MFO26/MFO26.tex

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\title{Application for a Workshop on Complex Analysis}
\author{Eleonora Di Nezza, Stefan Kebekus, Mihai Păun, Stefan Schreieder}

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\section{Description of the Workshop}


% Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules
%
%  
%- Singularities and Hodge ideals etc: Mustata-Popa, Park, Ruijie Yang, Schnell 
%
%- Lagrangian fibrations: Maulik-Shen, Shen-Yin, Schnell, Bakker
%
%- Singer-Hopf conjecture and fundamental groups of Kaehler manifolds: Arapura, Botong Wang, Maxim, Llosa-Isenrich—Py.


\subsection{Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules}

Over the last decade, Saito’s theory of Hodge modules has seen spectacular
applications in birational geometry. Over the last few years the theory has been
further developed and branched out to yield exciting applications to the
topology of algebraic varieties, singularity theory and commutative algebra.  
The following topics in this area will be of particular interest to our workshop.

\subsubsection{Singularities and Hodge ideals}

Hodge modules are used to define generalizations of well-known ideals of
singularities, such as multiplier ideals from analysis and algebraic geometry.
This theory has been put forward by Mustata and Popa, an alternative approach
was suggested by Schnell and Yang. These generalizations allow to study for
instance Bernstein-Sato polynomials, which are important commutative algebra
invariants of singularities that are typically hard to compute. Geometric
applications are given by the study of singularities of Theta divisors of
principally polarized abelian varieties, as pursued by Schnell and Yang. 

In most recent developments by Park and Popa, related methods have been used to
improve classical Lefschetz theorems for singular varieties due to Goresky-Mac
Pherson. Originally, Lefschetz theorems for singular varieties have been proven
via stratified Morse theory, while the recent improvements rely on perverse
sheaves and D-module theory. 

A related program put forward by Friedman and Laza aims at understanding the
Hodge structures of degenerating Calabi-Yau varieties. This led to the notions
of higher Du Bois and higher rational singularities which can be understood via
Hodge modules and will. 


\subsubsection{Lagrangian fibrations}

A Lagrangian fibration of a (not necessarily compact) hyperkaehler manifold $M$
is a proper holomorphic map $f:M\to B$ whose fibres are Langrangian
submanifolds. If $M$ is compact, then a well-known conjecture in the field
predicts that $B$ is projective space. This is known if $B$ is smooth by
celebrated work of Hwang. A Hodge theoretic proof of Hwang‘s result has recently
been found by Bakker and Schnell; the case where $B$ is allowed to be singular
remains open. 

In the non-compact setting, Lagrangian fibrations have been studied in the
framework of the so called P=W conjecture, which has recently been proven by
Maulik and Shen for the Hitchin fibration associated to the general linear group
and curves of arbitrary genus $g\geq 2$. Shen and Yin discovered a remarkable
symmetry of certain pushforward sheaves in the case of Lagrangian fibrations
over possibly non-compact bases. Recently, Schnell used Saito‘s theory of Hodge
modules to prove the conjecture of Shen and Yin in full generality.


\subsubsection{Singer-Hopf conjecture and fundamental groups of K\"ahler manifolds}

The Singer-Hopf conjecture says that a closed aspherical manifold of real
dimension $2n$ has positive signed Euler characteristic $(-1)^n\chi(X)\geq 0$.
This conjecture goes back to 1931, when Hopf formulated a related version for
Riemannian manifolds. Recently, Hodge-theoretic refinements of this conjecture
for K\"ahler manifolds have been put forward by Arapura, Maxim and Wang. Special
cases of these conjecture have recently been proven, but the statement remains
open in full generality.

In a related direction, Llosa-Isenrich and Py found recently an application of
complex geometry and Hodge theory to geometric group theory, thereby settling an
old question of of Brady  on finiteness properties of groups. As a byproduct,
the authors also obtain a proof of the classical Singer conjecture in an
important special case in the realm of K\"ahler manifolds.

Our goal in this workshop is to bring together several experts in geometric
group theory with experts on Hodge theory, and to explore further potential
applications of the methods from one field to problems in the other.


\subsection{Canonical metrics and Kobayashi hyperbolicity}

\subsubsection{Kähler-Einstein metrics with conic singularities and their limits}

In the proof of Donaldson-Tian-Yau conjecture -around 2015-, the Kähler-Einstein
metrics with conic singularities along a smooth divisor are playing a key role.
Since then, they have become an object of study in its own right. For example,
we currently dispose of results which are completely analog to Yau’s celebrated
solution of Calabi conjecture in conic setting, by the work of S. Brendle, S.
Donaldson, H. Guenancia, Y. Rubinstein,  among many others. 

An important number of the exciting recent developments in this field are
gravitating around the following general question: \emph{let $X$ be a projective
manifold, and let $D\subset $ be a non-singular divisor. We assume that for each
angle $0< \beta<< 1$ small enough, there exists a unique KE metric
$\omega_\beta$ with conic singularities of angle $2\pi\beta$ along $D$, i.e.
$$Ricci_{\omega_\beta}= \lambda \omega_{\beta}+ (1-\beta)[D],$$
where $\lambda$ is equal to -1 or 1. Can one extract a limit of $(\omega_\beta)$
as $\beta\to 0$, eventually after rescaling}?

The series of articles by Biquard-Guenancia —2022 and 2024-- settle many
interesting and technically challenging particular casses of this question:
toroidal compactifications of ball quotients -in which the limit mentioned above
is the hyperbolic metric- and the case of a Fano manifold together with a
divisor $D$ proportional to the anticanonical class -the limit of the rescaled
metric is the Tian-Yau metric. 
\smallskip

On the other hand, there has been increasing interest in the understanding of
Kähler-Einstein metrics on singular spaces. Perhaps one of the first result in
this direction is due to S. Kobayashi (construction of orbifold Kähler-Einstein
metrics), while a definitive existence result for a large class of singularities
was obtained by Eyssidieux-Guedj-Zeriahi by combining Yau's technique with S.
Kolodziej's $\mathcal C^0$ estimates. Recently Li-Tian-Wang extended
Chen-Donaldson-Sun’s solution of the Yau-Tian-Donaldson conjecture to general
$\mathbb Q$-Fano varieties. Thus, we now have several sources/motivations for
studying singular Kähler-Einstein metrics on normal varieties.

For applications it is desirable to have control of the geometry of these
metrics near the singularities, but so far little is known in general. The
continuity of their potential has only been established very recently (beginning
of 2024) by Y.-W- Luke and Y.-J. Choi. Beyond that, the main progress in this
direction is due to Hein-Sun, who showed that near a large class of smoothable
isolated singularities that are locally isomorphic to a Calabi-Yau cone, the
singular Calabi-Yau metric must be asymptotic in a strong sense to the
Calabi-Yau cone metric. Recently an analogous result was shown by Datar-Fu-Song
in the case of isolated log canonical singularities using the bounded geometry
method, and precise asymptotics were obtained shortly after by Fu-Hein-Jiang.
Important contributions in direct connection with these topics are due to S.-K.
Chiu,T. Delcroix, H.-J. Hein, C. Li, Y. Li, S. Sun, G. Székelyhidi, V. Tosatti
and K. Zhang.


\subsubsection{Complex hyperbolicity}

The Green-Griffiths conjecture (concerning the Zariski closure of holomorphic
entire curves or more generally, of families of holomorphic disks on varieties
of general type) continues to keep busy many complex geometers. Probably the
most complete result in this field is due to A. Bloch (more than 100 years ago),
who -in modern language- showed that the Zariski closure of a map $\varphi:
\mathbb C \to A$ to a complex tori $A$ is the translate of a sub-tori. A decade
ago, K.~Yamanoi established the Green-Griffiths conjecture for projective
manifolds general type, which admit a generically finite map into an Abelian
variety. This represents a very nice generalization of Bloch's theorem.

In the last couple of years the field is taking a very interesting direction, by
combining techniques from Hodge theory with the familiar Nevanlinna theory and
jet differentials, cf. the articles by D. Brotbek, Y. Deng, Y. Brunebarbe, B.
Cadorel and A. Javanpeykar.

Using recent advances in the theory of harmonic maps (due to
Daskalopoulos-Mese), Y. Deng and K. Yamanoi were able to confirm the
Green-Griffiths conjecture for manifolds whose fundamental group admits a
representation having certain natural properties (echoing the case of curves of
genus at least two). 

Techniques from birational geometry, in connection with the work of F.~Campana
are also present in the field via the -long awaited- work of E. Rousseau and its
collaborators. 


\subsubsection{Complex hyperbolicity. Mark II}

The 1979 Green-Griffiths-Lang conjecture asserts that every complex-projective
variety $X$ of general type contains a proper subvariety $Y \subsetneq X$, such
that every non-constant entire holomorphic curve $\mathbb C \to X$ takes its
values in $Y$.  Its beginnings date back to 1926, when André Bloch showed that
the Zariski closure of entire holomorphic curve $\varphi: \mathbb C \to A$ to a
complex torus $A$ is the translate of a sub-torus. Today, the conjecture still
drives much of the research in complex geometry. We highlight several advances
that will be relevant for our workshop.

\paragraph{Hypersurfaces in projective space}

A remarkable paper of Bérczi and Kirwan, \cite{MR4688701} published in September
last year, establishes hyperbolicity and proves the Green-Griffiths-Lang for
generic hypersurfaces of the projective space, $X \subsetneq \mathbb P^n$,
provided that the degree of $X$ is larger than an explicit polynomial in $n$.
These are significant improvements of earlier degree bounds, which involve
non-polynomial bounds of order $(\sqrt{n} \log n)^n$ or worse.  The proof builds
on a strategy of Diverio-Merker-Rousseau and combines non-reductive geometric
invariant theory with the ``Grassmannian techniques'' of Riedl-Yang.  A very
recent preprint of Cadorel simplifies the proof Bérczi-Kirwan substantially, but
still needs to undergo peer review, \cite{arXiv:2406.19003}.


\paragraph{Hyperbolicity and representations of fundamental groups}

Using recent advances in the theory of harmonic maps (due to
Daskalopoulos-Mese), Y. Deng and K. Yamanoi were able to confirm the
Green-Griffiths conjecture for manifolds whose fundamental group admits a
representation having certain natural properties (echoing the case of curves of
genus at least two). 


\paragraph{Material collections}



In the last couple of years the field is taking a very interesting direction, by
combining techniques from Hodge theory with the familiar Nevanlinna theory and
jet differentials, cf. the articles by D. Brotbek, Y. Deng, Y. Brunebarbe, B.
Cadorel and A. Javanpeykar.



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