# The Holomorphic Extension Property for $k$-du Bois Singularities

Benjamin Tighe  Department of Mathematics, University of Oregon
<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a normal complex variety and $\pi:\tilde{X}\to X$ a resolution of
singularities. We show that the inclusion morphism
$\pi_{*}\Omega_{\tilde{X}}^{p}\hookrightarrow\Omega_{X}^{[p]}$ is an
isomorphism for $p<\mathrm{codim}_{X}(X_{\mathrm{sing}})$ when $X$ has du Bois
singularities, giving an improvement on Flenner’s criterion for arbitrary
singularities. We also study the $k$-du Bois definition from the perspective
of holomorphic extension and compare how different restrictions on
$\mathscr{H}^{0}(\underline{\Omega}_{X}^{p})$ affect the singularities of $X$,
where $\underline{\Omega}_{X}^{p}$ is the $p^{th}$-graded piece of the du Bois
complex.

##  1\. Introduction

Let $X$ be a normal complex variety with regular locus $U$. We say that the
holomorphic extension property holds in degree $p$ if the natural inclusion

(1.1) $\pi_{*}\Omega_{\tilde{X}}^{p}\hookrightarrow j_{*}\Omega_{U}^{p}$

is an isomorphism. Understanding when (1.1) holds for all $p$ is an important
question in the Hodge theory of singularities and the minimal model program.

The holomorphic extension property is a fairly weak condition for small $p$,
as (1.1) always hold for $0\leq p<\mathrm{codim}_{X}(\Sigma)-1$, where
$\Sigma$ is the singular locus of $X$ [Fle88, Theorem]. For arbitrary $p$,
this condition is less topological and the inclusion can be strict.

(1.1) has been studied extensively for the singularities arising in the
minimal model program. For klt and rational singularities, the holomorphic
extension property holds for all $p$, see [GKKP11, Theorem 1.4] and [KS21,
Corollary 1.8]. As a consequence, varieties with these singularities satisfy
many useful properties, such as the existence of a canonical pullback morphism
for reflexive differentials [Keb13], and are known to satisfy the Zariski-
Lipman conjecture.

Of course, not all varieties satisfy (1.1) for every $p$. Almost by
definition, log-canonical singularities fail holomorphic extension for $p=\dim
X$. These singularities satisfy instead the logarithmic extension property:
for every $p$, the inclusion morphisms

(1.2) $\pi_{*}\Omega_{\tilde{X}}^{p}(\log E)\hookrightarrow
j_{*}\Omega_{U}^{p}$

are isomorphisms [GKKP11, Theorem 1.5].

In this paper, we wish to consider both the holomorphic extension and
logarithmic extension properties for du Bois singularities. Recall that a
variety has du Bois singularities if the natural morphism
$\mathscr{O}_{X}\to\underline{\Omega}_{X}^{0}$ is a quasi-isomorphism, where
$\underline{\Omega}_{X}^{0}=\mathrm{gr}_{F}^{0}\underline{\Omega}_{X}^{\bullet}$
is the $0^{th}$-graded piece of the du Bois complex. Since log-canonical
singularities are du Bois [KK10, Theorem 1.4], we consider the following
questions:

  1. 1.1.

Does logarithmic extension hold for every $p$?

  2. 1.2.

For which $p$ does holomorphic extension hold?

###  1.1. Holomorphic Extension for du Bois Singularities

In the algebraic Cohen-Macaulay setting, the answer to Question 1.1 is already
known: the sheaf $\pi_{*}\omega_{\tilde{X}}(E)$ is reflexive by [KSS10,
Theorem 1.1], and so $\pi_{*}\Omega_{\tilde{X}}^{p}(\log E)$ is reflexive for
every $p$ by [KS21, Theorem 1.5]. The proof shows that
$\pi_{*}\omega_{\tilde{X}}(E)$ is reflexive if and only if $X$ is du Bois in
the Cohen-Macaulay case. If $X$ is not Cohen-Macaulay, their proof essentially
follows with a few adjustments to show that $\pi_{*}\omega_{\tilde{X}}(E)$ is
reflexive for non-CM du Bois singularities. We give a slightly new proof of
this using ideas in [KS21]:

######  Theorem 1.1.

If $X$ has du Bois singularities, then (1.2) holds for every $p$.

As a consequence, we are able to study precisely when holomorphic extension
holds for du Bois singularities, giving a slight improvement of Flenner’s
extension criterion:

######  Theorem 1.2.

Suppose $X$ has at worst du Bois singularities. Then (1.1) holds for every
$0\leq p<\mathrm{codim}_{X}(\Sigma)$.

Theorem 1.2 easily follows from logarithmic extension for isolated du Bois
singularities: the differentiation morphism

$d:j_{*}\Omega_{U}^{n-1}/\pi_{*}\Omega_{\tilde{X}}^{n-1}\to
j_{*}\omega_{U}/\pi_{*}\omega_{\tilde{X}}(E)$

is injective by [vSS85, (1.4) Corollary], and so
$\pi_{*}\Omega_{\tilde{X}}^{n-1}\hookrightarrow j_{*}\Omega_{U}^{n-1}$ is an
isomorphism by Theorem 1.1. The goal is then to prove the general case by
cutting down by generic hyperplane sections.

There is a subtly we want to highlight here. The holomorphic extension
property behaves poorly when cutting down by a generic hyperplane $H$, as
$(\pi|_{\tilde{H}})_{*}\Omega_{\tilde{H}}^{p}$ can vanish for topological
reasons. Even when we assume $X$ is du Bois, it is difficult to see how an
isomorphism
$\pi_{*}\Omega_{\tilde{X}}^{p}\xrightarrow{\sim}\pi_{*}\Omega_{\tilde{X}}^{p}(\log
E)$ — which would prove holomorphic extension in degree $p$ by Theorem 1.1 —
can be predicted from an analogous isomorphism
$(\pi|_{\tilde{H}})_{*}\Omega_{\tilde{H}}^{p}\xrightarrow{\sim}(\pi|_{\tilde{H}})_{*}\Omega_{\tilde{H}}(\log
E|_{\tilde{H}})$111For $p=1$, this is the content of [GK13].. To remedy this
issue, we prove a stronger result for du Bois singularities using ideas from
[KS21] on the theory of mixed Hodge modules:

######  Theorem 1.3.

Let $X$ be a normal variety with at worst du Bois singularities. For
$p=\mathrm{codim}_{X}(\Sigma)-1$, we have

$\dim\mathrm{Supp}~{}\mathscr{H}^{j+n-p}\mathrm{gr}_{-(n-p)}^{F}\mathrm{DR}(\mathcal{IC}_{X})\leq-(j+2).$

Here, $\mathcal{IC}_{X}$ is the intersection $\mathscr{D}$-module supported on
$X$. By [KS21, Corollary 6.2], this implies holomorphic holds in degree $p$
(see §2.5).

###  1.2. Holomorphic Extension for $k$-du Bois Singularities

Since they were first investigated by Steenbrink, du Bois singularities have
been of interest to algebraic geometers as a general class of singularities
for which deformation theory and Hodge theory can be studied. A natural
generalization of du Bois singularities involves the higher graded pieces of
the du Bois complex: we say that $X$ has weakly $k$-du Bois singularities if
the natural map

$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}\xrightarrow{\sim}\underline{\Omega}_{X}^{p}$

is a quasi-isomorphism for every $0\leq p\leq k$, where
$\underline{\Omega}_{X}^{p}:=\mathrm{gr}_{F}^{p}\underline{\Omega}_{X}^{\bullet}[p]$.
This definition is a weakening of the $k$-du Bois property first appearing in
[MOPW21] and[JKSY21] which requires
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$ to be isomorphic to the sheaf of
Kähler $p$-forms.

The algebraic properties of the sheaf
$\mathscr{H}^{0}(\underline{\Omega}_{X}^{p})$ appear to be independent of the
morphism
$\mathscr{H}^{0}(\underline{\Omega}_{X}^{p})\to\underline{\Omega}_{X}^{p}$,
and the vanishing $\mathscr{H}^{q}\underline{\Omega}_{X}^{p}=0$ for $q>0$ does
not immediately imply anything about the reflexivity of
$\mathscr{H}^{0}(\underline{\Omega}_{X}^{p})$. Moreover, Theorem 1.2 appears
to be optimal even for weakly $k$-du Bois singularities, as we discuss in §5.

Instead, one could restrict $\mathscr{H}^{0}(\underline{\Omega}_{X}^{p})$ and
require these sheaves to be reflexive for $0\leq p\leq k$. While this is
definition is easy to check, this restriction seems to be an “unnatural”
definition compared to the $k$-du Bois property, as we often want the
complexes $\underline{\Omega}_{X}^{p}$ to be equipped with a left inverse.
Regardless, $\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$ does affect the
singularities of $X$ in meaningful ways:

######  Theorem 1.4.

Let $X$ be a normal variety with weakly $k$-du Bois singularities. If
$\mathscr{H}^{0}\underline{\Omega}_{X}^{k}$, then
$\pi_{*}\Omega_{\tilde{X}}^{k+1}$ is reflexive.

Our interest in studying the case $\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$
is reflexive is related to the original $k$-du Bois definition. While there is
a natural morphism $\Omega_{X}^{p}\to\underline{\Omega}_{X}^{p}$, the
following results outlines how restrictive it can be:

######  Theorem 1.5.

Suppose that $X$ has Gorenstein, rational, and $k$-du Bois singularities. Then
$X$ is smooth in codimension $k$. Moreover, $X$ is smooth if $k>\frac{1}{2}X$.

###  1.3. Some Hodge Theoretic Results

The du Bois complex was described in [DB81] as an object arising in Deligne’s
construction of a mixed Hodge structure on the cohomology of algebraic
varieties [Del74]. In particular, the du Bois complex generates a Hodge
filtration on the cohomology of a proper variety under the spectral sequence

$E_{1}^{p,q}=\mathbb{H}^{q}(X,\underline{\Omega}_{X}^{p})\Rightarrow
H^{p+q}(X,\mathbb{C}).$

It seems difficult to determine when this induces a pure Hodge structure on
$H^{k}(X,\mathbb{Q})$, even when $X$ has weakly $k$-du Bois singularities.
Instead, we consider another generalization studied in the literature: weakly
$k$-rational singularities.

Following [FL22a], we say that $X$ has weakly $k$-rational singularities if
the natural morphism

$j_{*}\Omega_{U}^{p}\to\mathbb{D}_{X}(\underline{\Omega}_{X}^{\dim X-p})$

is a quasi-isomorphism for every $0\leq p\leq k$, where $\mathbb{D}_{X}$ is
the Grothendieck duality functor. This definition should also be compared with
the notion of weakly $k$-rational appearing in the literature, which forces
the sheaf of Kähler $p$-forms to be reflexive. For $k=0$, this agrees with the
usual definition of rational singularities222The existence of this natural
morphism $j_{*}\Omega_{U}^{p}\to\mathbb{D}_{X}(\underline{\Omega}_{X}^{\dim
X-p})$ follows from $X$ having rational singularities..

######  Theorem 1.6.

If $X$ has weakly $k$-rational singularities and
$k<\mathrm{codim}_{X}(\Sigma)$, then $H^{k}(X,\mathbb{Q})$ carries a pure
Hodge structure.

###  1.4. Relationship to Previous Work

The du Bois complex has been used considerably in the last few years to study
nice singularities arising in the minimal model program. The notion of $k$-du
Bois singularities were first seen in [MOPW21] and [JKSY21] in the context of
hypersurface singularities and relates this property to the minimal exponent;
$k$-du Bois singularities and $k$-rational singularities were studied in the
case of local complete intersections and isolated singularities in work of
Kerr, Laza and Friedman [KL20], [FL22a], [FL22b]. The $k$-du Bois property has
also been the sandbox for constructing a good deformation theory for special
families singular Calabi-Yau and Fano varieties [FL22a] [FL23]. General
properties of the (weakly) $k$-du Bois property have been developed in
[SVV23]333Here, weakly $k$-du Bois is referred to as pre-$k$ du Bois, as well
as [Tig23].

One goal of this paper is to understand what properties $k$-du Bois varieties
share with rational singularities. Unlike local complete intersections, there
is a large gap between a singularity being 1-du Bois or rational (see §5).
Upcoming work of the author looks at $k$-du Bois singularities in the case of
symplectic singularities, which are rational — in this case, the failure of
$X$ being $k$-du Bois (or $k$-rational) is due to interesting geometry of the
singularities (see Example 5.1 for a discussion of this). This paper looks to
understand how $k$-du Bois singularities can admit desirable properties
despite not being rational.

###  1.5. Acknowledgements

Part of this paper is contained in the author’s PhD thesis. I want to thank
Benjamin Bakker for conversations regarding this topic. While preparing this
manuscript, Sung Gi Park independently proved Theorem 1.2 and Theorem 1.1
using also the theory of mixed Hodge modules [Par23].

##  2\. Preliminaries

###  2.1. Notation

Throughout, we let $X$ be a normal variety with singular locus $\Sigma$ of
dimension $n$. We will use $\pi:\tilde{X}\to X$ to denote a general resolution
of singularities.

If $\pi:\tilde{X}\to X$ is a log-resolution of singularities with exceptional
divisor $E$, we let $\Omega_{\tilde{X}}^{p}(\log E)$ be the sheaf of log
$p$-forms, and we let

$\Omega_{\tilde{X}}^{p}(\log E)(-E):=\Omega_{\tilde{X}}^{p}(\log
E)\otimes\mathscr{I}_{E}.$

For any local embedding $X|_{V}\hookrightarrow Y$ into a smooth complex
variety, we define the sheaf of Kähler differentials

$\Omega_{X}^{1}|_{V}:=\Omega_{Y}^{1}/\langle df_{1},df_{2},...,df_{m}\rangle,$

where $\Omega_{Y}^{1}$ is the sheaf of holomorphic 1-forms on $Y$ and
$f_{1},...,f_{m}$ are some defining equations for the open set $V$. We let
$\Omega_{X}^{p}:=\wedge^{p}\Omega_{X}^{1}$.

If $f:X^{\prime}\to X$ is a morphism, we write $\mathbf{R}f_{*}$ for the
derived pushforward, and $\mathbf{R}\mathscr{H}om$ for the derived hom. We
will use $H^{k}$ for the cohomology of a sheaf, $\mathscr{H}^{k}$ the
cohomology sheaf of a complex, and $\mathbb{H}^{k}$ to denote hypercohomology
of a complex. Finally, we use $\mathbb{D}_{X}$ for Grothendieck duality
functor.

###  2.2. The du Bois Complex

The du Bois complex $(\underline{\Omega}_{X}^{\bullet},F)$ is an object in the
derived category of filtered complexes generalizing the holomorphic de Rham
complex for algebraic varieties over $\mathbb{C}$. We denote its graded pieces
by
$\underline{\Omega}_{X}^{p}:=\mathrm{gr}_{F}^{p}\underline{\Omega}_{X}^{\bullet}[p]$.
Studied by du Bois [DB81] from Deligne’s construction of the mixed Hodge
structure [Del74], the du Bois complex is constructed by simplicial or cubical
hyperresolutions of $X$. We will not need this construction in this paper but
will only use the formal consequences of this construction. The interested
reader should consult [PS08] for a good treatment of this construction.

######  Theorem 2.1.

For $X$ a complex scheme of finite type and $\underline{\Omega}_{X}^{\bullet}$
its du Bois complex, we have

  1. (i)

[DB81, §3.2]
$\underline{\Omega}_{X}^{\bullet}\cong_{\mathrm{qis}}\mathbb{C}_{X}$.

  2. (ii)

[DB81, (3.2.1)]If $f:Y\to X$ is a proper morphism of finite type schemes, then
there is a morphism
$f^{*}:\underline{\Omega}_{X}^{\bullet}\to\mathbf{R}f_{*}\underline{\Omega}_{Y}^{\bullet}$
in $\mathscr{D}_{\mathrm{filt}}^{b}(X)$.

  3. (iii)

[DB81, 3.10 Corollaire] If $U\subset X$ is an open subscheme then
$\underline{\Omega}_{X}^{\bullet}|_{U}\cong_{\mathrm{qis}}\underline{\Omega}_{U}^{\bullet}$.

  4. (iv)

[DB81, §3.2] There is a natural morphism
$\Omega_{X}^{\bullet}\to\underline{\Omega}_{X}^{\bullet}$, where
$\Omega_{X}^{\bullet}$ is the complex of Kähler differentials. Moreover, this
morphism is a quasi-isomorphism if $X$ is smooth.

  5. (v)

[Kol95, (12.6.2)] If $H\subset X$ is a general member of a basepoint free
linear system, then
$\underline{\Omega}_{H}^{\bullet}\cong_{\mathrm{qis}}\underline{\Omega}_{X}^{\bullet}\otimes\mathscr{O}_{H}$.

  6. (vi)

[DB81, 4.11 Proposition] There is an exact triangle

$\underline{\Omega}_{X}^{p}\to\underline{\Omega}_{\Sigma}^{p}\oplus\mathbf{R}\pi_{*}\Omega_{\tilde{X}}^{p}\to\mathbf{R}\pi_{*}\underline{\Omega}_{E}^{p}\xrightarrow{+1}$

where $\pi:\tilde{X}\to X$ is a morphism of complex schemes of finite type
with singular locus $\Sigma\subset X$ and $E=\pi^{-1}(\Sigma)$. In particular,
$\underline{\Omega}_{X}^{n}\cong_{\mathrm{qis}}\mathbf{R}\pi_{*}\omega_{\tilde{X}}\cong_{\mathrm{qis}}\pi_{*}\omega_{\tilde{X}}$,
if $\pi:\tilde{X}\to X$ is a resolution of singularities.

  7. (vii)

[DB81, 4.5 Théorème] If $X$ is a proper variety, there is a spectral sequence

$E_{1}^{p,q}:=\mathbb{H}^{q}(X,\underline{\Omega}_{X}^{p})\Rightarrow
H^{p+q}(X,\mathbb{C})$

which degenerates at $E_{1}$ for every $p,q$. Moreover, the filtration induced
by this degeneration is equal to the Hodge filtration on the underlying mixed
Hodge structure.

  8. (viii)

[Kov11, §3.C] If $\pi:\tilde{X}\to X$ is a log-resolution of singularities
with exceptional divisor $E$, there is a right triangle

$\mathbf{R}\pi_{*}\Omega_{\tilde{X}}^{p}(\log
E)(-E)\to\underline{\Omega}_{X}^{p}\to\underline{\Omega}_{\Sigma}^{p}\xrightarrow{+1}$

which is independent of the choice of $\pi$.

######  Definition 2.2.

We say that $X$ has (at worst) du Bois singularities if the natural morphism
$\mathscr{O}_{\tilde{X}}\to\underline{\Omega}_{X}^{0}$ is a quasi-isomorphism.
More generally, a variety $X$ is said to have $k$-du Bois singularities if the
morphism $\Omega_{X}^{p}\to\underline{\Omega}_{X}^{p}$ is a quasi-isomorphism
for every $0\leq p\leq k$. Finally, we say that $X$ has weakly $k$-du Bois
singularities if
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}\xrightarrow{\sim}\underline{\Omega}_{X}^{p}$
for every $q>0$ and $0\leq p\leq k$.

###  2.3. Mixed Hodge Modules

We will use the theory of mixed Hodge modules to study the holomorphic
extension property for du Bois singularities. We review the relevant
definition and properties here and follow the ideas of Kebekus-Schnell [KS21].
We point the reader to [Sch14b] for a nice overview.

We assume that our complex variety $X$ has an embedding into a smooth complex
manifold $Y$ of dimension $n+c$. A pure Hodge module on $Y$ is an object
$M=(\mathcal{M},F^{\bullet},\mathrm{rat}~{}M)$ consisting of

  1. 2.1.

A regular holonomic left $\mathscr{D}_{Y}$-module $\mathcal{M}$;

  2. 2.2.

An increasing good filtration $F^{\bullet}\mathcal{M}$ of coherent
$\mathscr{O}_{Y}$-modules, called the Hodge filtration;

  3. 2.3.

a perverse sheaf $\mathrm{rat}~{}M$ of $\mathbb{Q}$-vector spaces;

  4. 2.4.

An isomorphism

$\mathrm{rat}~{}M\otimes\mathbb{C}\cong\mathrm{DR}(\mathcal{M}),$

where $\mathrm{DR}(\mathcal{M})$ is the de Rham complex

$\mathrm{DR}(\mathcal{M}):=\big{[}\mathcal{M}\to\Omega_{Y}^{1}\otimes_{\mathscr{O}_{Y}}\mathcal{M}\to...\to\Omega_{Y}^{n+c}\otimes_{\mathscr{O}_{Y}}\mathcal{M}\big{]}[n+c]$

associated to the $\mathscr{D}_{Y}$-module $\mathcal{M}$.

Since the complexes $\mathrm{DR}(\mathcal{M})$ are perverse, we have

(2.1) $\dim\mathrm{Supp}~{}\mathscr{H}^{j}\mathrm{DR}(\mathcal{M})\leq-j$

for all $j$.

Pure Hodge modules generalize variations of pure Hodge structures in the
presence of singularities (in particular, variations of pure Hodge structures
form objects in the category of pure Hodge structures). The data defining a
pure Hodge module is subject to a number of compatibility conditions: a Hodge
module is completely determined by a collection of variations of pure Hodge
structures $\mathcal{H}_{Z_{i}}$, supported on a stratification $Z_{i}$, of
weights

$w-\dim Z_{i},$

where $w$ is the weight of the Hodge module. From this perspective, we can
make sense of a polarized Hodge module. We refer to the category of polarized
pure Hodge modules of weight $w$ by $\mathrm{HM}(Y,w)$.

In the same vein, a mixed Hodge module generalizes variations of mixed Hodge
modules. Such an object $M$ consists of a $\mathscr{D}_{Y}$-module structure
$\mathcal{M}$, a perverse structure $\mathrm{rat}(M)$, along with an
increasing weight filtration $W_{\bullet}$ on these structures such that
$\mathrm{gr}_{k}^{W}M=W_{k}M/W_{k-1}M$ are polarizable pure Hodge modules.

As part of the formal construction of the category of mixed Hodge modules,
there are several constructions on the $\mathscr{D}_{Y}$-module structure and
perverse-sheaf structure we want to agree. The first is the dual Hodge module
$\mathbb{D}M$: this mixed Hodge module satisfies the property that
$\mathbb{D}(\mathrm{gr}_{k}^{W}M)=\mathrm{gr}_{-k}^{W}\mathbb{D}M$. The
underlying perverse sheaf $\mathrm{rat}(\mathbb{D}M)$ is the Verdier dual of
$\mathrm{rat}(M)$, while the underlying $\mathscr{D}_{Y}$ is the holonomic
dual
$R^{n+c}\mathrm{Hom}_{\mathscr{D}_{Y}}(\omega_{Y}\otimes_{\mathscr{O}_{Y}}\mathcal{M},\mathscr{D}_{Y})$.
The dual Hodge module appears in Saito’s duality theorem [Sai88, 2.4.3]:

(2.2)
$\mathbf{R}\mathscr{H}om_{\mathscr{O}_{Y}}(\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{M}),\omega_{Y}^{\bullet})\cong\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathbb{D}\mathcal{M}).$

If $M\in\mathrm{HM}(Y,w)$, there is an isomorphism $\mathbb{D}M\cong M(w)$,
where $M(w)$ is the Tate-twist in degree $w$. In this case,

(2.3)
$\mathbf{R}\mathscr{H}om_{\mathscr{O}_{Y}}(\mathrm{gr}_{p}^{F}\mathrm{DR}(M),\omega_{Y}^{\bullet})\cong\mathrm{gr}_{-p-w}^{F}\mathrm{DR}(\mathcal{M}).$

Next is the relationship between a Hodge module and the data retained by a
generic hyperplane section – or non-characteristic hypersurfaces [KS21,
Definition 4.15]. Given such a section $H$ and a mixed Hodge module
$M=(\mathcal{M},F^{\bullet}\mathcal{M})$, there is a mixed Hodge module
$M_{H}$ with underlying $\mathscr{D}_{H}$-module

$(\mathscr{O}_{H}\otimes_{\iota_{H}^{-1}\mathscr{O}_{Y}}\iota_{H}^{-1}\mathcal{M},\mathscr{O}_{H}\otimes_{\iota_{H}^{-1}\mathscr{O}_{Y}}\iota_{H}^{-1}F^{\bullet}\mathcal{M})$

with de Rham complex

$\iota_{H}^{-1}\mathrm{DR}(\mathcal{M})[-1],$

where $\iota_{H}:H\hookrightarrow Y$ is the inclusion [Sai90, Lemma 2.25]. The
relationship between a mixed Hodge module and the restricted Hodge module
$\mathcal{M}_{H}$ is observed in the following exact sequence [Sch14a,
(13.3)]:

(2.4) $0\to
N^{*}_{H|Y}\otimes_{\mathscr{O}_{H}}\mathrm{gr}_{p+1}^{F}\mathrm{DR}(\mathcal{M}_{H})\to\mathscr{O}_{H}\otimes_{\mathscr{O}_{Y}}\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{M})\to\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{M}_{H})[1]\to
0,$

where $N_{H|Y}^{*}$ is the conormal bundle.

###  2.4. Cutting-Down Properties

Let $H$ be a general member of a base-point-free linear system. There is a
quasi-isomorphism

(2.5)
$\underline{\Omega}_{H}^{0}\cong_{\mathrm{qis}}\mathscr{O}_{H}\otimes\underline{\Omega}_{X}^{0}$

which depends on the choice of complex representing
$\underline{\Omega}_{X}^{0}$ [Kol95, (12.6.2)]. If $X$ has du Bois
singularities, then so will $H$. By repeatedly cutting-down $X$ by successive
(generic) hyperplane sections $H_{k}\hookrightarrow
H_{k-1}\hookrightarrow...\hookrightarrow H_{1}\hookrightarrow X$, where
$k=\dim\Sigma$, then $H_{k}$ has at worst isolated du Bois singularities.

The isomorphism (2.5) can be seen from the mixed Hodge module structure of
$\underline{\Omega}_{X}^{p}$, whose underlying $\mathbb{Q}$-complex is the
structure sheaf $\mathbb{Q}_{X}$. Specifically, there is a holomorphic
$\mathscr{D}_{Y}$-module $M_{X}=(\mathcal{M}_{X},F_{\bullet},W_{\bullet})$,
where $X\hookrightarrow Y$ is a smooth embedding, such that

$\underline{\Omega}_{X}^{p}\cong\mathrm{gr}_{F}^{p}\mathrm{DR}(\mathcal{M}_{X})[p].$

This, combined with (2.4), implies (2.5). In fact, one can use this Hodge-
module interpretation to prove that the $k$-du Bois property is preserved
under general hyperplane sections; we omit this proof and point the reader to
[SVV23, Theorem A] for a proof that depends on the choice of simplicial
hyperresolution:

######  Proposition 2.3.

Let $X$ be a complex analytic space and $H\hookrightarrow X$ a generic
hyperplane section. For some integer $k$, suppose
$\mathscr{H}^{q}\underline{\Omega}_{X}^{p}=0$ for every $q>0$ and $0\leq p\leq
k$. Then $\mathscr{H}^{q}\underline{\Omega}_{H}^{p}=0$ in this range.

More classically, the following singularity types are preserved by

###  2.5. An Extension Criterion

In [KS21], Kebekus-Schnell show that the holomorphic extension property in
degree $p$ implies the property in degree $p-1$ using the theory of Hodge
modules, as well as an extension criterion for sections of complexes, which we
describe below.

Given a complex manifold $Y$ and a subspace $Z\subset Y$, the sections of an
$\mathscr{O}_{Y}$-module $\mathscr{F}$ extend uniquely across $Z$ if and only
if

$\dim\mathrm{Supp}~{}(Z\cap
R^{k}\mathscr{H}om_{\mathscr{O}_{Y}}(\mathscr{F},\omega_{Y}^{\bullet}))\leq-(k+2)$

for every $k$. In particular, the sections of $\mathscr{F}$ extends across any
$Z$ with $\dim Z\leq\dim Y-2$ if and only if

$\dim\mathrm{Supp}~{}R^{k}\mathscr{H}om_{\mathscr{O}_{Y}}(\mathscr{F},\omega_{Y}^{\bullet})\leq-(k+2)$

for every $k$ [KS21, Corollary 6.2]. This can be applied to sections of a
complex. Indeed, suppose that $K\in D^{b}(Y)$ with $\mathscr{H}^{j}K=0$ for
$j<0$. If

(2.6) $\dim\mathrm{Supp}~{}(Z\cap
R^{k}\mathscr{H}om_{\mathscr{O}_{Y}}(K,\omega_{Y}^{\bullet}))\leq-(k+2)$

for every $k$, then the sections of the sheaf $\mathscr{H}^{0}K$ extend
uniquely across $Z$ [KS21, Proposition 6.4]. This condition is not
biconditional, but we emphasize that if $\mathscr{H}^{0}K\cong K$, and
$\mathscr{H}^{0}K$ extends uniquely across $Z$, then we get the support
condition (2.6).

###  2.6. The Intersection Hodge Module

The most important example of a pure Hodge module supported on a singular
space is the intersection Hodge module, whose underlying perverse sheaf is the
intersection complex.

First, let $X$ be a reduced complex variety of dimension $n$ which is embedded
into our smooth complex manifold $\iota:X\hookrightarrow Y$. Let
$\pi:\tilde{X}\to X$ be a resolution of singularities, and write
$f:\tilde{X}\to Y$ be the induced morphism.

On the resolution of singularities $\tilde{X}$, the locally constant sheaf
$\mathbb{Q}_{\tilde{X}}$ admits a structure as a pure variation of Hodge
structures (and therefore a pure Hodge module). Its underlying
$\mathscr{D}_{\tilde{X}}$-module structure is given by
$\mathscr{O}_{\tilde{X}}$; the Hodge filtration is the trivial filtration
$F_{p}\mathscr{O}_{\tilde{X}}=\mathscr{O}_{\tilde{X}}$ for $p\geq 0$ and $0$
otherwise. The perverse sheaf is the de Rham complex

$\mathbb{C}_{\tilde{X}}[n]\cong_{\mathrm{qis}}\mathrm{DR}(\mathscr{O}_{\tilde{X}})=\big{[}\mathscr{O}_{\tilde{X}}\xrightarrow{d}\Omega_{\tilde{X}}^{1}\xrightarrow{d}...\xrightarrow{d}\Omega_{\tilde{X}}^{n}\big{]}[n],$

and the graded components of the complex are
$\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathscr{O}_{\tilde{X}})\cong_{\mathrm{qis}}\Omega_{\tilde{X}}^{p}[n-p]$.

Now consider the regular locus $U$. As before, there is the constant variation
of Hodge structures $\mathbb{Q}_{U}[n]$. By Saito [Sai90], this determines a
Hodge module $\mathrm{IC}_{X}$ of weight $n$ on the embedding $Y$ whose
support is contained in $X$. This is the intersection Hodge module. We denote
the underlying $\mathscr{D}_{Y}$-module by $\mathcal{IC}_{X}$.

The intersection Hodge module is uniquely determined by the following
property. By Saito’s direct image theorem [Sai88, §5.3], there are Hodge
modules $(M_{l},\mathcal{M}_{l},F^{\bullet})$ supported on the singularities
of $X$ and an isomorphism of complexes (on $Y$)

(2.7)
$\mathbf{R}f_{*}\Omega_{\tilde{X}}^{p}[n-p]\cong_{\mathrm{qis}}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{IC}_{X})\oplus\bigoplus_{l\in\mathbb{Z}}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{M}_{l})[-l].$

As a consequence of the decomposition theorem, Kebekus-Schnell show that the
sheaves $f_{*}\Omega_{\tilde{X}}^{p}$ are related to the intersection complex:

(2.8)
$f_{*}\Omega_{\tilde{X}}^{p}\cong\mathscr{H}^{-(n-p)}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{IC}_{X}),$

see [KS21, Proposition 8.1], which follows since the cohomology groups
associated to the $\mathcal{M}_{l}$ must be torsion. The complexes
$\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{IC}_{X})$ are related by duality by
(2.3):

$\mathbf{R}\mathscr{H}om_{\mathscr{O}_{Y}}(\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{I}C_{X}),\omega_{Y}^{\bullet})\cong\mathrm{gr}_{-(n-p)}^{F}\mathrm{DR}(\mathcal{I}C_{X}).$

We therefore get the following criterion for holomorphic extension by §2.5.

######  Proposition 2.4.

If

$\mathrm{dim}~{}\mathrm{Supp}~{}\mathscr{H}^{j}\mathrm{gr}_{-(n-p)}\mathrm{DR}(\mathcal{IC}_{X})\leq-(j-p+2)$

for $p+j\geq-n+1$, then $\pi_{*}\Omega_{\tilde{X}}^{p}$ is reflexive.

###  2.7. The Logarithmic Mixed Hodge Module

Now suppose that $\pi:\tilde{X}\to X$ is a log-resolution of singularities
with exceptional divisor $E$. Associated to the complement $\tilde{X}\setminus
E\cong U$ is the variation of Hodge structures $\mathbb{Q}_{\tilde{X}\setminus
E}[n]$. Instead of considering the induced pure Hodge module on the embedding
$Y$, we get an extended mixed Hodge module
$j_{*}\mathbb{Q}_{\tilde{X}\setminus E}[n]$ on $\tilde{X}$ [Sai90, Theorem
3.27], where $j:\tilde{X}\setminus E\hookrightarrow\tilde{X}$ is the
inclusion.

The underlying perverse sheaf is the complex
$\mathbf{R}j_{*}\mathbb{Q}_{\tilde{X}\setminus E}[n]$; the underlying
$\mathscr{D}_{\tilde{X}}$-module $\mathscr{O}_{\tilde{X}}(*E)$ of meromorphic
functions which are regular outside of $E$. It comes with a natural Hodge
filtration, and its de Rham complex is

$\mathrm{DR}(\mathscr{O}_{\tilde{X}}(*E))=\big{[}\mathscr{O}_{\tilde{X}}(*E)\xrightarrow{d}\Omega_{\tilde{X}}^{1}(*E)\xrightarrow{d}...\xrightarrow{d}\Omega_{\tilde{X}}^{n}(*E)\big{]}[n].$

By [Sai90, Proposiion 3.11], there is a filtered quasi-isomorphism

$\Omega_{\tilde{X}}^{p}(\log
E)[n-p]\cong_{\mathrm{qis}}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathscr{O}_{\tilde{X}}(*E))$

induced by the inclusion $\Omega_{\tilde{X}}^{\bullet}(\log
E)[n]\hookrightarrow\mathrm{DR}(\mathscr{O}_{\tilde{X}}(*E))$.

The Hodge module $j_{*}\mathbb{Q}_{\tilde{X}\setminus E}[n]$ produces a family
of mixed Hodge modules under the morphism $f:\tilde{X}\to Y$, defined by the
cohomology

$N_{l}:=H^{l}f_{*}(j_{*}\mathbb{Q}_{\tilde{X}}[n])$

for $l\in\mathbb{Z}$. For $l=0$, $N_{0}$ is supported in $X$ while $N_{l}$ is
supported in $X_{\mathrm{sing}}$ for $l\neq 0$.

The mixed Hodge module
$N_{0}=(\mathcal{N}_{0},F^{\bullet}\mathcal{N}_{0},W_{\bullet}\mathcal{N}_{0})$
is related to the logarithmic differentials is an important way:

(2.9) $f_{*}\Omega_{\tilde{X}}^{p}(\log
E)\cong\mathscr{H}^{-(n-p)}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{N}_{0}),$

see [KS21, Proposition 9.5]. We can therefore predict logarithmic extension by
looking at the supports of the logarithmic complexes.

######  Proposition 2.5.

Let $N^{\prime}=\mathbb{D}(N_{0})(-n)$ be the dual mixed Hodge module of
$N_{0}$ twisted by $-n$. If

$\dim\mathrm{Supp}~{}\mathscr{H}^{j}\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{N}^{\prime})\leq-(j+p+2)$

for $p+j\geq-n+1$, then $\pi_{*}\Omega_{\tilde{X}}^{n-p}(\log E)$ is
reflexive.

In what will be useful later, we have the following special relationship
between the intersection and logarithmic Hodge modules, see [KS21, Proposition
9.8]:

(2.10)
$\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathrm{gr}_{n}^{W}\mathcal{N}_{0})\cong\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathcal{IC}_{X}).$

##  3\. Logarithmic Extension for du Bois Singularities

It is well known that a variety with at worst du Bois singularities will not
satisfy the holomorphic extension property in all degrees. If $X$ is the
affine cone of a smooth projective Calabi-Yau variety, then the inclusion
$\pi_{*}\omega_{\tilde{X}}\hookrightarrow j_{*}\omega_{U}$ will be strict.
Logarithmic extension (1.2) is known to hold for Cohen-Macaulay varieties by
[KSS10, Theorem 1.1] and [KS21, Theorem 1.5]. The key input is to use
Schwede’s criterion for du Bois singularities, along with some vanishing input
for CM varieties. Using the extension criterion of Kebekus-Schnell, we can
extend this result to non-CM varieties, giving an analytic approach to
logarithmic extension.

######  Theorem 3.1.

Let $X$ be a normal variety with at worst du Bois singularities, and let
$\pi:\tilde{X}\to X$ be a log-resolution of singularities with exceptional
divisor $E$. The sheaf $\pi_{*}\omega_{\tilde{X}}(E)$ is reflexive; in
particular, logarithmic extension holds in all degrees.

###### Proof.

We note that the claim is independent of the resolution of singularities
chosen. Assume $X$ is a closed subvariety of a smooth variety $Y$. We consider
an embedded resolution of singularities $\pi:\tilde{Y}\to Y$ from a smooth
variety $\tilde{Y}$ such that $\pi$ is an isomorphism outside
$X\setminus\Sigma$, and $\pi^{-1}(\Sigma)\subset\tilde{Y}$ is an snc divisor
which intersects the strict transform $\tilde{X}$ in an snc divisor. Such
resolutions exist in the analytic category [BM91], [Wło09]. We let
$\overline{X}=\pi^{-1}(X)_{\mathrm{red}}$ and
$E=\pi^{-1}(\Sigma)_{\mathrm{red}}$. By [Sch07, Theorem 4.6], $X$ has du Bois
singularities if and only if the natural morphism
$\mathscr{O}_{X}\to\mathbf{R}\pi_{*}\mathscr{O}_{\overline{X}}$ is a quasi-
isomorphism.

Consider the exact triangle

(3.1)
$\omega_{E}^{\bullet}\to\omega_{\overline{X}}^{\bullet}\to\omega_{\tilde{X}}^{\bullet}\otimes\mathscr{O}_{\tilde{Y}}(E)\xrightarrow{+1}.$

obtained from dualizing the divisor sequence. Applying $\mathbf{R}\pi_{*}$ and
taking cohomology, we get the exact sequence

$R^{-n}\pi_{*}\omega_{E}^{\bullet}\to
R^{-n}\pi_{*}\omega_{\overline{X}}^{\bullet}\to
R^{-n}\pi_{*}(\omega_{\tilde{X}}^{\bullet}\otimes\mathscr{O}_{\tilde{Y}}(E))\to
R^{-n+1}\pi_{*}\omega_{E}^{\bullet}.$

By [KSS10, Corollary 3.7], the first and last terms of this sequence vanish:
since $E$ is an snc divisor,
$\mathbf{R}\pi_{*}\mathscr{O}_{E}\cong\underline{\Omega}_{\Sigma}^{0}$, whence

$\mathbf{R}\pi_{*}\omega_{E}^{\bullet}\cong\mathbf{R}\mathscr{H}om_{\mathscr{O}_{Y}}(\underline{\Omega}_{\Sigma}^{0},\omega_{Y}^{\bullet}).$

The latter sheaf has no cohomology in degree $j<-\dim\Sigma$, which is shown
in [KSS10, Corollary 3.6] (this also follows from the mixed Hodge module
structure $\underline{\Omega}_{\Sigma}^{0}$ inherits). We then get
$\mathscr{H}^{0}\mathbf{R}\pi_{*}\omega_{\overline{X}}^{\bullet}[-n]\cong
R^{-n}\pi_{*}(\omega_{\tilde{X}}^{\bullet}\otimes\mathscr{O}_{\tilde{Y}}(E))\cong\pi_{*}\omega_{\tilde{X}}(E|_{\tilde{X}})$.

To finish, note that

$\dim\mathrm{Supp}~{}R^{j}\mathscr{H}om_{\mathscr{O}_{Y}}(R\pi_{*}\omega_{\overline{X}}^{\bullet}[-n],\omega_{Y}^{\bullet})=\dim\mathrm{Supp}~{}R^{j}\pi_{*}\mathscr{O}_{\overline{X}}<n-j-2$

by Schwede’s criterion. Therefore
$R^{-n}\pi_{*}\omega_{\overline{X}}^{\bullet}\cong\pi_{*}\omega_{\tilde{X}}(E|_{\tilde{X}})$
is reflexive for our normal variety $X$ with du Bois singularities by the
support criterion of Kebekus-Schnell (§2.5). This implies that
$\pi_{*}\Omega_{\tilde{X}}^{p}(\log E)$ is reflexive for every $p$ by [KS21,
Theorem 1.5]. ∎

##  4\. Holomorphic Extension for du Bois Singularities

Now we turn to the holomorphic extension property for du Bois singularities.
We emphasize again that the sheaves are $\pi_{*}\Omega_{\tilde{X}}^{p}$ are
reflexive for $p<\mathrm{codim}_{X}(\Sigma)-1$. Therefore we consider the next
interesting case.

###  4.1. Holomorphic Extension

######  Theorem 4.1.

Let $X$ be a normal variety with du Bois singularities. Then

$\dim\mathrm{Supp}~{}\mathscr{H}^{j+n-p}\mathrm{gr}_{-(n-p)}^{F}\mathrm{DR}(\mathcal{IC}_{X})\leq-(j+2)$

for each $p<\mathrm{codim}_{X}(\Sigma)$.

###### Proof.

Recall by Theorem 1.1 that $\pi_{*}\omega_{\tilde{X}}(E)$ is reflexive. By
[KS21, Proposition 9.10], we get a family of inequalities

(4.1)
$\dim\mathrm{Supp}~{}\mathscr{H}^{j}\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathbb{D}(\mathcal{N}_{0})(-n))\leq-(j+p+2)$

for $p+j\geq-n+1$. This give the important vanishing

(4.2)
$\mathscr{H}^{0}\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathbb{D}(\mathcal{N}_{0})(-n))=0.$

Write $\mathcal{N}^{\prime}=\mathbb{D}(\mathcal{N}_{0})(-n)$. Let
$W_{\bullet}$ denote the increasing weight filtration on
$\mathcal{N}^{\prime}$. We note that
$W_{n}\mathcal{N}^{\prime}=\mathcal{N}^{\prime}$ in this case since
$\mathrm{gr}_{n}^{W}\in\mathrm{HM}(Y,n)$. Consider then the exact sequence

$\mathscr{H}^{0}\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathcal{N}^{\prime})\to\mathscr{H}^{0}\mathrm{gr}_{-1}^{F}(\mathrm{gr}_{n}^{W}\mathcal{N}^{\prime})\to\mathscr{H}^{1}\mathrm{gr}_{-1}^{F}\mathrm{DR}(W_{n-1}\mathcal{N}^{\prime}).$

The last term vanishes as the terms involved must have cohomology in negative
degree only. Therefore (4.2) gives us the additional vanishing
$\mathscr{H}^{0}\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathrm{gr}_{n}^{W}\mathcal{N}^{\prime})=0$.
By definition, we have an isomorphism
$\mathrm{gr}_{n}^{W}\mathcal{N}^{\prime}\cong\mathrm{gr}_{n}^{W}\mathcal{N}_{0}$
since $\mathrm{gr}_{n}^{W}\mathcal{N}_{0}$ by (2.3). By [KS21, Proposition
9.8], there is an isomorphism

(4.3)
$\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathrm{gr}_{n}^{W}\mathcal{N}_{0})\cong\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathcal{\mathcal{missing}}{IC}_{X}).$

It follows immediately that
$\mathscr{H}^{0}\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathcal{IC}_{X})=0$.

The above vanishing demonstrates that the theorem holds in the case of
isolated singularities, as the non-zero cohomology sheaves are supported on
$\Sigma$. To prove the theorem in general, we proceed by induction on the
dimension of $\Sigma$. Let $H$ be a very general (or non-characteristic)
hyperplane section of the ambient smooth variety $Y$. Consider the exact
triangle

$0\to
N^{*}_{H|Y}\otimes_{\mathscr{O}_{H}}\mathrm{gr}_{p+1}^{F}\mathrm{DR}(\mathcal{IC}_{X\cap
H})\to\mathscr{O}_{H}\otimes_{\mathscr{O}_{Y}}\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{IC}_{X})\to\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{IC}_{X\cap
H})[1]\to 0$

associated to (2.4) and the intersection $\mathscr{D}$-module
$\mathcal{IC}_{X}$. Since $H$ is very general, the restricted
$\mathscr{D}$-module $\mathcal{IC}_{H}$ is in fact that intersection
$\mathscr{D}$-module of $X\cap H$.444For the underlying intersection complex,
this is[dCM09, §2.6]. This induces the statement on the underlying
$\mathscr{D}$-module. If $X$ has du Bois singularities, then so does $X\cap H$
by Theorem 2.1(v). Now the exact sequence and induction immediately imply

$\dim\mathrm{Supp}~{}\mathscr{H}^{j}\mathrm{gr}_{-(n-p)}^{F}\mathrm{DR}(\mathcal{IC}_{X})\leq
n-j-p-2$

for $p<\mathrm{codim}_{X}(\Sigma)$, using the fact that $H$ is very general,
except possibly in the case that $j+n-p>-2$. On the other hand, the proof
regards $p=n-\dim\Sigma$, and higher cohomology sheaves are supported on
$\Sigma$. This immediately implies that $j>-2$. Therefore the only cases are
when $p=n$ (which never holds) or $p=n-1$, which we have already shown. This
proves the theorem. ∎

######  Theorem 4.2.

If $X$ be a normal complex variety with at worst du Bois singularities and
$\pi:\tilde{X}\to X$ a resolution of singularities, then the sheaves
$\pi_{*}\Omega_{\tilde{X}}^{p}$ are reflexive for every
$p<\mathrm{codim}_{X}(\Sigma)$.

###### Proof.

Recalling that
$\pi_{*}\Omega_{\tilde{X}}^{p}\cong\mathscr{H}^{-(n-p)}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{IC}_{X})$
from (2.8), the theorem follows from Theorem 4.1. ∎

######  Example 4.3.

This holomorphic extension property for du Bois singularities is as optimal as
can be. For isolated singularities, we can again take affine cones of smooth
projective Calabi-Yau varieties. More generally, let $Z$ be a smooth
projective Calabi-Yau variety of dimension $k-1$ with affine cone $C(Z)$. For
any smooth variety $V$ of dimension $n-k$, the product $X=C(Z)\times V$ will
be a singular variety with singular locus of codimension $k$. The variety $X$
will have du Bois (in fact log-canonical) singularities, but
$\pi_{*}\Omega_{\tilde{X}}^{k}$ will not be reflexive.

We note that the du Bois assumption can be weakened in Theorem 4.1, using the
fact that the reflexivity of $\pi_{*}\omega_{\tilde{X}}(E)$ is equivalent to
the corresponding support condition on $\mathcal{N}^{\prime}$:

######  Corollary 4.4.

If $\pi_{*}\omega_{\tilde{X}}(E)$ is reflexive, then
$\pi_{*}\Omega_{\tilde{X}}^{\mathrm{codim}_{X}(\Sigma)-1}=0$.

###### Proof.

The proof is the same as the case of du Bois singularities. Note that we again
have the inequalities (4.1) by assumption, which proves the case of isolated
singularities. Now we remark that the property $\pi_{*}\omega_{\tilde{X}}(E)$
is preserved by general hyperplane. ∎

###  4.2. Log Forms to Holomorphic Forms

In the proof of Theorem 1.1, we use the fact that holomorphic extension holds
in degree $p\leq\mathrm{codim}_{X}(\Sigma)-1$ by relying on the logarithmic
extension property holding in degree $p$. In particular, we show that the
composition

$\pi_{*}\Omega_{\tilde{X}}^{p}\hookrightarrow\pi_{*}\Omega_{\tilde{X}}^{p}(\log
E)\hookrightarrow j_{*}\Omega_{U}^{p}$

is an isomorphism.

We think it is interesting to know when precisely the inclusion morphism
$\pi_{*}\Omega_{\tilde{X}}^{p}\hookrightarrow\pi_{*}\Omega_{\tilde{X}}^{p}(\log
E)$ is an isomorphism. For instance, this holds for $p=1$ when $X$ is normal
by [GK13, Theorem 3.1]. Our interest in this morphism comes from the fact that
Flenner’s criterion cannot be improved even for logarithmic forms: If $X$ has
isolated singularities, then
$\pi_{*}\Omega_{\tilde{X}}^{n-1}\xrightarrow{\sim}\pi_{*}\Omega_{\tilde{X}}^{n-1}(\log
E)$, see [vSS85, Proof of (1.3) Theorem]. This problem then seems to go beyond
holomorphic/logarithmic extension.

A variant of this problem is understanding when logarithmic extension in
degree $p$ implies holomorphic extension in degree $p^{\prime}<p$. Certainly
this is true for isolated singularities, but it is unclear how this affect the
support of the complexes
$\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{N}^{\prime})$, and cutting down by
general hyperplane needs a stronger assumption. The following is a
generalization of Theorem 4.1 and gives a partial answer to this question.

######  Proposition 4.5.

Suppose that
$\dim\mathrm{Supp}~{}\mathscr{H}^{j}\mathrm{gr}_{p}^{F}\mathrm{DR}(\mathcal{N}^{\prime})\leq-(j+p+2)$
for $p=\mathrm{codim}_{X}(\Sigma)-n-1$, then
$\pi_{*}\Omega_{\tilde{X}}^{\mathrm{codim}_{X}(\Sigma)-1}$ is reflexive.

###### Proof.

We proceed by induction on $\dim X$, noting that the claim is true for
isolated singularities by the preceding discussion. In the case of surfaces,
we also get
$\mathscr{H}^{0}\mathrm{gr}_{-1}^{F}\mathrm{DR}(\mathcal{I}C_{X})=0$ by the
proof of Theorem 4.1.

Consider the exact sequence

$\mathscr{O}_{H}\otimes\mathscr{H}^{j}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{N}^{\prime})\to\mathscr{H}^{j+1}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{N}^{\prime}|_{H})\to
N_{H|Y}^{*}\otimes\mathscr{H}^{j+1}\mathscr{H}^{j+1}\mathrm{gr}_{-p+1}^{F}\mathrm{DR}(\mathcal{N}^{\prime}|_{H}).$

Since $H$ is very general, the mixed Hodge module $N^{\prime}|_{H}$ is the
dual Hodge module associated to the logarithmic Hodge module on $H$. This
gives inequalities
$\dim\mathrm{Supp}~{}\mathscr{H}^{j+1}\mathrm{gr}_{-p+1}^{F}\mathrm{DR}(\mathcal{N}^{\prime}|_{H})\leq-(j-p+3)$
since we choose $p=\mathrm{codim}_{X}(\Sigma)-1$. Our assumption then implies
that
$\dim\mathrm{Supp}~{}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{N}^{\prime}|_{H})\leq-(j-p+3)$.
The induction hypothesis then implies that
$\dim\mathrm{Supp}~{}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{I}C_{H})\leq-(j-p+2)$.
To conclude, taking cohomology of the triangle (4.3) implies now that
$\mathrm{dim}\mathrm{Supp}~{}\mathscr{H}^{j}\mathrm{gr}_{-p}^{F}\mathrm{DR}(\mathcal{I}C_{X})<0$
except possibly when $j-p+2=0$. But by the proof of Theorem 4.1, this can only
happen in the isolated singularities case by assumption. ∎

##  5\. Holomorphic Extension for Weakly $k$-du Bois Singularities

###  5.1. Rational v.s. $k$-du Bois

A fundamental aspect of the Hodge theory of singularities is the relationship
between the du Bois property and the singularities of the MMP: Log-canonical
singularities [KK10] and rational singularities[Kov99, Corollary 2.6], [Sai00,
Theorem 5.4] are du Bois; du Bois singularities are very close to log-
canonical singularities, as both agree in the normal, quasi-Gorenstein case.
The gap between rational and du Bois singularities is much larger. Recall by
Kempf’s criterion that a variety has rational singularities if and only if $X$
is Cohen-Macaulay $\pi_{*}\omega_{\tilde{X}}$ is reflexive. We have already
seen a du Bois singularity failing holomorphic extension, but they can also
fail the Cohen-Macaulay property (take the affine cone of a compact
hyperkähler manifold of dimension $\geq 4$).

Therefore, it seems interesting to ask how closely (weakly) $k$-du Bois
singularities are from having rational singularities. Even for $k>0$, there is
a disconnect. The following examples were described in [Tig23], see also
[SVV23, §7.5].

######  Example 5.1.

Let $Y$ be a projective K3 surface. For any ample bundle $L$, the affine cone
$X$ is Cohen-Macaulay and du Bois by Kodaira vanishing. Suppose further that
$Y$ has Picard rank 1 and the degree $L^{2}\geq 24$. Then the affine cone is
also weakly 1-du Bois. Indeed, it is sufficient to show that
$H^{1}(Y,\Omega_{Y}^{1}\otimes L^{m})=0$ for every $m>0$. This follows from
[Tot20, Theorem 3.2 and Theorem 3.5]. The first theorem states that, under
these assumptions, the vanishing holds for $H^{1}(Y,\Omega_{Y}^{1}\otimes
L)=0$, considering the pair $(Y,L)$ as a polarized K3 surface. The second
theorem verifies the vanishing $H^{1}(Y,\Omega_{Y}^{1}\otimes L^{m})=0$ for
$m>1$ assuming the first vanishing. Since $H^{0}(Y,\Omega_{Y}^{1})=0$, this
implies that $X$ is weakly 1-du Bois. But
$H^{0}(Y,\omega_{Y})\cong\mathbb{C}$, and so $X$ does not have rational (or
2-du Bois) singularities.

######  Example 5.2.

Here is an example of an affine cone singularity which is rational but is not
$k$-du Bois for some $1\leq k\leq n-1$. The example is essentially given in
[BTLM97, §4.1] and is related to the failure of Bott vanishing for non-
projective spaces. Let $Y\subset\mathbb{P}^{4}$ be a smooth quadric
hypersurface. Let $L=\mathscr{O}_{Y}(1)$. Then the affine cone $X$ has
rational singularities since $Y$ is a Fano variety. The cone is also weakly
1-du Bois. To see this555Since $Y$ is a hypersurface, then so will the affine
cone $X$. Therefore this follows from [MOPW21], we consider the cohomology
sequence of

$0\to\mathscr{O}_{Y}(2+m)\to\Omega_{\mathbb{P}^{4}}^{1}\otimes\mathscr{O}(m)\otimes\mathscr{O}_{Y}\to\Omega_{Y}^{1}\otimes\mathscr{O}_{Y}(m)\to
0.$

The vanishing $H^{i}(Y,\Omega_{Y}^{1}\otimes\mathscr{O}_{Y}(m))=0$ follows
then by Bott vanishing on $\mathbb{P}^{4}$ and the rationality of $(X,v)$. On
the other hand, it was observed in loc. cit. that
$H^{1}(Y,\Omega_{Y}^{2}\otimes\mathscr{O}_{Y}(1))\cong\mathbb{C}$. This
implies that $X$ is not weakly $2$-du Bois.

The $k$-du Bois definition seems to be a useful classification of
singularities. On the one hand, the weakly $k$-du Bois property makes it easy
to understand some support conditions on the singularities and gives a nice
description of the Hodge filtration on the cohomology of proper varieties; on
the other hand, we can have still a large family of singularities if we allow
no restriction on the sheaves $\mathscr{H}^{0}(\underline{\Omega}_{X}^{p})$.
In fact, it is interesting to know how the sheaves
$\mathscr{H}^{0}(\underline{\Omega}_{X}^{p})$ record the geometry of the
singularities.

###  5.2. Holomorphic Extension

We now wish to consider the holomorphic extension property for weakly $k$-du
Bois singularities. Even though the isomorphism
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}\to\underline{\Omega}_{X}^{p}$ is
strong, in view of the extension criterion (§2.5) it only places a restriction
on the support of the sheaves $R^{j}\pi_{*}\Omega_{\tilde{X}}^{p}(\log
E)(-E)$, which is related to the reflexivity of the sheaves of log-forms.
Since $X$ is du Bois, we already know these should be reflexive.

To highlight the issue, suppose $X$ has isolated singularities. If $X$ is
weakly $k$-du Bois, we have an exact sequence

$0\to\mathscr{H}^{0}\underline{\Omega}_{X}^{p}\to\mathscr{\pi}_{*}\Omega_{\tilde{X}}^{p}\to\pi_{*}\underline{\Omega}_{E}^{p}\to
0.$

There is no reason for $\pi_{*}\underline{\Omega}_{E}^{p}$ to vanish, as
Example 5.1 highlights. In particular,
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$ may not be reflexive for $0\leq
p\leq n-2$, nor should $\pi_{*}\Omega_{\tilde{X}}^{n-1}$ be reflexive. We
therefore restrict the singularities by requiring
$\mathrm{depth}(\mathscr{O}_{X})\geq k+2$. In this case, it is well known that
$R^{p}\pi_{*}\mathscr{O}_{X}=0$ for $p\leq k$. Since $X$ is du Bois, and so
$R^{p}\pi_{*}\mathscr{O}_{\tilde{X}}(-E)=0$ for $p>0$, we get that
$R^{p}\pi_{*}\mathscr{O}_{E}=0$ for $p\leq k$. By shrinking $X$ to a single
isolated singularity, this implies
$H^{p}(E,\mathscr{O}_{E})\cong\mathrm{gr}_{F}^{p}H^{p}(E,\mathbb{C})=0$. On
the other hand, the Hodge theory of $E$ implies that
$H^{0}(E,\underline{\Omega}_{E}^{p})\cong\mathrm{gr}_{p}^{W}H^{p}(E,\mathbb{C})^{p,0}$,
so this must vanish by Hodge symmetry. In particular,
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$ must be reflexive for $p\leq
k<n-1$ since $\pi_{*}\Omega_{\tilde{X}}^{p}$ is reflexive in this range (this
gives a different proof of [MP21, Theorem G]).

We can generalize the above argument slightly to higher dimensions under a
slightly different assumption, but we note that this does not immediately
restrict the depth of the singularities:

######  Proposition 5.3.

Suppose that $X$ is weakly $k$-du Bois
$\mathscr{H}^{n-1}\mathbb{D}_{X}(\underline{\Omega}_{X}^{k})=0$. Then
$\mathscr{H}^{0}\underline{\Omega}_{X}^{k}$ is reflexive.

###### Proof.

The assumption implies by §2.5 that
$\mathscr{H}^{0}\underline{\Omega}_{X}^{k}$ is reflexive for isolated
singularities. Assume by induction that we have shown that
$\mathscr{H}^{0}\underline{\Omega}_{X}^{k}$ is reflexive in dimension $n-1$.
Since
$\mathscr{H}^{0}\underline{\Omega}_{X}^{k}\cong\underline{\Omega}_{X}^{k}$,
extension is equivalent to the inequalities
$\dim\mathrm{Supp}~{}\mathscr{H}^{j}\mathbb{D}_{X}(\underline{\Omega}_{X}^{p})<n-j-3$.
By Theorem 2.1(v), this implies that

$\dim\mathrm{Supp}~{}\mathscr{H}^{j}\mathbb{D}_{X}(\underline{\Omega}_{X}^{p})<n-j-2$

except possibly for $j=n-1$, which is our assumption. ∎

As Example 5.1 illustrates, holomorphic extension does not hold if $X$ is
weakly $k$-du Bois for all $k$. Instead, we consider the relationship between
the reflexivity of $\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$ and
$\pi_{*}\Omega_{\tilde{X}}^{p+1}$. The following gives an optimal results with
respect to holommorphic extension:

######  Proposition 5.4.

Suppose $X$ is weakly $k$-du Bois and
$\mathscr{H}^{0}\underline{\Omega}_{X}^{k}$ is reflexive. Then
$\pi_{*}\Omega_{\tilde{X}}^{k+1}$ is reflexive.

###### Proof.

By assumption, $\mathscr{H}^{n-1}\mathbb{D}(\underline{\Omega}_{X}^{k})=0$
since $\underline{\Omega}_{X}^{k}$ has no higher cohomology. Consider the
triangle

$\mathbf{R}\pi_{*}\Omega_{\tilde{X}}^{k}(\log
E)(-E)\to\underline{\Omega}_{X}^{k}\to\underline{\Omega}_{\Sigma}^{k}\xrightarrow{+1}$

of Theorem 2.1(viii). Dualizing and taking cohomology, this implies
$R^{n-1}\pi_{*}\Omega_{\tilde{X}}^{n-k}(\log E)=0$, and so
$R^{n-1}\pi_{*}\Omega_{E_{i}}^{n-k-1}=0$ for each smooth component $E_{i}$ of
$E$. The problem we are proving is local, so we may assume that
$H^{n-1}(E_{i},\Omega_{E_{i}}^{n-k-1})=0$. By Hard Lefschetz, this implies
$H^{0}(E_{i},\Omega_{E_{i}}^{k})=0$, and so
$H^{0}(\tilde{X},\Omega_{\tilde{X}}^{k+1})\xrightarrow{\sim}H^{0}(\tilde{X},\Omega_{\tilde{X}}^{k+1}(\log
E))$ by the logarithmic sheaf sequence. This prove the claim by Theorem 1.1. ∎

######  Corollary 5.5.

Suppose $X$ is weakly $k$-du Bois for $k\geq\mathrm{codim}_{X}(\Sigma)-1$ and
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$ is reflexive for each $0\leq p\leq
k$. Then $\dim\mathrm{Supp}~{}R^{j}\pi_{*}\mathscr{O}_{\tilde{X}}<n-j-2$ for
$j\leq k$.

###### Proof.

For isolated singularities, Proposition 5.4 implies that
$\pi_{*}\omega_{\tilde{X}}$ is reflexive, and so
$\dim\mathrm{Supp}~{}R^{j}\pi_{*}\mathscr{O}_{\tilde{X}}<n-j-2$ by Grauert-
Rimenschneider vanishing and §2.5. By cutting down by a general hyperplane
$H$, using Theorem 2.1(v), and considering the isomorphism
$\mathscr{O}_{H}\otimes\mathbf{R}\pi_{*}\mathscr{O}_{\tilde{X}}\cong\mathbf{R}(\pi|_{H})_{*}\mathscr{O}_{\widetilde{X\cap
H}}[1]$, we see the claim is true except possibly in the case $k=n-1$. But
Proposition 5.4 again implies $\pi_{*}\omega_{\tilde{X}}$ is reflexive. ∎

The following is immediate:

######  Corollary 5.6.

If $X$ is Cohen-Macaulay, weakly $k$-du Bois for $k=n-1$ and
$\mathscr{H}^{0}\underline{\Omega}_{X}^{n-1}$ is reflexive, then $X$ has
rational singularities.

###### Proof.

Use Proposition 5.4 and Kempf’s criterion. ∎

###  5.3. Remark on the Functorial Pullback Morphism

Recall that if $f:Z\to X$ is a morphism of complex spaces, there is a
functorial pullback morphism $f^{*}\Omega_{X}^{p}\to\Omega_{Z}^{p}$ between
the sheaves of Kähler $p$-forms. In general, this pullback does not extend to
the sheaves of reflexive differentials. Work of Kebekus [Keb13] describes a
process of constructing a natural reflexive pullback morphism which agrees
with the Kähler pullback morphism on the regular locus (see the discussion
below for a list of functorial properties). The pullback morphism was
originally constructed for morphisms of algebraic varieties with at worst klt
singularities but was extended to arbitrary complex spaces with rational
singularities [KS21, §14]. There are two major inputs for a pullback morphism
$f^{*}\Omega_{X}^{[p]}\to\Omega_{Z}^{[p]}$ to exist in degree $p$, which hold
for rational singularities:

  1. 5.1.

The varieties $X$ and $Z$ satisfy the holomorphic extension property in degree
$p$ [KS21, Corollary 1.8]

  2. 5.2.

If $E_{t}=\pi^{-1}(t)$ is the fiber of a resolution of singularities
$\pi:\tilde{X}\to X$, then $H^{0}(E_{t},\Omega_{E_{t}}^{p}/\mathrm{tor})=0$
[Nam00, Lemma 1.2].

Both of these properties hold when $X$ for $p\leq k$ when $X$ is weakly $k$-du
Bois and $\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$ is reflexive for each $p$
by (the proof of) Proposition 5.4. On the other hand, the
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$ agree with the sheaves of
$h$-differentials, the sheafification of the sheaf of Kähler differentials in
the $h$-topology [HJ13], and so $\mathscr{H}^{0}\underline{\Omega}_{X}^{p}$
satisfies the functorial pullback morphism. Therefore we deduce a partial
strengthening of Proposition 5.3:

######  Proposition 5.7.

Suppose $X$ is weakly $k$-du Bois for $k\geq\mathrm{codim}_{X}(\Sigma)-1$.
Then $\mathscr{H}^{0}\underline{\Omega}_{X}^{k}$ is reflexive if and only if
$\pi_{*}\Omega_{E_{i}}^{k}=0$ for every smooth component $E_{i}$ of $E$.

###### Proof.

The forward direction was shown in the proof of Proposition 5.4. For the other
direction, notice that $\pi_{*}\Omega_{\tilde{X}}^{p}$ is reflexive by Theorem
1.2. Therefore the proof of [KS21, Theorem 14.1] implies the reverse
direction. ∎

###  5.4. Reflexivity of Kähler $p$-Forms

The $k$-du Bois property is most naturally stated using the sheaves of Kähler
$p$-forms. There is a natural morphism
$\Omega_{X}^{p}\to\underline{\Omega}_{X}^{p}$ which is necessary for
naturality arguments concerning the du Bois complex; this is the perspective
taken in [MOPW21], [JKSY21], [MP21], [FL22a], and [FL22b] in establishing the
$k$-du Bois property. These papers consider lci singularities, in which case
the $k$-du Bois properties implies $\Omega_{X}^{p}$ is reflexive for every
$p$; this is particularly special for lci singularities and corresponds to the
codimension of the singular locus. For instance, if $Y$ is a variety with lci
singularities, $\Omega_{Y}^{1}$ is reflexive if and only if $Y$ is smooth in
codimension 3 [Kun86].

######  Example 5.8.

Consider the Gorenstein terminal singularities $(\mathbb{C}^{4}/\langle\pm
1\rangle,0)$. Since $X$ has quotient singularities, the sheaf of reflexive
1-forms is isomorphic to $(\Omega_{\mathbb{C}^{4}}^{1})^{\langle\pm
1\rangle}$, the sheaf of invariant 1-forms under this action. At the singular
point $0$, there are 16 generators corresponding to the 1-forms $x_{i}dx_{j}$,
where the $x_{i}$ are the coordinates for $\mathbb{C}^{4}$. On the other hand,
the rank of $\Omega_{X,x}^{1}$ is equal to the embedding dimension of the
singularity $x$; the invariant ring is generated by

$\mathbb{C}[x_{i}^{2},x_{i}x_{j}]_{1\leq i\leq 4,i\neq j},$

and so the embedding dimension is at most 10. Therefore $\Omega_{X}^{1}$ must
have cotorsion.

The $k$-du Bois property seems to be very special, even among Gorenstein
rational singularities. It is worth noting that this definition is fairly
restrictive for hypersurface singularities, as it implies smoothness for large
$k>\frac{1}{2}\dim X$ [JKSY21]. This property also holds under much broader
singularities. The following proof was inspired by ideas in [Gra15].

######  Proposition 5.9.

If $X$ has at worst Gorenstein $k$-du Bois singularities for some
$k>\frac{1}{2}\dim X$, then $X$ is smooth.

###### Proof.

By assumption, $X$ is weakly $k$-du Bois, and so $\Omega_{X}^{p}$ is reflexive
for $0\leq p\leq k$ since $\mathscr{H}^{0}(\underline{\Omega}_{X}^{p})$ agrees
with the sheaves of $h$-differentials [HJ13, Theorem 7.14], and thus
isomorphic to $\Omega_{X}^{[p]}$ by [KS21, Corollary 1.12]. There is then a
pairing

$\Omega_{X}^{k}\times\Omega_{X}^{\dim X-k}\to\omega_{X},$

where $\omega_{X}=j_{*}\omega_{U}$. This pairing is perfect by [Gra15, Lemma
3.2].

Consider the corank function
$\varphi_{p}(x):=\dim\Omega_{X,x}^{p}/\mathfrak{m}_{x}\Omega_{X,x}^{p}$ which
gives the minimal number of generators of $\Omega_{X,x}^{p}$ and satisfies the
property $\varphi_{p}(x)=\binom{\varphi_{1}(x)}{p}$ by [Gra15, Lemma 4.3].
Since $X$ is Gorenstein, the perfect pairing implies that $\varphi_{\dim
X-k}(x)=\varphi_{k}(x)$ for any $x$. But this implies

$\binom{\varphi_{1}(x)}{k}=\binom{\varphi_{1}(x)}{\dim X-k},$

which can happen if and only if $\varphi_{1}(x)=\dim X$. Therefore
$\Omega_{X}^{1}$ is a vector bundle of rank $\dim X$, which implies $X$ is
smooth. ∎

######  Corollary 5.10.

If $X$ is Gorenstein $k$-du Bois, then $X$ is smooth in codimension $k$.

###### Proof.

Let $H$ be a general hyperplane section. It is easy to see from properties of
Kähler differentials that if $\Omega_{X}^{p}$ is reflexive, then so will
$\Omega_{X|H}^{p}$. By Proposition 2.3, the $k$-du Bois property is preserved
if we cut down by $H$. If $X$ has a singular locus supported with a
$(n-k)$-dimensional stratum, then we can cut down by $(n-k)$ hyperplanes to
produce a singular variety $Z$ which is $k$-du Bois for $k>\dim Z$. This
contradicts Proposition 5.9. Therefore $X$ must be smooth in codimension $k$.
∎

##  6\. Hodge Structures in Low Degree

Let $X$ be a proper variety. Recall from Theorem 2.1(vii) that there is a
spectral sequence

$E_{1}^{p,q}=\mathbb{H}^{q}(X,\underline{\Omega}_{X}^{p})\Rightarrow
H^{p+q}(X,\mathbb{C})$

which degenerates at the first page and generates the Hodge filtration on the
associated mixed Hodge structure. When $X$ has weakly $k$-du Bois
singularities, there is a direct sum decomposition

(6.1)
$H^{k}(X,\mathbb{C})=\bigoplus_{p+q=k}H^{q}(X,\mathscr{H}^{0}\underline{\Omega}_{X}^{p}).$

When
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}\xrightarrow{\sim}\pi_{*}\Omega_{\tilde{X}}^{p}$,
we get a decomposition which is akin to quotient singularities [Ste76, (1.11)
Lemma], but it is unclear if this filtration induces a pure Hodge structure on
$H^{k}(X,\mathbb{C})$. For instance, $H^{2}(X,\mathbb{Z})$ carries a pure
Hodge structure if $R^{1}\pi_{*}\mathbb{Z}=0$ [BL16, Lemma 2.1], which holds
if $R^{1}\pi_{*}\mathscr{O}_{\tilde{X}}=0$. For surfaces, this requires $X$ to
have rational singularities; in higher dimensions, this is a different
condition than holomorphic extension. We therefore consider what happens for
weakly $k$-rational singularities.

Recall that a variety has weakly $k$-rational singularities if the natural
morphism

$\Omega_{X}^{[p]}\to\mathbb{D}_{X}(\underline{\Omega}_{X}^{\dim X-p})$

is an isomorphism for $0\leq p\leq k$, where $\mathbb{D}_{X}$ is the
Grothendieck duality functor. By Grauert-Rimenschneider vanishing,
$0$-rational is the same as having rational singularities. This explains the
natural morphism from the sheaf of reflexive differentials, as
$\mathscr{H}^{0}\underline{\Omega}_{X}^{p}\xrightarrow{\sim}\Omega_{X}^{[p]}$
by [KS21, Corollary 1.12] and [HJ13, Theorem 7.14]. By [SVV23, Theorem B],
(weakly) $k$-rational singularities are (weakly) $k$-du Bois.

Suppose that $X$ is projective with isolated weakly $k$-rational
singularities. Since

$\underline{\Omega}_{X}^{p}\cong\mathbf{R}\pi_{*}\Omega_{\tilde{X}}^{p}(\log
E)(-E)$

for each $p$, then $\mathbb{D}_{X}(\underline{\Omega}_{X}^{\dim
X-p})\cong\mathbf{R}\pi_{*}\Omega_{\tilde{X}}^{p}(\log E)$. If $X$ is
$k$-rational, then

$H^{q}(\tilde{X},\Omega_{\tilde{X}}^{p}(\log E)(-E))\cong
H^{q}(\tilde{X},\Omega_{\tilde{X}}^{p}(\log E))$

for $q\geq 0$ and $0\leq p\leq k$. Indeed, $\pi_{*}\Omega_{\tilde{X}}^{p}(\log
E)(-E)\hookrightarrow\pi_{*}\Omega_{\tilde{X}}^{p}(\log E)$ is an isomorphism
by Theorem LABEL:theorem_log_zero_extension, and so the isomorphism follows
from the vanishing cohomology sheaves of $\underline{\Omega}_{X}^{p}$ and
$\mathbb{D}_{X}(\underline{\Omega}_{X}^{\dim X-p})$. Recalling the following
decompositions

(6.2)
$H^{k}(U,\mathbb{C})=\bigoplus_{p+q=k}H^{q}(\tilde{X},\Omega_{\tilde{X}}^{p}(\log
E)),\quad
H_{c}^{k}(U,\mathbb{C})=\bigoplus_{p+q=k}H^{q}(\tilde{X},\Omega_{\tilde{X}}^{p}(\log
E)(-E))$

where $U=X_{\mathrm{reg}}$ and the isomorphism
$H_{c}^{k}(X,\mathbb{Q})\xrightarrow{\sim}H^{k}(X,\mathbb{Q})$ for $k\geq 2$,
we get the following:

######  Proposition 6.1.

Let $X$ be a projective variety with isolated singularities. If $X$ has weakly
$k$-rational singularities, then $H^{p}(X,\mathbb{Q})$ carries a pure Hodge
structure for each $p\leq k$.

More generally, if $k<\mathrm{codim}_{X}(\Sigma)$ and $X$ is weakly
$k$-rational, then $H^{p}(X,\mathbb{Q})$ carries a pure Hodge structure for
$p\leq k$.

###### Proof.

The first statement follows from the decompositions (6.2). These
decompositions come from the Hodge filtration on $H^{k}(U,\mathbb{C})$ and
$H_{c}^{k}(U,\mathbb{C})$, respectively, where we think of $\tilde{X}$ as a
smooth compactification of $U\cong\tilde{X}\setminus E$. Therefore the natural
morphism of mixed Hodge structures $H_{c}^{k}(U,\mathbb{C})\to
H^{k}(U,\mathbb{C})$ is an isomorphism; but $H^{k}(U,\mathbb{C})$ carries a
pure Hodge structure if $X$ has isolated singularities for $k<\dim X$, and so
does $H_{c}^{k}(U,\mathbb{C})\xrightarrow{\sim}H^{k}(X,\mathbb{C})$ for $k\geq
2$ (for $k\leq 1$, we refer to Proposition
LABEL:proposition_k_du_Bois_hodge_structure). This leaves us with the case
$k=n$. But
$\mathrm{im}(H^{n}(X,\mathbb{C})\xrightarrow{\sim}H^{n}(U,\mathbb{C}))$
carries a pure Hodge structure, so $H^{k}(X,\mathbb{C})$ must be pure, too,
since $H^{k}(X,\mathbb{C})\xrightarrow{\sim}H^{k}(U,\mathbb{C})$.

In general, there is a natural morphism $H^{k}(X,\mathbb{Q})\to
H^{k}(U,\mathbb{Q})$ of mixed Hodge structures. Note that for
$k<\mathrm{codim}_{X}(\Sigma)$, $H^{k}(U,\mathbb{Q})$ carries a pure Hodge
structure, so it’s enough to show that this morphism is injective (in fact an
isomorphism) when $X$ is $k$-rational. In this case, we have a quasi-
isomorphism

$\pi_{*}\Omega_{\tilde{X}}^{p}\cong\underline{\Omega}_{X}^{p}\cong\mathbb{D}_{X}(\underline{\Omega}_{X}^{\dim
X-p})\cong\mathbf{R}\pi_{*}\Omega_{\tilde{X}}^{p}(\log E),$

and in particular

$\mathbb{H}^{q}(X,\underline{\Omega}_{X}^{p})\cong\mathbb{H}^{q}(X,\mathbf{R}\pi_{*}\Omega_{\tilde{X}}^{p}(\log
E))\cong H^{q}(\tilde{X},\Omega_{\tilde{X}}^{p}(\log E)).$

Therefore $H^{k}(X,\mathbb{C})\cong H^{k}(U,\mathbb{C})$, and
$H^{k}(X,\mathbb{C})$ is a pure Hodge structure for $p<k$. The case $p=k$
follows once again from the fact that $\mathrm{im}~{}(H^{k}(X,\mathbb{C})\to
H^{k}(U,\mathbb{C}))$ carries a pure Hodge structure for
$k=\mathrm{codim}_{X}(\Sigma)$. ∎

######  Example 6.2.

The $k$-rational implies something stronger than the cohomology carrying a
pure Hodge structure — it implies that it agrees with the cohomology of the
regular locus $U$. If $X$ has rational singularities and
$\mathrm{codim}_{X}(\Sigma)\geq 3$, then $H^{2}(U,\mathbb{Q})$ will carry a
pure Hodge structure, and the natural morphism $H^{2}(X,\mathbb{Q})\to
H^{k}(U,\mathbb{C})$ will be injective. However, this injection can be strict
if $X$ admits a semismall resolution of singularities.

For instance, take a projective symplectic variety with
$\mathrm{codim}_{X}(\Sigma)\geq 4$ (by [Nam01, Corollary 1], $X$ is terminal).
By [KM92, Lemma 12.5.6], there is an isomorphism
$\mathrm{Ext}^{1}(\Omega_{X}^{1},\mathscr{O}_{X})\xrightarrow{\sim}H^{1}(U,T_{U})$.
The first group parametrizes first-order flat deformations of $X$ while the
second parametrizes locally trivial deformations of $U$. Since $U$ is
symplectic, it is isomorphic to $H^{1}(U,\Omega_{U}^{1})$. By the codimension
restriction, the Hodge-to-de Rham spectral sequence for $U$ degenerates in the
range $p+q\leq 2$ (see for example [Nam00, Lemma 2.6]). In particular
$\mathrm{gr}_{F}^{1}H^{2}(U,\mathbb{C})\cong H^{1}(U,\Omega_{U}^{1})$. If
$H^{2}(X,\mathbb{C})\hookrightarrow H^{2}(U,\mathbb{C})$ is an isomorphism,
then
$\mathrm{gr}_{F}^{1}H^{2}(X,\mathbb{C})\xrightarrow{\sim}H^{1}(U,\Omega_{U}^{1})$.
The first group is isomorphic to $H^{1}(X,j_{*}\Omega_{U}^{1})$ by [BL16,
Lemma 2.1] and parametrizes locally trivial deformations of $X$ [BL18, Lemma
4.5]. Summarizing, if
$H^{2}(X,\mathbb{Q})\xrightarrow{\sim}H^{2}(U,\mathbb{Q})$, the first order
deformations of $X$ are locally trivial. In particular, $X$ cannot admit a
symplectic resolution [Nam05, Corollary 2].

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# Blow-up of solutions to relaxed compressible Navier-Stokes equations in
divergence form

Johannes Bärlin

##  1 Compressible Navier-Stokes equations with relaxation in divergence form

We consider a hyperbolic system of balance laws with relaxation given by

(1.1) $\displaystyle\left\\{\begin{aligned} \rho_{t}+(\rho u)_{x}&=0\\\ (\rho
u)_{t}+(\rho u^{2}+p(\rho))_{x}&=S_{x}\\\ \tau((\rho S)_{t}+(\rho
uS)_{x})-u_{x}&=-S\end{aligned}\right.$

where $\tau>0$ and the pressure $p$ satisfies the constitutive relation

(1.2) $\displaystyle p(\rho)=\rho^{\gamma}$

for $\gamma>1$. System (1.1) is to be solved on $[0,T)\times\mathbb{R}$ for
some $T\in(0,\infty]$ with $(\rho,u,S)$ taking values in the state space
$(0,\infty)\times\mathbb{R}\times\mathbb{R}$. It is equivalent to

(1.3) $\displaystyle\rho_{t}+(\rho u)_{x}$ $\displaystyle=0$ (1.4)
$\displaystyle\rho u_{t}+\rho uu_{x}+p(\rho)_{x}$ $\displaystyle=S_{x}$ (1.5)
$\displaystyle\tau\rho(S_{t}+uS_{x})-u_{x}$ $\displaystyle=-S.$

System (1.1) is a delayed version of the Navier-Stokes equations with a
Maxwell-type law as a relaxation equation. A divergence-form variant of
systems recently considered by Hu et al. [HW19, HRW22], it is a prototypical
version of a model proposed by Ruggeri [Rug83] as discussed by Freistühler in
[Fre22]. In the latter the question is raised whether, like the systems
considered in [HW19, HRW22], also divergence-form relaxation systems such as
(1.1) possess a dichotomy in the sense that for small perturbations of a
homogeneous reference state the Cauchy problem for (1.1) has unique global
solutions which time-asymptotically decay to the reference state, while for
(some) large data there is a blow-up of solutions, i.e. physically reasonable
solutions can only exist on a finite time interval. This note serves to
confirm the second part of the mentioned dichotomy.

In proving the claimed blow-up we closely follow the procedure given by Hu,
Racke and Wang [HRW22]. In fact we show that system (1.1) has all the
necessary features in order to apply the identical steps as in [HRW22] from a
certain point on. Our system is some what similar to the one considered in
[HW19], and in this note one may find many analogies to [HW19] including a
dissipative-entropy equation and a finite speed of propagation result for the
system (1.1).

The general strategy has its origins in two remarkable papers of T. Sideris.
In [Sid84] he establishes a finite speed of propagation result for symmetric
hyperbolic conservation laws which he uses in [Sid85] to show the “formation
of singularities in three-dimensional compressible fluids“. The blow-up occurs
for a certain averaged quantity involving $\rho u$. The strategy in [Sid85] is
by now well-established and for instance used in [HW19] and [HRW22].

The system (1.1) belongs to a general class of quasi-linear equations
considered by Godunov [God61] and Boillat [Boi74]. Equations in this Godunov-
Boillat class have the property of possessing a dissipative entropy equation.
In this note we will use, and shortly derive, the follwing mathematical
entropy equation

(1.6) $\displaystyle 0=$
$\displaystyle\left[\frac{p(\rho)-p(\overline{\rho})-\gamma(\rho-\overline{\rho})}{\gamma-1}+\frac{\tau\rho
S^{2}}{2}+\frac{\rho u^{2}}{2}\right]_{t}$
$\displaystyle+\left[\frac{\gamma}{\gamma-1}up(\rho)+\frac{\gamma}{\gamma-1}\rho
u+\frac{\rho u^{3}}{2}+\frac{\tau\rho uS^{2}}{2}-uS\right]_{x}$
$\displaystyle+S^{2}$

where $\overline{\rho}>0$ is some reference density. Assume we are given a
solution $(\rho,u,S)\in C^{1}(\Omega)$ on some open subset
$\Omega\subset(0,\infty)\times\mathbb{R})$. To obtain (1.6) multiply (1.3) by
$p^{\prime}(\rho)$ to get

$\displaystyle 0=p(\rho)_{t}+\gamma
u_{x}p(\rho)+up(\rho)_{x}=p(\rho)_{t}+[\gamma
up(\rho)]_{x}+(1-\gamma)up(\rho)_{x}.$

In this way we find

(1.7) $\displaystyle
up(\rho)_{x}=\left[\frac{1}{\gamma-1}p(\rho)\right]_{t}+\left[\frac{\gamma}{\gamma-1}up(\rho)\right]_{x}.$

From (1.3) and (1.4) we deduce

(1.8) $\displaystyle-up(\rho)_{x}+uS_{x}=\rho uu_{t}+\rho
u^{2}u_{x}+\frac{u^{2}}{2}(\rho_{t}+(\rho u)_{x})=\left[\frac{\rho
u^{2}}{2}\right]_{t}+\left[\frac{\rho u^{3}}{2}\right]_{x}$

and similarly from (1.5)

(1.9) $\displaystyle-S^{2}+u_{x}S=\tau(\rho S_{t}S+\rho
uS_{x}S)=\left[\frac{\tau\rho S^{2}}{2}\right]_{t}+\left[\frac{\tau\rho
uS^{2}}{2}\right]_{x}.$

Combining (1.7)-(1.9) gives the entropy dissipation equality (1.6) where we
have added an affine linear expression in the time-derivative term and
compensated for this appropriately in the space-derivative term.

##  2 Blow-up of solutions for some large data

We begin with two properties of the Cauchy problem for system (1.3)-(1.5). On
the one hand we have existence of physically reasonable solutions of said
Cauchy problem on a maximal time interval for suitable inital values since
(1.3)-(1.5) may be written as a first-order symmetrizable hyperbolic system of
quasi-linear differential equations. On the other hand there is a finite speed
of propagation result in the spirit of [Sid84].

######  Lemma 2.1.

Let $\overline{\rho}>0$. Let
$(\rho_{0},u_{0},S_{0}):\mathbb{R}\rightarrow\mathbb{R}^{3}$ satisfy
$(\rho_{0}-\overline{\rho},u_{0},S_{0})\in H^{2}(\mathbb{R})$ and
$\rho_{0}(x)>0$ for all $x\in\mathbb{R}$. Then there exists a maximal
$T\in(0,\infty]$ such that there is a unique solution $(\rho,u,S)$ of
(1.3)-(1.5) satisfying

$\displaystyle(\rho-\overline{\rho},u,S)\in C^{0}([0,T),H^{2}(\mathbb{R}))\cap
C^{1}([0,T),H^{1}(\mathbb{R})),$

and

$\displaystyle(\rho(0),u(0),S(0))=(\rho_{0},u_{0},S_{0}),$

and for all $(t,x)\in[0,T)\times\mathbb{R}$

$\displaystyle\rho(t,x)>0.$

###### Proof.

See Chapter 2 in [Maj84]. ∎

######  Lemma 2.2.

Let $T,R>0$ and $\overline{\rho}>0$. Set

(2.1)
$\displaystyle\sigma:=\sqrt{p^{\prime}(\overline{\rho})+\frac{1}{\tau\overline{\rho}^{2}}}>0.$

Suppose $(\rho,u,S)\in C^{1}([0,T)\times\mathbb{R})$ solves (1.3)-(1.5) and
satisfies

  * •

$(\rho,u,S)\in C^{1}_{b}([0,t]\times\mathbb{R})$ for all $t\in[0,T)$;

  * •

$\textnormal{supp}\,(\rho(0,\cdot)-\overline{\rho},u(0),S(0))\subset(-R,R)$.

Then for all $t\in[0,T)$ and $x\in D_{t}:=\\{x\in\mathbb{R}:\;|x|\geq R+\sigma
t\\}$ it holds

(2.2) $\displaystyle(\rho(t,x)-\overline{\rho},u(t,x),S(t,x))=(0,0,0).$

###### Proof.

Write $U=(\rho,u,S)$ and define

$\displaystyle A_{0}(U)=\begin{pmatrix}1&0&0\\\ 0&\rho&0\\\
0&0&\tau\rho\end{pmatrix},\;A_{1}(U)=\begin{pmatrix}u&\rho&0\\\
p^{\prime}(\rho)&u&-1\\\ 0&-1&\tau\rho
u\end{pmatrix},\;B=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&1\end{pmatrix}.$

Set $\overline{U}:=(\overline{\rho},0,0)$ and consider for $V:=U-\overline{U}$
the system

$\displaystyle
V_{t}+A_{0}(\overline{U})^{-1}A_{1}(\overline{U})V_{x}+A_{0}(\overline{U})^{-1}BV=G(\overline{U},U,U_{t},U_{x})$

with

$\displaystyle
G(\overline{U},U,U_{t},U_{x})=A_{0}(\overline{U})^{-1}[(A_{0}(\overline{U})-A_{0}(U))U_{t}+(A_{1}(\overline{U})-A_{1}(U))U_{x}].$

The eigenvalues of $A_{0}(\overline{U})^{-1}A_{1}(\overline{U})$ are

$\displaystyle\lambda_{1}(\overline{U},\tau)=-\sqrt{p^{\prime}(\overline{\rho})+\tfrac{1}{\tau\overline{\rho}^{2}}},\;\lambda_{2}(\overline{U},\tau)=0,\;\lambda_{3}(\overline{U},\tau)=\sqrt{p^{\prime}(\overline{\rho})+\tfrac{1}{\tau\overline{\rho}^{2}}}.$

Set $\sigma:=\lambda_{3}$ and proceed like in [HW19] which follows the idea of
[Sid84] to show the claim of the lemma. ∎

The data for which solutions of (1.3)-(1.5) experience a blow-up is
constructed from the function in the following lemma taken from [HR14] (see
[HRW22], too):

######  Lemma 2.3.

Let $L>0$ and $M>2$. Then the function

$\displaystyle u_{L,M}(x):=\left\\{\begin{aligned} 0,&&x\in(-\infty,-M]\\\
\tfrac{L}{2}\cos(\pi(x+M))-\tfrac{L}{2},&&x\in(-M,-M+1]\\\
-L,&&x\in(-M+1,-1]\\\ L\cos(\pi(x+M)),&&x\in(-1,1]\\\ L,&&x\in(1,M-1]\\\
\tfrac{L}{2}\cos(\pi(x+M))+\tfrac{L}{2},&&x\in(M-1,M]\\\
0,&&x\in(M,\infty)\end{aligned}\right.$

is an element of $H^{2}(\mathbb{R})\cap C^{1}(\mathbb{R})$ with

(2.3) $\displaystyle||u_{L,M}||_{L^{2}}^{2}\leq 2L^{2}M.$

Now we state and prove the theorem on blow-up of solutions to (1.3)-(1.5) for
some large data.

######  Theorem 2.4.

Let $R>0,\overline{\rho}=1$ and let $(\rho_{0},S_{0})\in C^{1}(\mathbb{R})$.
Suppose $\rho_{0}(x)>0$ for all $x\in\mathbb{R}$ and
$\textnormal{supp}\;(\rho_{0}-1,S_{0})\subset(-R,R)$. Assume

(2.4) $\displaystyle\int_{\mathbb{R}}\rho_{0}(x)-1dx\geq 0.$

and set $\sigma:=\sqrt{\gamma+\tau^{-1}}$. Choose $\tilde{\sigma},L\ >0$ such
that

(2.5)
$\displaystyle\tilde{\sigma}^{2}=\max\\{\sigma^{2},(8\max\rho_{0})^{-1}\\}$

and

(2.6)
$\displaystyle\frac{L}{2}\min\rho_{0}>\max\\{\sqrt{8\max\rho_{0}},16\tilde{\sigma}\max\rho_{0}\\}.$

Define

(2.7) $\displaystyle
H_{0}:=\int_{\mathbb{R}}\frac{p(\rho_{0}(x))-1-\gamma(\rho_{0}(x)-1)}{\gamma-1}+\frac{\tau\rho_{0}(x)S^{2}_{0}(x)}{2}dx$

and choose $M\geq\max\\{4,R\\}$ such that

(2.8) $\displaystyle H_{0}+\max\rho_{0}L^{2}M\leq
2\tilde{\sigma}M^{2}\max\rho_{0}.$

Let $(\rho,u,S)$ be the solution of (1.3)-(1.5) with initial data
$(\rho_{0},u_{L,M},S_{0})$ and maximal time of existence $T>0$ as given by
Lemma 2.1. Then $T$ is finite and the critical averaged quantity initially
satisfies

(2.9)
$\displaystyle\int_{\mathbb{R}}x\cdot\rho_{0}(x)u_{0}(x)dx>\max\\{16\tilde{\sigma}M^{2}\max\rho_{0},M^{2}\sqrt{8\max\rho_{0}}\\}.$

######  Remark 2.5.

Before we enter the proof let us comment on where the above choices of
constants enter in the proof. The relations (2.5) and (2.6) are needed to
establish two a-priori estimates which appear in the deduction of the central
Riccati-type differential inequality. Then again (2.6) together with (2.8)
serves to find that the quantity on the right of (2.9) plays the role of a
critical treshold for the Riccati-type inequality thereby allowing the
conclusion of a finite maximal time of existence of the considered solution of
(1.3)-(1.5).

###### Proof.

The proof is very close to section 3 of [HRW22], also with respect to
notation. In fact after establishing the Riccati-type estimate the claim
follows like in [HRW22]. The general strategy follows [Sid85]. First define
for $t\in[0,T)$ the averaged quantities

(2.10) $\displaystyle m(t):=\int_{\mathbb{R}}\rho(t,x)-1dx$

and

(2.11) $\displaystyle F(t):=\int_{\mathbb{R}}x\cdot\rho(t,x)u(t,x)dx.$

$m$ and $F$ are well-defined and continuous functions because of the compact
supports of $u(t,\cdot)$ and $\rho(t,\cdot)-1$ by Lemma 2.2.

By conservation of density (1.3), the compact support of $\rho(t,\cdot)-1$ and
by (2.4) we have for $t\in[0,T)$

(2.12) $\displaystyle m(t)=m(0)\geq 0.$

Setting

$\displaystyle B_{t}:=\\{x\in\mathbb{R}:\;|x|<M+\tilde{\sigma}t\\}\subset
D_{t}^{c}$

and using Jensen’s inequality (see p.846 [WH18]) implies

(2.13)
$\displaystyle\int_{B_{t}}p(\rho(t,x))dx\geq\int_{B_{t}}p(\overline{\rho})dx.$

By Lemma 2.2, partial integration and (1.3) and (1.4) we have

$\displaystyle F^{\prime}(t)=\int_{\mathbb{R}}(\rho
u^{2})(t,x)dx+\int_{\mathbb{R}}p(\rho(t,x))-p(\overline{\rho})dx-\int_{\mathbb{R}}S(t,x).$

Apply estimate (2.13) and Young’s inequality to conclude

(2.14) $\displaystyle F^{\prime}(t)\geq\int_{\mathbb{R}}(\rho
u^{2})(t,x)dx-\frac{1}{2}\int_{\mathbb{R}}S^{2}(t,x)dx-(M+\tilde{\sigma}t).$

Hölder’s inequality and (2.12) yield

(2.15) $\displaystyle F^{2}(t)$
$\displaystyle\leq\int_{B_{t}}x^{2}\rho(t,x)dx\int_{B_{t}}(\rho u^{2})(t,x)dx$
$\displaystyle\leq 2(M+\tilde{\sigma}t)^{3}\max\rho_{0}\int_{B_{t}}(\rho
u^{2})(t,x)dx.$

Let

$\displaystyle c_{2}$ $\displaystyle:=\frac{\tilde{\sigma}}{M},$
$\displaystyle c_{3}$ $\displaystyle:=\frac{1}{2\max\rho_{0}M^{3}},$
$\displaystyle c_{1}$ $\displaystyle:=\frac{2c_{2}}{c_{3}},$

and assume for all $t\in[0,T)$ the a-priori estimates

(2.16) $\displaystyle F(t)\geq c_{1}>0$

and

(2.17) $\displaystyle
M+\tilde{\sigma}t=M(1+c_{2}t)\leq\frac{c_{3}}{2(1+c_{2}t)^{3}}F^{2}(t).$

Combining the estimate (2.14) for $F^{\prime}$ with the estimate (2.15) for
$F^{2}$ and using the a-priori estimates (2.16) and (2.17) one finds the
Riccati-type inequality

(2.18)
$\displaystyle\frac{F^{\prime}(t)}{F^{2}(t)}\geq\frac{c_{3}}{2(1+c_{2}t)^{3}}-\frac{1}{2c_{1}}\int_{\mathbb{R}}S^{2}(t,x)dx.$

One uses the entropy equality (1.6) to treat the $S^{2}$-term above:

(2.19) $\displaystyle\int_{\mathbb{R}}S^{2}(t,x)dx$
$\displaystyle\leq\int_{\mathbb{R}}\frac{p(\rho_{0})-1-\gamma(\rho_{0}-1)}{\gamma-1}+\frac{\rho_{0}u_{L,M}^{2}}{2}+\frac{\tau\rho_{0}S^{2}}{2}dx$
$\displaystyle\leq H_{0}+\frac{\max\rho_{0}}{2}||u_{L,M}||_{L^{2}}^{2}.$

######  Remark 2.6.

By Taylor’s Theorem one has for $\rho>0$

$\displaystyle
p(\rho)=1+\gamma(\rho-1)+\frac{p^{\prime\prime}}{2}(\xi)(\rho-1)^{2}$

for some $\xi>0$. Note $p^{\prime\prime}(\rho)>0$ for $\rho>0$. These facts
were used for the estimate in (2.19) (see also p.829 [HW19]).

Now we are in the exact same situation as in [HRW22] section 3, and all
further steps repeat theirs. Collecting the constants once more in

$\displaystyle
c_{4}:=\frac{H_{0}}{2c_{1}^{2}}\;\textnormal{and}\;c_{5}:=\frac{\max\rho_{0}}{4c_{1}^{2}}$

and noting

$\displaystyle
c_{4}+c_{5}||u_{L,M}||_{L^{2}}^{2}\leq\frac{c_{3}^{2}}{8c_{2}^{2}}(H_{0}+\max\rho_{0}L^{2}M)\leq\frac{c_{3}}{8c_{2}}$

by (2.3), the definition of $c_{1}$ and (2.8) we finally find by integrating
(2.18) that for all $t\in[0,T)$

(2.20) $\displaystyle F(0)^{-1}$ $\displaystyle\geq F(0)^{-1}-F(t)^{-1}$
$\displaystyle\geq-\frac{c_{3}}{4c_{2}(1+c_{2}t)^{2}}+\frac{c_{3}}{4c_{2}}-c_{4}-c_{5}||u_{L_{M}}||_{L^{2}}^{2}$
$\displaystyle\geq-\frac{c_{3}}{4c_{2}(1+c_{2}t)^{2}}+\frac{c_{3}}{8c_{2}}.$

But by definition of $u_{L,M}$ (note its symmetry $u_{L,M}(x)=-u_{L,M}(-x)$),
$M\geq 4$ and by (2.6) it holds

(2.21) $\displaystyle F(0)>\frac{L}{2}\min\rho_{0}M^{2}\geq
16\frac{\tilde{\sigma}}{M}\max\rho_{0}M^{3}=\frac{8c_{2}}{c_{3}}$

which implies $T<\infty$ since otherwise there exists a $t>0$ such that (2.20)
violates (2.21).

It remains to verify the a-priori estimates. We have

$\displaystyle F(0)\geq 2c_{1}$

by (2.21). If for some $T^{\ast}\in(0,T]$ we have $F\geq c_{1}$ on
$[0,T^{\ast})$ then we get from (2.20)

$\displaystyle F(t)\geq\frac{4c_{2}(1+c_{2}t)^{2}}{c_{3}}\geq 2c_{1}.$

for all $t\in[0,T^{\ast})$. Since $F$ is continuous these two facts imply
$F\geq c_{1}$ on $[0,T)$ (see p.10 [HR14]).

We show

(2.22) $\displaystyle
M+\tilde{\sigma}t=M(1+c_{2}t)\leq\frac{c_{3}}{4(1+c_{2}t)^{3}}F^{2}(t)$

which implies (2.17). Proceeding in a similar fashion like before we have

$\displaystyle F^{2}(0)\geq\frac{L^{2}M^{4}\min\rho_{0}^{2}}{2}\geq
4M^{4}\max\rho_{0}=\frac{4M}{c_{3}}=2\frac{2M}{c_{3}}$

by (2.21) and (2.6). Hence the estimate (2.22) holds for $t=0$. Again if for
some $T^{\ast}\in(0,T]$ the estimate (2.17) holds for all $t\in[0,T^{\ast})$
then by the Riccati-type inequality (2.20) and assumption (2.5) we find

$\displaystyle
F^{2}(t)\geq\frac{16c_{2}^{2}}{c_{3}^{2}}(1+c_{2}t)^{4}\geq\frac{4M}{c_{3}}(1+c_{2}t)^{4}$

closing the a-priori estimate (2.17) by a continuity argument. ∎

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