\documentclass[a4paper, british]{scrartcl}

%
% Local font definitions -- need to come first
%
\usepackage{libertine}
\usepackage[libertine]{newtxmath}


\usepackage{xcolor}
\usepackage{longtable}
%\usepackage{ccfonts,color,comment}
\usepackage[T1]{fontenc}
\usepackage{hyperref}
\usepackage[utf8]{inputenc}
\newcounter{saveenum}
\newenvironment{itemize-compact}{\begin{itemize}\itemsep -2pt}{\end{itemize}}
\usepackage{colortbl}
\usepackage{pdflscape}

\sloppy

% Colours for hyperlinks
\definecolor{lightgray}{RGB}{220,220,220}
\definecolor{gray}{RGB}{180,180,180}
\definecolor{linkred}{rgb}{0.7,0.2,0.2}
\definecolor{linkblue}{rgb}{0,0.2,0.6}

% Limit table of contents to section titles
\setcounter{tocdepth}{1}

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

\makeatletter
\hypersetup{
  pdfauthor={Philippe Eyssidieux, Jun-Muk Hwang, Stefan Kebekus, Mihai Paun},
  pdftitle={\@title},
  pdfstartview={Fit},
  pdfpagelayout={TwoColumnRight},
  pdfpagemode={UseOutlines},
  colorlinks,
  linkcolor=linkblue,
  citecolor=linkred,
  urlcolor=linkred}
\makeatother

\newcommand\young[1]{{\textbf{#1}}}

\begin{document}
\maketitle







\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 generalisation 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 Nevalinna 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. 


\end{document}
