# On vanishing theorems for analytic spaces

Osamu Fujino  Department of Mathematics, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan<EMAIL_ADDRESS>
(Date: 2023/7/18, version 0.01)

###### Abstract.

This is a short report on our new vanishing theorems for projective morphisms
between complex analytic spaces. We established a complex analytic
generalization of Kollár’s torsion-freeness and vanishing theorem for analytic
simple normal crossing pairs. Although our results may look artificial, they
have already played a crucial role for the study of minimal models in the
complex analytic setting.

###### Key words and phrases:

vanishing theorems, complex analytic spaces, mixed Hodge modules, minimal
model program

###### 2010 Mathematics Subject Classification:

Primary 32L20; Secondary 14E30

##  1\. Introduction

In his monumental paper [K], Kollár generalized the Kodaira vanishing theorem
for complex projective varieties. His results consist of injectivity, torsion-
free, and vanishing theorems. We have already had a powerful generalization of
Kollár’s package for reducible algebraic varieties (for the details, see, for
example, [F3, Chapter 5]), which plays a crucial role for the study of log
canonical pairs, semi-log canonical pairs, and quasi-log schemes in the theory
of minimal models of algebraic varieties (see [F1], [F2], [F3, Chapter 6], and
so on). Hence it was highly desirable to establish an analytic generalization
(see [F3, Remark 5.8.3] and [F4, 1.10]). Roughly speaking, from the Hodge
theoretic viewpoint, Kollár’s original result in [K] is pure and the
generalization in [F3, Chapter 5] is mixed. Recently, in [F5], we established
an appropriate generalization of [F3, Chapter 5] for projective morphisms of
complex analytic spaces. By this new generalization, we can translate the
results in [F1], [F2], and [F3, Chapter 6] into the ones for projective
morphisms between complex analytic spaces (see [F6] and [F7]). More precisely,
in [F6], we proved the cone and contraction theorem of normal pairs for
projective morphisms between complex analytic spaces as an application of
[F5]. Then, in [F7], we discussed quasi-log structures for complex analytic
spaces. We have already established the theory of minimal models for
projective morphisms of complex analytic spaces with mild singularities in
[F4], which is an analytic generalization of the great work of
Birkar–Cascini–Hacon–McKernan. We note that [F4] does not need our new
vanishing theorems. The Kawamata–Viehweg vanishing theorem for projective
morphisms of complex analytic spaces is sufficient for [F4]. Finally, we
recommend the reader who is interested in vanishing theorems and the minimal
model program to see [F3, Chapter 3].

In this paper, every complex analytic space is assumed to be Hausdorff and
second-countable. We will freely use the standard notation in [F1], [F3],
[F4], and so on. Let us prepare various definitions in order to explain our
new vanishing theorems.

######  1.1Analytic globally embedded simple normal crossing pairs.

Let $X$ be a simple normal crossing divisor on a smooth complex analytic space
$M$ and let $B$ be an $\mathbb{R}$-divisor on $M$ such that the support of
$B+X$ is a simple normal crossing divisor on $M$ and that $B$ and $X$ have no
common irreducible components. Then we put $D:=B|_{X}$ and consider the pair
$(X,D)$. The pair $(X,D)$ is called an analytic globally embedded simple
normal crossing pair.

Analytic globally embedded simple normal crossing pairs naturally appear when
we use the resolution of singularities.

######  1.2Analytic simple normal crossing pairs.

If the pair $(X,D)$ is locally isomorphic to an analytic globally embedded
simple normal crossing pair and the irreducible components of $X$ and $D$ are
all smooth, then $(X,D)$ is called an analytic simple normal crossing pair.
When $(X,D)$ is an analytic simple normal crossing pair, $X$ has an invertible
dualizing sheaf $\omega_{X}$.

######  1.3Strata.

Let $(X,D)$ be an analytic simple normal crossing pair. Let $\nu\colon
X^{\nu}\to X$ be the normalization. Let $\Theta$ be the union of
$\nu^{-1}_{*}D$ and the inverse image of the singular locus of $X$. If $W$ is
an irreducible component of $X$ or the $\nu$-image of some log canonical
center of $(X^{\nu},\Theta)$, then $W$ is called a stratum of $(X,D)$. Note
that $X^{\nu}$ is smooth and the support of $\Theta$ is a simple normal
crossing divisor on $X^{\nu}$. We also note that if the coefficients of $D$
are in $[0,1]$ then $(X^{\nu},\Theta)$ is log canonical.

In the theory of minimal models, the notion of $\mathbb{R}$-line bundles is
indispensable.

######  1.4$\mathbb{R}$-line bundles and $\mathbb{Q}$-line bundles.

Let $X$ be a complex analytic space and let $\operatorname{Pic}(X)$ be the
group of line bundles on $X$, that is, the Picard group of $X$. An element of
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}$ (resp.
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{Q}$) is called an
$\mathbb{R}$-line bundle (resp. a $\mathbb{Q}$-line bundle) on $X$. As usual,
in this paper, we write the group law of
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}$ additively for
simplicity of notation.

We need Siu’s theorem to state our result.

######  1.5Associated subvarieties, see [Si].

Let $\mathcal{F}$ be a coherent sheaf on a complex analytic space $X$. Then
there exists a locally finite family $\\{Y_{i}\\}_{i\in I}$ of complex
analytic subvarieties of $X$ such that

$\operatorname{Ass}_{\mathcal{O}_{X,x}}(\mathcal{F}_{x})=\\{\mathfrak{p}_{x,1},\ldots,\mathfrak{p}_{x,r(x)}\\}$

holds for every point $x\in X$, where
$\mathfrak{p}_{x,1},\ldots,\mathfrak{p}_{x,r(x)}$ are the prime ideals of
$\mathcal{O}_{X,x}$ associated to the irreducible components of the germs
$Y_{i,x}$ of $Y_{i}$ at $x$ with $x\in Y_{i}$. We note that each $Y_{i}$ is
called an associated subvariety of $\mathcal{F}$.

The following theorem is the main result of [F5], which is obviously an
analytic generalization of [F3, Theorem 5.6.2].

######  Theorem 1.6 (Main theorem, [F5, Theorem 1.1]).

Let $(X,\Delta)$ be an analytic simple normal crossing pair such that the
coefficients of $\Delta$ are in $[0,1]$. Let $f\colon X\to Y$ be a projective
morphism to a complex analytic space $Y$ and let $\mathcal{L}$ be a line
bundle on $X$. Let $q$ be an arbitrary nonnegative integer. Then we have the
following properties.

  * (i)

$($Strict support condition$)$. If $\mathcal{L}-(\omega_{X}+\Delta)$ is
$f$-semiample, then every associated subvariety of $R^{q}f_{*}\mathcal{L}$ is
the $f$-image of some stratum of $(X,\Delta)$.

  * (ii)

$($Vanishing theorem$)$. If
$\mathcal{L}-(\omega_{X}+\Delta)\sim_{\mathbb{R}}f^{*}\mathcal{H}$ holds for
some $\pi$-ample $\mathbb{R}$-line bundle $\mathcal{H}$ on $Y$, where
$\pi\colon Y\to Z$ is a projective morphism to a complex analytic space $Z$,
then we have $R^{p}\pi_{*}R^{q}f_{*}\mathcal{L}=0$ for every $p>0$.

Since we treat complex analytic spaces, we have to be careful about some basic
definitions.

######  1.7Globally $\mathbb{R}$-Cartier divisors.

In Theorem 1.6, we always implicitly assume that $\Delta$ is globally
$\mathbb{R}$-Cartier, that is, $\Delta$ is a finite $\mathbb{R}$-linear
combination of Cartier divisors on $X$. We note that if the number of the
irreducible components of the support of $\Delta$ is finite then $\Delta$ is
globally $\mathbb{R}$-Cartier. This condition is harmless to applications
because the restriction of $\Delta$ to a relatively compact open subset of $X$
has only finitely many irreducible components in its support. Under the
assumption that $\Delta$ is globally $\mathbb{R}$-Cartier, we can obtain an
$\mathbb{R}$-line bundle $\mathcal{N}$ naturally associated to
$\mathcal{L}-(\omega_{X}+\Delta)$, which is a hybrid of line bundles
$\mathcal{L}$ and $\omega_{X}$ and a globally $\mathbb{R}$-Cartier divisor
$\Delta$. The assumption in Theorem 1.6 (i) means that $\mathcal{N}$ is a
finite positive $\mathbb{R}$-linear combination of $f$-semiample line bundles
on $X$. The assumption in Theorem 1.6 (ii) means that
$\mathcal{N}=f^{*}\mathcal{H}$ holds in
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}$.

######  1.8.

Let $X=\mathbb{C}$ and let $\\{P_{n}\\}_{n=1}^{\infty}$ be a set of mutually
distinct discrete points of $X$. Then
$\Delta=\sum_{n=1}^{\infty}\frac{1}{n}P_{n}$ is a $\mathbb{Q}$-Cartier
$\mathbb{Q}$-divisor on $X$. However, it is not a finite $\mathbb{R}$-linear
combination of Cartier divisors on $X$. Hence it is not a globally
$\mathbb{R}$-Cartier divisor.

Theorem 1.6 (ii) can be generalized as follows. It is an analytic
generalization of [F3, Theorem 5.7.3].

######  Theorem 1.9 (Vanishing theorem of Reid–Fukuda type, [F5, Theorem
1.2]).

Let $(X,\Delta)$ be an analytic simple normal crossing pair such that the
coefficients of $\Delta$ are in $[0,1]$. Let $f\colon X\to Y$ and $\pi\colon
Y\to Z$ be projective morphisms between complex analytic spaces and let
$\mathcal{L}$ be a line bundle on $X$. If
$\mathcal{L}-(\omega_{X}+\Delta)\sim_{\mathbb{R}}f^{*}\mathcal{H}$ holds such
that $\mathcal{H}$ is an $\mathbb{R}$-line bundle, which is nef and log big
over $Z$ with respect to $f\colon(X,\Delta)\to Y$, then
$R^{p}\pi_{*}R^{q}f_{*}\mathcal{L}=0$ holds for every $p>0$ and every $q$.

The definition of nef and log big line bundles in Theorem 1.9 is as follows.

######  1.10Nef and log big line bundles.

In Theorem 1.9, we note that $\mathcal{H}$ is said to be nef and log big over
$Z$ with respect to $f\colon(X,\Delta)\to Y$ if $\mathcal{H}\cdot C\geq 0$
holds for every projective integral curve $C$ on $Y$ such that $\pi(C)$ is a
point and $\mathcal{H}|_{f(W)}$ can be written as a finite positive
$\mathbb{R}$-linear combination of $\pi$-big line bundles on $f(W)$ for every
stratum $W$ of $(X,\Delta)$.

It is more or less well known that Theorem 1.6 follows from Theorem 1.11 (i)
and (ii) below and that Theorem 1.11 (iii) is an easy consequence of Theorem
1.11 (i) and (ii). Hence all we have to do is to establish Theorem 1.11 (i)
and (ii).

######  Theorem 1.11 (Kollár’s package for analytic simple normal crossing
pairs).

Let $(X,D)$ be an analytic simple normal crossing pair such that $D$ is
reduced and let $f\colon X\to Y$ be a projective morphism of complex analytic
spaces. Then we have the following properties.

  * (i)

$($Strict support condition$)$. Every associated subvariety of
$R^{q}f_{*}\omega_{X}(D)$ is the $f$-image of some stratum of $(X,D)$ for
every $q$.

  * (ii)

$($Vanishing theorem$)$. Let $\pi\colon Y\to Z$ be a projective morphism
between complex analytic spaces and let $\mathcal{A}$ be a $\pi$-ample line
bundle on $Y$. Then

$R^{p}\pi_{*}\left(\mathcal{A}\otimes R^{q}f_{*}\omega_{X}(D)\right)=0$

holds for every $p>0$ and every $q$.

  * (iii)

$($Injectivity theorem$)$. Let $\mathcal{L}$ be an $f$-semiample line bundle
on $X$. Let $s$ be a nonzero element of $H^{0}(X,\mathcal{L}^{\otimes k})$ for
some nonnegative integer $k$ such that the zero locus of $s$ does not contain
any strata of $(X,D)$. Then, for every $q$, the map

$\times s\colon R^{q}f_{*}\left(\omega_{X}(D)\otimes\mathcal{L}^{\otimes
l}\right)\to R^{q}f_{*}\left(\omega_{X}(D)\otimes\mathcal{L}^{\otimes
k+l}\right)$

induced by $\otimes s$ is injective for every positive integer $l$.

Kollár’s original result in [K] is a special case of Theorem 1.11.

######  1.12Kollár’s original statement.

If $X$ is a smooth projective variety with $D=0$ and $f\colon X\to Y$ is a
projective surjective morphism onto a projective variety $Y$ in Theorem 1.11
(i), then the strict support condition is nothing but Kollár’s torsion-
freeness of $R^{q}f_{*}\omega_{X}$ (see [K, Theorem 2.1 (i)]). We further
assume that $Z$ is a point in Theorem 1.11 (ii). Then we can recover Kollár’s
vanishing theorem (see [K, Theorem 2.1 (iii)]). If $X$ is a smooth projective
variety, $D=0$, and $Y$ is a point, then Theorem 1.11 (iii) coincides with
Kollár’s original injectivity theorem (see [K, Theorem 2.2]). Hence Theorem
1.11 generalizes Kollár’s original statement in [K].

Our approach to Theorem 1.11 in [F5], which is completely different from the
argument in [F3, Chapter 5], is very simple. By using a spectral sequence
coming from Saito’s theory of mixed Hodge modules (see Theorem 2.3 below), we
can reduce Theorem 1.11 to a well-known simpler case due to Takegoshi (see
Theorem 2.1 below). Hence our proof of Theorem 1.11 in [F5] uses the
semisimplicity of polarizable Hodge modules. The advantage of this approach is
to clarify the meaning of the strict support condition in Theorem 1.11 (i). We
note that the above semisimplicity comes from the semisimplicity of
polarizable variations of pure Hodge structure since polarizable Hodge modules
are uniquely determined by their generic variations of pure Hodge structure.
We note that the reader can find an alternative approach to Theorem 1.11,
which is free from Saito’s theory of mixed Hodge modules and only depends on
the semisimplicity of polarizable variations of pure Hodge structure, in [FF].

##  2\. Sketch of Proof

In this section, we will briefly discuss how to prove the theorems in Section
1. As we have already explained in Section 1, we reduce the problem to a well-
known simpler case due to Takegoshi by using a spectral sequence coming from
the theory of mixed Hodge modules.

The following theorem is a special case of Takegoshi’s result (see [T]). It is
a complex analytic generalization of Kollár’s torsion-freeness and vanishing
theorem.

######  Theorem 2.1 (see [T]).

Let $f\colon X\to Y$ be a projective surjective morphism from a smooth
irreducible complex analytic space $X$. Then $R^{q}f_{*}\omega_{X}$ is a
torsion-free coherent sheaf on $Y$ for every $q$.

Furthermore, let $\pi\colon Y\to Z$ be a projective morphism between complex
analytic spaces and let $\mathcal{A}$ be a $\pi$-ample line bundle on $Y$.
Then $R^{p}\pi_{*}\left(\mathcal{A}\otimes R^{q}f_{*}\omega_{X}\right)=0$
holds for every $p>0$ and every $q$.

Although Takegoshi’s complex analytic approach to Kollár’s theorems in [T] is
interesting, we do not discuss it here since the statement of Theorem 2.1 is
sufficient for our purposes in this paper.

######  2.2.

Let $(X,D)$ be an analytic simple normal crossing pair such that $D$ is
reduced. For any positive integer $k$, we put

$X^{[k]}:=\left\\{x\in X|\operatorname{mult}_{x}X\geq k\right\\}^{\sim},$

where $Z^{\sim}$ denotes the normalization of $Z$. Then $X^{[k]}$ is the
disjoint union of the intersections of $k$ irreducible components of $X$, and
is smooth. We have a reduced simple normal crossing divisor $D^{[k]}\subset
X^{[k]}$ defined by the pull-back of $D$ by the natural morphism $X^{[k]}\to
X$. For any nonnegative integer $l$, we put

$D^{[k,l]}:=\left\\{\left.x\in
X^{[k]}\right|\operatorname{mult}_{x}D^{[k]}\geq l\right\\}^{\sim}.$

We note that $D^{[k,0]}=X^{[k]}$ holds by definition. We also note that $\dim
D^{[k,l]}=n+1-k-l$, where $n=\dim X$. In this situation, $W$ is a stratum of
$(X,D)$ if and only if $W$ is the image of an irreducible component of
$D^{[k,l]}$ for some $k>0$ and $l\geq 0$.

One of the main ingredients of [F5] is the following result coming from
Saito’s theory of mixed Hodge modules (see [Sa1] and [Sa2]).

######  Theorem 2.3 ([FFS, Corollary 1 and 4.7. Remark]).

Let $(X,D)$ be an analytic simple normal crossing pair with $\dim X=n$ such
that $D$ is reduced and let $f\colon X\to Y$ be a projective morphism to a
smooth complex analytic space $Y$. Then there is the weight spectral sequence

${}_{F}E^{-q,i+q}_{1}=\bigoplus_{k+l=n+q+1}R^{i}f_{*}\omega_{D^{[k,l]}/Y}\Rightarrow
R^{i}f_{*}\omega_{X/Y}(D),$

degenerating at $E_{2}$, and its $E_{1}$-differential $d_{1}$ splits so that
the ${}_{F}E^{-q,i+q}_{2}$ are direct factors of ${}_{F}E^{-q,i+q}_{1}$.

For the details of the weight spectral sequence of mixed Hodge modules on $Y$
necessary for Theorem 2.3, see [FFS]. Once we know Theorems 2.1 and 2.3, it is
easy to prove Theorem 1.11.

###### Sketch of Proof of Theorem 1.11.

First we prove (i). We may assume that $Y$ is a closed analytic subspace of a
polydisc since the problem is local. Therefore, it is sufficient to prove (i)
under the extra assumption that $Y$ itself is a polydisc. In this case,
$\omega_{Y}\simeq\mathcal{O}_{Y}$. By Theorem 2.1 and the description in 2.2,
$\bigoplus R^{i}f_{*}\omega_{D^{[k,l]}}$ satisfies the strict support
condition. Hence, by using the spectral sequence in Theorem 2.3, we obtain
(i). Next, we treat (ii). We may assume that $Z$ is a polydisc and that $Y$ is
a closed analytic subspace of $Z\times\mathbb{P}^{n}$. By Theorem 2.1, we see
that $R^{p}\pi_{*}\left(\mathcal{A}\otimes
R^{i}f_{*}\omega_{D^{[k,l]}}\right)=0$ holds for every $p>0$. By applying
Theorem 2.3 to $f\colon X\to Y\hookrightarrow Z\times\mathbb{P}^{n}$, we
obtain the following spectral sequence:

$E^{-q,i+q}_{1}=\bigoplus_{k+l=n+q+1}R^{i}f_{*}\omega_{D^{[k,l]}}\Rightarrow
R^{i}f_{*}\omega_{X}(D),$

which degenerates at $E_{2}$ such that its $E_{1}$-differential $d_{1}$
splits. Hence, we can easily check that (ii) holds true. Finally, it is not
difficult to see that (iii) is a direct easy consequence of (i) and (ii). We
note that (iii) is not necessary for the proof of Theorems 1.6 and 1.9. ∎

###### Sketch of Proof of Theorem 1.6.

For (i), we take an arbitrary point $y\in Y$. We may shrink $Y$ around $y$
suitably. By perturbing $\Delta$, we may further assume that $\Delta$ is a
$\mathbb{Q}$-divisor. By repeatedly taking suitable finite covers, we can
reduce the problem to Theorem 1.11 (i). For (ii), we can shrink $Z$ suitably
and may assume that $\Delta$ is a $\mathbb{Q}$-divisor. As for (i), we
repeatedly take suitable finite covers. Then we see that (ii) follows from
Theorem 1.11 (ii). ∎

###### Sketch of Proof of Theorem 1.9.

We need no new ideas for the proof of Theorem 1.9. The proof of [F3, Theorem
5.7.3] can work with some suitable modifications. Theorem 1.9 can be seen as a
corollary of Theorem 1.6. ∎

######  2.4Traditional approach versus new approach.

Here we briefly review the main difference between the approach adopted in
[F3, Chapter 5] and the one explained in this paper. Note that in [F3, Chapter
5] everything is assumed to be algebraic. In [F3, Chapter 5], we first
establish an injectivity theorem by using the theory of mixed Hodge structures
on cohomology with compact support (see [F3, Sections 5.4 and 5.5]). By this
Hodge theoretic injectivity theorem, we can check that the injectivity theorem
like Theorem 1.11 (iii) holds true in the algebraic setting (see, for example,
[F3, Theorem 5.6.1]). It is well known that we can quickly recover Theorem
1.11 (i) and (ii) once we obtain Theorem 1.11 (iii). When $X$ is algebraic, we
can always take a compactification $\overline{X}$ of $X$ and use the mixed
Hodge structures on $\overline{X}$. On the other hand, if $X$ is not
algebraic, then we can not always compactify $X$. Thus, the argument in [F3,
Chapter 5] does not work when $X$ is not algebraic. Anyway, a key result in
the approach of [F3, Chapter 5] is the injectivity theorem coming from the
theory of mixed Hodge structures on cohomology with compact support. In the
approach explained in this paper, injectivity theorems do not play an
important role.

######  Acknowledgments.

The author thanks Professor Taro Fujisawa very much. He was partially
supported by JSPS KAKENHI Grant Numbers JP19H01787, JP20H00111, JP21H00974,
JP21H04994.

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