275 lines
12 KiB
TeX
275 lines
12 KiB
TeX
\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={Eleonora Di Nezza, Stefan Kebekus, Mihai Păun, Stefan Schreieder},
|
||
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. More recent developments connect the
|
||
theory to singularity theory, commutative algebra, and the topology of algebraic
|
||
varieties. The following topics in this area will be of particular interest to
|
||
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. An alternative approach towards similar ends was recently suggested in
|
||
the preprint \cite{arXiv:2309.16763} of Schnell and Yang. First applications
|
||
pertain to Bernstein--Sato polynomials and their zero sets; these are important
|
||
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.
|
||
|
||
Very recently, Park and Popa 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.
|
||
|
||
If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
|
||
projective space. In case where $B$ is smooth, the conjecture has been
|
||
established more than 16 years ago in a celebrated work of Hwang. Today, there
|
||
is new insight, as Bakker--Schnell recently found a purely Hodge theoretic proof
|
||
of Hwang's result, \cite{arXiv:2311.08977}. There is hope that these methods
|
||
give insight into the singular setting, which remains open to date.
|
||
|
||
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--Shen and Hausel--Mellit--Minets--Schiffmann, \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. These conjectures have recently been established by Schnell,
|
||
\cite{arXiv:2303.05364}, who 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, Hodge-theoretic refinements
|
||
of this conjecture for Kähler manifolds have been put forward by
|
||
Arapura--Maxim--Wang, \cite{arXiv:2310.14131}. Special cases of these conjecture
|
||
have been proven, but 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 finiteness properties of groups, \cite{zbMATH07790946}. As a byproduct,
|
||
the authors also obtain a proof of the classical Singer conjecture in an
|
||
important special 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 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.
|
||
|
||
|
||
|
||
\bibstyle{alpha}
|
||
\bibliographystyle{alpha}
|
||
\bibliography{general}
|
||
|
||
\end{document}
|