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.  More recent developments, which are of
significant importance, connect the theory to singularity theory, commutative
algebra, and the topology of algebraic varieties.  The following topics in this
area will particularly interest our workshop.


\subsubsection{Singularities and Hodge Ideals}

In a series of influential papers starting with \cite{MR4081135}, Mustaţă and
Popa used Hodge modules to refine and generalize well-known invariants of
singularities, most notably the multiplier ideals used in analysis and algebraic
geometry. Schnell and Yang’s recent preprint \cite{arXiv:2309.16763} suggested
an alternative approach toward similar ends.  The first applications pertain to
Bernstein--Sato polynomials and their zero sets; these are essential invariants
of singularities originating from commutative algebra that are hard to compute.
Schnell and Yang apply their results to conjectures of
Debarre--Casalaina-Martin--Grushevsky concerning the Riemann--Schottky problem
and the singularities of Theta divisors of principally polarized Abelian
varieties. 

Park and Popa recently applied perverse sheaves and D-module theory to improve
Goresky--MacPherson's classic Lefschetz theorems in the singular setting. A
program put forward by Friedman--Laza aims at understanding the Hodge structures
of degenerating Calabi--Yau varieties. 


\subsubsection{Lagrangian fibrations}

A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map
$f : M \to B$ whose generic fibers are Langrangian. 


\paragraph{Compact Setting}

If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
projective space.  In the case where $B$ is smooth, Hwang established the
conjecture more than 16 years ago in a celebrated paper.  There is new insight
today, as Bakker--Schnell recently found a purely Hodge theoretic proof of
Hwang's result in \cite{arXiv:2311.08977}.  Hopefully, these methods will give
insight into the singular setting, which remains open to date.


\paragraph{Non-compact Setting}

Geometers study Lagrangian fibrations over non-compact bases in the framework of
the ``$P=W$ conjecture,'' which Maulik--Shen and Hausel--Mellit--Minets--Schiffmann
have recently proved \cite{arXiv:2209.02568, arXiv:2209.05429}.  In the same
setting, Shen–Yin discovered a remarkable symmetry of certain pushforward
sheaves and conjectured that more general symmetries exist. Schnell has recently
established these conjectures in \cite{arXiv:2303.05364} and also proved two
conjectures of Maulik–Shen–Yin on the behavior of certain perverse sheaves near
singular fibers.


\subsubsection{Singer--Hopf conjecture and fundamental groups of Kähler manifolds}

The Singer-Hopf conjecture asserts that a closed aspherical manifold of real
dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot
\chi(X)\geq 0$. This conjecture goes back to 1931 when Hopf formulated a related
version for Riemannian manifolds. Recently, Arapura–Maxim–Wang suggested
Hodge-theoretic refinements of this conjecture for Kähler manifolds in
\cite{arXiv:2310.14131}. While the methods of \cite{arXiv:2310.14131} suffice to
show particular cases, the statement remains open in full generality. 

In a related direction, Llosa-Isenrich--Py found an application of complex
geometry and Hodge theory to geometric group theory, settling an old question of
Brady on the finiteness properties of groups \cite{zbMATH07790946}. As a
byproduct, the authors also obtain a proof of the classical Singer conjecture in
an essential particular case in the realm of Kähler 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 Hyperbolicity}

\subsubsection{Kähler--Einstein Metrics with Conic Singularities and Their Limits}

In the proof of the Donaldson--Tian--Yau conjecture, which Chen--Donaldson--Sun
gave in a series of papers around 2015, Kähler--Einstein metrics with conic
singularities along a smooth divisor emerged to play a vital role. Since then,
these metrics have become an object of study in their own right.  The work of
Brendle, Donaldson, Guenancia, Rubinstein, and many others provides a complete
package of results that generalize Yau's celebrated solution of the Calabi
conjecture to the conic setting.  Today, many exciting recent developments in
this field gravitate around the following general question.

\begin{q}
  Let $X$ be a projective manifold, and let $D\subset $ be a non-singular
  divisor.  Assume that for every sufficiently small angle $0< \beta << 1$,
  there exists a unique Kähler--Einstein metric $\omega_\beta$ with conic
  singularities of angle $2\pi\beta$ along $D$.  In other words, assume that
  \[
    Ricci_{\omega_\beta}= \lambda \cdot \omega_{\beta}+ (1-\beta)\cdot [D],
    \quad
    \text{where } \lambda \in \{ \pm 1\}.
  \]
  Is there a meaningful limit of $\omega_\beta$ as $\beta\to 0$, perhaps after
  rescaling?
\end{q}

Starting with \cite{zbMATH07615186}, a series of articles by Biquard--Guenancia
settles many relevant (and technically challenging!) particular cases of this
question.
\begin{itemize}
  \item If $(X,D)$ is the toroidal compactification of a ball quotient, then the
  limit of the metric exists and equals the hyperbolic metric.

  \item If $X$ is Fano and $D$ is a divisor whose class is proportional to the
  anticanonical class, then the limit of the rescaled metric exists and equals
  the Tian--Yau metric. 
\end{itemize}
More work is ongoing, and we expect to report on substantial progress by the
time our workshop takes place.


\subsubsection{Kähler--Einstein Metrics on Singular Spaces}

Motivated by progress in the Minimal Model Program, there has been increasing
interest in Kähler--Einstein metrics on singular spaces.  While one of the first
results in this direction dates back to the early 1970s when Kobayashi
constructed orbifold Kähler--Einstein metrics, a definitive existence result for
a relevant class of singularities was obtained by Eyssidieux--Guedj--Zeriahi
about 15 years ago in \cite{zbMATH05859416}, by combining Yau's technique with
Kolodziej's $\mathcal C^0$ estimates.  Much more recently, Li--Tian--Wang
extended Chen-Donaldson-Sun's solution of the Yau--Tian--Donaldson conjecture to
general $\mathbb Q$-Fano varieties \cite{zbMATH07382001, zbMATH07597119}.

For most applications, it is essential to control the geometry of these metrics
near the singularities.  Despite the problem's obvious importance, little is
known so far.  The continuity of the metric's potential has been established
quite recently in the preprint \cite{arXiv:2401.03935} of Cho--Choi. Beyond
that, the main progress in this direction is due to Hein--Sun
\cite{zbMATH06827885}, 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.  Using the bounded geometry method, Datar--Fu--Song recently showed an
analogous result in the case of isolated log canonical singularities
\cite{zbMATH07669617}.  Fu–Hein–Jiang obtained precise asymptotics shortly
after, \cite{zbMATH07782497}. Essential contributions directly connected to
these topics are due to Chiu, Delcroix, Hein, C.~Li, Y.~Li, Sun, Székelyhidi,
Tosatti, and Zhang.

\bigskip

{\color{red}\textbf --- DO NOT READ ANYTHING BELOW THIS LINE ---}


\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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