2024-07-16 15:18:00 +02:00
|
|
|
|
\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{
|
2024-07-16 22:10:19 +02:00
|
|
|
|
pdfauthor={Eleonora Di Nezza, Stefan Kebekus, Mihai Păun, Stefan Schreieder},
|
2024-07-16 15:18:00 +02:00
|
|
|
|
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}
|
2024-07-16 15:22:24 +02:00
|
|
|
|
|
2024-07-19 11:00:51 +02:00
|
|
|
|
Over the last decade, Saito's theory of Hodge modules has seen spectacular
|
2024-07-19 13:16:42 +02:00
|
|
|
|
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.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
|
|
|
|
|
2024-07-19 13:16:42 +02:00
|
|
|
|
\subsubsection{Singularities and Hodge Ideals}
|
2024-07-16 15:22:24 +02:00
|
|
|
|
|
2024-07-19 11:00:51 +02:00
|
|
|
|
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
|
2024-07-19 13:16:42 +02:00
|
|
|
|
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
|
2024-07-19 11:00:51 +02:00
|
|
|
|
Debarre--Casalaina-Martin--Grushevsky concerning the Riemann--Schottky problem
|
|
|
|
|
and the singularities of Theta divisors of principally polarized Abelian
|
|
|
|
|
varieties.
|
2024-07-16 15:22:24 +02:00
|
|
|
|
|
2024-07-19 13:16:42 +02:00
|
|
|
|
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.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\subsubsection{Lagrangian fibrations}
|
|
|
|
|
|
2024-07-19 11:00:51 +02:00
|
|
|
|
A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map
|
|
|
|
|
$f : M \to B$ whose generic fibers are Langrangian.
|
|
|
|
|
|
2024-07-19 13:16:42 +02:00
|
|
|
|
|
|
|
|
|
\paragraph{Compact Setting}
|
|
|
|
|
|
2024-07-19 11:00:51 +02:00
|
|
|
|
If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
|
2024-07-19 13:16:42 +02:00
|
|
|
|
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}
|
|
|
|
|
|
|
|
|
|
In the non-compact setting, geometers study Lagrangian fibrations 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
|
2024-07-19 11:00:51 +02:00
|
|
|
|
symmetry of certain pushforward sheaves and conjectured that more general
|
2024-07-19 13:16:42 +02:00
|
|
|
|
symmetries exist. Schnell has recently established these conjectures in
|
|
|
|
|
\cite{arXiv:2303.05364} and also proved two conjectures of Maulik–Shen–Yin on
|
2024-07-19 11:00:51 +02:00
|
|
|
|
the behavior of certain perverse sheaves near singular fibers.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\subsubsection{Singer--Hopf conjecture and fundamental groups of Kähler manifolds}
|
|
|
|
|
|
2024-07-19 13:16:42 +02:00
|
|
|
|
The Singer-Hopf conjecture asserts that a closed aspherical manifold of real
|
2024-07-19 11:00:51 +02:00
|
|
|
|
dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot
|
2024-07-19 13:16:42 +02:00
|
|
|
|
\chi(X)\geq 0$. 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.
|
2024-07-19 11:00:51 +02:00
|
|
|
|
|
|
|
|
|
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
|
2024-07-19 13:16:42 +02:00
|
|
|
|
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.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
2024-07-16 15:22:24 +02:00
|
|
|
|
Our goal in this workshop is to bring together several experts in geometric
|
2024-07-19 13:16:42 +02:00
|
|
|
|
group theory with experts on Hodge theory and to explore further potential
|
2024-07-16 15:22:24 +02:00
|
|
|
|
applications of the methods from one field to problems in the other.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\subsection{Canonical metrics and Kobayashi hyperbolicity}
|
|
|
|
|
|
|
|
|
|
\subsubsection{Kähler-Einstein metrics with conic singularities and their limits}
|
|
|
|
|
|
2024-07-16 15:22:24 +02:00
|
|
|
|
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.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
$$Ricci_{\omega_\beta}= \lambda \omega_{\beta}+ (1-\beta)[D],$$
|
|
|
|
|
where $\lambda$ is equal to -1 or 1. Can one extract a limit of $(\omega_\beta)$
|
|
|
|
|
as $\beta\to 0$, eventually after rescaling}?
|
|
|
|
|
|
2024-07-16 15:22:24 +02:00
|
|
|
|
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.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
\smallskip
|
|
|
|
|
|
2024-07-16 15:22:24 +02:00
|
|
|
|
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.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\subsubsection{Complex hyperbolicity}
|
|
|
|
|
|
2024-07-16 15:22:24 +02:00
|
|
|
|
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),
|
2024-07-16 16:48:07 +02:00
|
|
|
|
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.
|
2024-07-16 15:22:24 +02:00
|
|
|
|
|
|
|
|
|
In the last couple of years the field is taking a very interesting direction, by
|
2024-07-16 16:48:07 +02:00
|
|
|
|
combining techniques from Hodge theory with the familiar Nevanlinna theory and
|
2024-07-16 15:22:24 +02:00
|
|
|
|
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).
|
|
|
|
|
|
2024-07-16 16:48:07 +02:00
|
|
|
|
Techniques from birational geometry, in connection with the work of F.~Campana
|
2024-07-16 15:22:24 +02:00
|
|
|
|
are also present in the field via the -long awaited- work of E. Rousseau and its
|
|
|
|
|
collaborators.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
|
|
|
|
|
2024-07-16 15:22:24 +02:00
|
|
|
|
\subsubsection{Complex hyperbolicity. Mark II}
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
2024-07-16 16:48:07 +02:00
|
|
|
|
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.
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
2024-07-16 16:48:07 +02:00
|
|
|
|
\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.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\bibstyle{alpha}
|
|
|
|
|
\bibliographystyle{alpha}
|
|
|
|
|
\bibliography{general}
|
2024-07-16 15:18:00 +02:00
|
|
|
|
|
|
|
|
|
\end{document}
|