrt{-1}\left(-\nabla\nabla^{c}_{h}\Lambda_{\omega}+\Lambda_{\omega}\nabla^{c}_{h}\nabla-\nabla^{c}_{h}\nabla\Lambda_{\omega}+\Lambda_{\omega}\nabla\nabla^{c}_{h}\right)$
$\displaystyle=[\Lambda_{\omega},\sqrt{-1}\Theta_{h}(\nabla)].$

∎

######  Lemma 4.18.

$\Theta_{e^{\varphi}h}(\nabla)=\Theta_{h}(\nabla)+dd^{c}\varphi$, where
$d^{c}=\bar{\partial}-\partial$.

###### Proof.

For every $C^{\infty}$ sections $\alpha$, $\beta$ of $\mathcal{V}$ there are
equalities

(4.21) $\displaystyle
d\left((\alpha,\beta)_{h}e^{\varphi}\right)=\left((\nabla^{1,0}+\delta^{\prime\prime}_{e^{\varphi}h})\alpha,\beta\right)_{h}e^{\varphi}+\left(\alpha,(\nabla^{1,0}+\delta^{\prime\prime}_{e^{\varphi}h})\beta\right)_{h}e^{\varphi}$

and

(4.22) $\displaystyle
d\left((\alpha,\beta)_{h}\right)=\left((\nabla^{1,0}+\delta^{\prime\prime}_{h})\alpha,\beta\right)_{h}+\left(\alpha,(\nabla^{1,0}+\delta^{\prime\prime}_{h})\beta\right)_{h}.$

Combining (4.21) and (4.21) we get

$\displaystyle\delta^{\prime\prime}_{e^{\varphi}h}=\delta^{\prime\prime}_{h}+\bar{\partial}\varphi.$

Similarly,

$\displaystyle\delta^{\prime}_{e^{\varphi}h}=\delta^{\prime}_{h}+\partial\varphi.$

As a consequence

$\displaystyle\Theta_{e^{\varphi}h}(\nabla)$
$\displaystyle=\nabla(\delta^{\prime\prime}_{e^{\varphi}h}-\delta^{\prime}_{e^{\varphi}h})+(\delta^{\prime\prime}_{e^{\varphi}h}-\delta^{\prime}_{e^{\varphi}h})\nabla$
$\displaystyle=\nabla(\nabla^{c}_{h}+d^{c}\varphi)+(\nabla^{c}_{h}+d^{c}\varphi)\nabla$
$\displaystyle=\Theta_{h}(\nabla)+dd^{c}\varphi.$

∎

######  Definition 4.19.

$(\mathbb{V},h)$ or $(\mathcal{V},\nabla,h)$ is called a pluriharmonic bundle
if $\Theta_{h}(\nabla)=0$. In this case $h$ is called a pluriharmonic metric.

The notion of harmonic bundle is used by Simpson [Simpson1992] to establish a
correspondence between local systems and semistable higgs bundles with
vanishing Chern classes over a compact Kähler manifold. A typical example of
pluriharmonic bundles comes from a polarized variation of Hodge structure
(loc. cit.). A pluriharmonic bundle produces a $\lambda$-connection structure
which gives a $\mathscr{D}$-module on $\lambda=1$ and a higgs bundle on
$\lambda=0$. This is the main subject of non-abelian Hodge theory. We will not
go further on this splendid topic. Readers may consult [Simpson1988,
Simpson1990, Simpson1992, Sabbah2005, Mochizuki20072, Mochizuki20071] for more
details.

####  4.5.2. $0$-tame harmonic bundles

The $0$-tame harmonic bundle is a subclass of the tame harmonic bundles in the
sense of Simpson [Simpson1990] and Mochizuki [Mochizuki20072, Mochizuki20071].

######  Definition 4.20.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. A pluriharmonic bundle $(\mathbb{V},h)$ on $X^{o}$ is called a
$0$-tame harmonic bundle on $(X,X^{o})$ if locally at every point $x\in X$ one
has

(4.23)
$\displaystyle(\sum_{i=1}^{r}\|f_{i}\|^{2})^{\epsilon}\lesssim|v|_{h}\lesssim(\sum_{i=1}^{r}\|f_{i}\|^{2})^{-\epsilon},\quad\forall\epsilon>0$

for every (multivalued) flat section $v$. Here $\\{f_{1},\dots,f_{r}\\}$ is a
local generator of the ideal sheaf defining $X\backslash X^{o}\subset X$.

The $0$-tameness is independent of the choice of the local generators
$f_{1},\dots,f_{r}$. Let $\pi:\widetilde{X}\to X$ be a desingularization of
$X\backslash X^{o}\subset X$ so that the preimage of $X\backslash X^{o}$ is
supported on a simple normal crossing divisor $E$. Then (4.23) is equivalent
to the estimate

(4.24) $\displaystyle\|z_{1}\cdots
z_{r}\|^{\epsilon}\lesssim|v|_{h}\lesssim\|z_{1}\cdots
z_{r}\|^{-\epsilon},\quad\forall\epsilon>0$

where $(z_{1},\dots,z_{n})$ is an arbitrary holomorphic local coordinate of
$\widetilde{X}$ such that $E=\\{z_{1}\cdots z_{r}=0\\}$.

One has a more general notion of tame harmonic bundles, with (4.23) replaced
by

$\displaystyle|v|_{h}\lesssim(\sum_{i=1}^{r}\|f_{i}\|^{2})^{-b}$

for some $b>0$. Or equivalently, the eigenvalues of the higgs field are
multivalued holomorphic one-forms with poles along (some resolution of)
$X\backslash X^{o}$ with order at most 1. For a tame harmonic bundle $(V,h)$
on $(X,X^{o})$ and a holomorphic disc $\gamma:\Delta\to X$ so that
$\gamma(0)\in X$ and $\gamma(\Delta^{\ast})\subset X^{o}$, the restriction
$(\gamma^{\ast}\mathbb{V},\gamma^{\ast}h)$ is a tame harmonic bundle on
$\Delta^{\ast}$ which satisfies the estimate
$|v|_{\gamma^{\ast}h}\lesssim\|w\|^{-b}$ for some $b>0$. Here $w$ is the
coordinate of $\Delta$. According to [Simpson1990, §3] there is a canonical
filtration of bundles $F_{\alpha}$ consisting of the holomorphic sections of
$\gamma^{\ast}\mathcal{V}$ (the associated vector bundle of
$\gamma^{\ast}\mathbb{V}$) with the estimate
$|v|_{\gamma^{\ast}h}\lesssim\|w\|^{\alpha-\epsilon}$, $\forall\epsilon>0$.
The $0$-tameness is equivalent to that: for each such $\gamma:\Delta\to X$,
$F_{0}$ is the only object that matters in the filtration for both
$(\gamma^{\ast}\mathbb{V},\gamma^{\ast}h)$ and
$(\gamma^{\ast}\mathbb{V}^{\ast},\gamma^{\ast}h^{\ast})$. This is the origin
of the name ”$0$-tame”. Using notions in [Simpson1990, Mochizuki20072],
$0$-tame harmonic bundles are exactly the tame harmonic bundles without
parabolic structures.

The asymptotic behavior of the $0$-tame harmonic metric coincides with the
Hodge metric on a variation of Hodge structure. Let $(\mathbb{V},h)$ be a
$0$-tame harmonic bundle on

$(\Delta^{\ast})^{n}\times\Delta^{m}=\\{(z_{1},\dots,z_{n},w_{1},\dots,w_{m})|0<\|z_{1}\|,\dots,\|z_{n}\|<1,\|w_{1}\|,\dots,\|w_{m}\|<1\\}.$

Let $(E,\theta)$ be the associated higgs bundle with regular singularities
([Simpson1990, Mochizuki20072]) and $N_{i}$ the nilpotent part of the residue
${\rm Res}_{D_{i}}\theta$ along $D_{i}:=\\{z_{i}=0\\}$.

######  Theorem 4.21 (Mochizuki, [Mochizuki20072], Part 3, Chapter 13).

Let $(\mathbb{V},h)$ be a $0$-tame harmonic bundle on
$(\Delta^{\ast})^{n}\times\Delta^{m}$. Let

$p:\mathbb{H}^{n}\times\Delta^{m}\to(\Delta^{\ast})^{n}\times\Delta^{m},$
$(z_{1},\dots,z_{n},w_{1},\dots,w_{m})\mapsto(e^{2\pi\sqrt{-1}z_{1}},\dots,e^{2\pi\sqrt{-1}z_{n}},w_{1},\dots,w_{m})$

be the universal covering. Let
$W^{(1)}=W(N_{1}),\dots,W^{(n)}=W(N_{1}+\cdots+N_{n})$ be the residue weight
filtrations on
$V:=\Gamma(\mathbb{H}^{n},p^{\ast}\mathcal{V})^{p^{\ast}\nabla}$. Then for any
$v\in V$ such that $0\neq[v]\in{\rm Gr}_{l_{n}}^{W^{(n)}}\cdots{\rm
Gr}_{l_{1}}^{W^{(1)}}V$, one has

(4.25)
$\displaystyle|v|_{h_{\mathbb{V}}}\sim\left(\frac{\log|s_{1}|}{\log|s_{2}|}\right)^{l_{1}}\cdots\left(-\log|s_{n}|\right)^{l_{n}}$

over any region of the form

$\left\\{(s_{1},\dots,s_{n},w_{1},\dots,w_{m})\in(\Delta^{\ast})^{n}\times\Delta^{m}\bigg{|}\frac{\log|s_{1}|}{\log|s_{2}|}>\epsilon,\dots,-\log|s_{n}|>\epsilon,(w_{1},\dots,w_{m})\in
K\right\\}$

for any $\epsilon>0$ and an arbitrary compact subset $K\subset\Delta^{m}$.

######  Remark 4.22.

There are two ways to get a $0$-tame harmonic bundle.

  1. (1)

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. Let $(\mathcal{V},\nabla,\mathcal{F}^{\bullet},h_{\mathbb{V}})$ be a
polarized $\mathbb{Z}$-variation of Hodge structure on $X^{o}$, or more
generally, an $\mathbb{R}$-polarized variation of Hodge structure with quasi-
unipotent local monodromies. Then $(\mathbb{V}={\rm
Ker}\nabla,h_{\mathbb{V}})$ is a $0$-tame harmonic bundle on $(X,X^{o})$.
Indeed by desingularization locally we may resolve the subspace $X\backslash
X^{o}$ to a simple normal crossing divisor $E$ of some smooth manifold. By
[Cattani_Kaplan_Schmid1986, Theorem 5.21] the Hodge metric satisfies the
estimate (4.25).

  2. (2)

Let $X$ be a compact Kähler manifold and $X^{o}=X\backslash D$ for some normal
crossing divisor $D\subset X$. Let $\mathbb{V}$ be a local system on $X^{o}$
which does not necessarily have quasi-unipotent local monodromy. Depending on
the work by Jost and Zuo [Jost_Zuo1997], Mochizuki ([Mochizuki20072, Theorem
1.19]) shows that the $0$-tame harmonic metric exists on $\mathbb{V}$ if and
only if $\mathbb{V}$ is semisimple. Moreover the $0$-tame harmonic metric is
unique up to flat automorphisms. In fact this metric shares many other good
properties similar to the Hodge metric on a variation of Hodge structure.

For a compact Kähler space $X$, a dense Zariski open subset $X^{o}\subset
X_{\rm reg}$ and a semisimple local system $\mathbb{V}$ on $X^{o}$, there is a
$0$-tame harmonic metric on $\mathbb{V}$ (unique up to flat automorphisms)
since one can blowup $X\backslash X^{o}$ to a simple normal crossing divisor.

######  Proposition 4.23.

Let $X$ be a complex space and $X^{o}\subset X_{\rm reg}$ a dense Zariski open
subset. Let $(\mathbb{V},h)$ be a $0$-tame harmonic bundle on $(X,X^{o})$. Let
$(\\{X_{i}\\},\pi:\widetilde{X}\to X)$ be a Whitney stratified
desingularization of $(X,X^{o})$. Then $(\mathbb{V},h)$ is stratified along
any bimeromorphic fibering neighborhood with respect to the Whitney stratified
desingularization.

To be precise, let $x\in S$ be a point in a stratum $S$ of $\\{X_{i}\\}$ and
let $(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$ be an arbitrary bimeromorphic
fibering neighborhood of $x\in S$ with respect to the Whitney stratified
desingularization. Then $(\mathbb{V},h)$ is stratified along
$(U_{x},U_{S,x},C_{x},V,\pi_{x},\alpha_{x})$.

Moreover, $(\mathbb{V}|_{C_{x}},h|_{C_{x}})$ is a $0$-tame harmonic bundle on
$(C_{x},C_{x}\cap X^{o})$.

###### Proof.

Let $x\in S$ for some stratum $S$ of dimension $d$ and let
$(U_{x},U_{S,x},C_{x},V,\pi,\alpha_{x})$ be a bimeromorphic fibering
neighborhood of $x\in S$ with respect to the Whitney stratified
desingularization. Denote by $E$ the exceptional divisor of $\pi$. Let
$y\in\pi^{-1}\\{x\\}$ and let $(z^{\prime}_{1},\dots,z^{\prime}_{n-d})$ be a
holomorphic coordinate of $x^{\prime}\in\pi^{-1}C_{x}$ so that
$E\cap\pi^{-1}C_{x}:=\\{z^{\prime}_{1}\cdots z^{\prime}_{r}=0\\}$ for some
$r\leq n-d$. Let $(z^{\prime}_{n-d+1},\dots,z^{\prime}_{n})$ be a holomorphic
coordinate of $x\in U_{S,x}$, then the diffeomorphism