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.vscode/ltex.dictionary.en-US.txt vendored
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@ -79,3 +79,9 @@ Delcroix
Székelyhidi Székelyhidi
Tosatti Tosatti
Chiu Chiu
Oberwolfach
Conlon
Daskalopoulos
Mese
Nevanlinna
arithmetics

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MFO26.tex
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@ -54,9 +54,6 @@
\maketitle \maketitle
\section{Workshop Title} \section{Workshop Title}
Komplexe Analysis --- Differential and Algebraic methods in Kähler spaces Komplexe Analysis --- Differential and Algebraic methods in Kähler spaces
@ -122,15 +119,18 @@ Secondary & 14 &--& Algebraic geometry \\
\section{Description of the Workshop} \section{Description of the Workshop}
The proposed workshop presents recent results in Complex Geometry and surveys relations to other fields. The proposed workshop presents recent results in Complex Geometry and surveys
For 2026, we would like to emphasize the fields described below. relations to other fields. For 2026, we would like to emphasize the fields
described below.
Each relates to complex analysis differently. Each relates to complex analysis differently.
Each has seen substantial progress recently, producing results that will be of importance for years to come. Each has seen substantial progress recently, producing results that will be of importance for years to come.
%The bullet items list some of the latest developments that have attracted our attention. %The bullet items list some of the latest developments that have attracted our attention.
%We plan to include at least one broader overview talk for each of the three subjects, as well as more specialized presentations by senior experts and junior researchers. %We plan to include at least one broader overview talk for each of the three subjects, as well as more specialized presentations by senior experts and junior researchers.
We will account for new developments that arise between the time of submission of this proposal and the time of the workshop. We will account for new developments that arise between the time of submission
Following good Oberwolfach tradition, we will keep the number of talks small to provide ample opportunity for informal discussions. of this proposal and the time of the workshop. Following good Oberwolfach
tradition, we will keep the number of talks small to provide ample opportunity
for informal discussions.
%After so many months of the pandemic, this will be more than welcome! %After so many months of the pandemic, this will be more than welcome!
@ -140,16 +140,16 @@ Following good Oberwolfach tradition, we will keep the number of talks small to
In the proof of the Donaldson--Tian--Yau conjecture, which Chen--Donaldson--Sun 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 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, singularities along a smooth divisor emerged to play a vital role. The work of
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 Brendle, Donaldson, Guenancia, Rubinstein, and many others provides a complete
package of results that generalize Yau's celebrated solution of the Calabi package of results that generalize Yau's celebrated solution of the Calabi
conjecture to the conic setting. Today, many exciting recent developments in conjecture to the conic setting. Since then, these metrics have become an
object of study in their own right. Today, many exciting recent developments in
this field gravitate around the following general question. this field gravitate around the following general question.
\begin{q} \begin{q}
Let $X$ be a projective manifold, and let $D\subset $ be a non-singular Let $X$ be a projective manifold, and let $D \subsetneq X$ be a non-singular
divisor. Assume that for every sufficiently small angle $0< \beta << 1$, divisor. Assume that for every sufficiently small angle $0 < \beta \ll 1$,
there exists a unique Kähler--Einstein metric $\omega_\beta$ with conic there exists a unique Kähler--Einstein metric $\omega_\beta$ with conic
singularities of angle $2\pi\beta$ along $D$. In other words, assume that singularities of angle $2\pi\beta$ along $D$. In other words, assume that
\[ \[
@ -161,9 +161,9 @@ this field gravitate around the following general question.
rescaling? rescaling?
\end{q} \end{q}
Starting with \cite{zbMATH07615186}, a series of articles by Biquard--Guenancia In \cite{zbMATH07615186} and the very recent preprint \cite{arXiv:2407.01150},
settles many relevant (and technically challenging!) particular cases of this Biquard--Guenancia begin settling relevant (and technically challenging!)
question. particular cases of this question.
\begin{itemize} \begin{itemize}
\item If $(X,D)$ is the toroidal compactification of a ball quotient, then the \item If $(X,D)$ is the toroidal compactification of a ball quotient, then the
limit of the metric exists and equals the hyperbolic metric. limit of the metric exists and equals the hyperbolic metric.
@ -200,43 +200,11 @@ metric. Using the bounded geometry method, Datar--Fu--Song recently showed an
analogous result in the case of isolated log canonical singularities analogous result in the case of isolated log canonical singularities
\cite{zbMATH07669617}. Fu–Hein–Jiang obtained precise asymptotics shortly \cite{zbMATH07669617}. Fu–Hein–Jiang obtained precise asymptotics shortly
after, \cite{zbMATH07782497}. Essential contributions directly connected to after, \cite{zbMATH07782497}. Essential contributions directly connected to
these topics are due to Chiu, Delcroix, Hein, C.~Li, Y.~Li, Sun, Székelyhidi, these topics are due to Chiu--Székelyhidi \cite{zbMATH07810677}, Delcroix,
Tosatti, and Zhang. Conlon--Hein \cite{zbMATH07858206}, C.~Li, Y.~Li, Tosatti and Zhang.
\bigskip
{\color{red}\textbf --- DO NOT READ ANYTHING BELOW THIS LINE ---}
\subsubsection{Complex hyperbolicity} \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 The 1979 Green-Griffiths-Lang conjecture asserts that every complex-projective
variety $X$ of general type contains a proper subvariety $Y \subsetneq X$, such variety $X$ of general type contains a proper subvariety $Y \subsetneq X$, such
@ -244,10 +212,15 @@ 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 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 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 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 drives substantial research in complex geometry. Several authors, including
that will be relevant for our workshop. Brotbek, Brunebarbe, Deng, Cadorel, and Javanpeykar, opened a new research
direction with relation to arithmetic, by combining techniques from Hodge theory
with Nevanlinna theory and jet differentials, \cite{arXiv:2007.12957,
arXiv:2207.03283, arXiv:2305.09613}. Besides, we highlight two additional
advances that will be relevant for our workshop.
\paragraph{Hypersurfaces in projective space}
\paragraph{Hypersurfaces in Projective Space}
A remarkable paper of Bérczi and Kirwan, \cite{MR4688701} published in September A remarkable paper of Bérczi and Kirwan, \cite{MR4688701} published in September
last year, establishes hyperbolicity and proves the Green-Griffiths-Lang for last year, establishes hyperbolicity and proves the Green-Griffiths-Lang for
@ -261,43 +234,28 @@ recent preprint of Cadorel simplifies the proof Bérczi-Kirwan substantially, bu
still needs to undergo a peer review, \cite{arXiv:2406.19003}. still needs to undergo a peer review, \cite{arXiv:2406.19003}.
\paragraph{Hyperbolicity and representations of fundamental groups} \paragraph{Representations of Fundamental Groups}
Using recent advances in the theory of harmonic maps (due to Using recent advances in the theory of harmonic maps due to Daskalopoulos--Mese
Daskalopoulos-Mese, cf. \cite{arXiv:2112.13961}), B. Cadorel, Y. Deng K. Yamanoi were able to confirm the \cite{arXiv:2112.13961}, Deng--Yamanoi were able to confirm the Green--Griffiths
Green-Griffiths conjecture for manifolds whose fundamental group admits a conjecture for manifolds whose fundamental group admits a representation having
representation having certain natural properties (echoing the case of curves of certain natural properties, in direct analogy to the case of general-type
genus at least two), cf. \cite{arXiv:2212.12225}. curves.
\paragraph{Material collections} \subsection{Topology and Hodge Theory of Kähler spaces}
Ever since its invention, Hodge theory has been one of the most powerful tools
in studying the geometry and topology of Kähler spaces. More recent
In the last couple of years the field is taking a very interesting direction, by developments connect the theory to singularity theory, commutative algebra, and
combining techniques from Hodge theory with the familiar Nevanlinna theory and the topology of algebraic varieties. The following topics in this area will
jet differentials, cf. the articles by D. Brotbek, Y. Deng, Y. Brunebarbe, B. particularly interest our workshop.
Cadorel and A. Javanpeykar, cf. \cite{arXiv:2007.12957}, \cite{arXiv:2305.09613}, \cite{arXiv:2207.03283}.
\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} \subsubsection{Singularities and Hodge Ideals}
In a series of influential papers starting with \cite{MR4044463}, % \cite{MR4081135} is not the first one In a series of influential papers starting with \cite{MR4081135}, Mustaţă and
Mustaţă and Popa used Hodge modules to refine and generalize well-known invariants of Popa used Hodge modules to refine and generalize well-known invariants of
singularities, most notably the multiplier ideals used in analysis and algebraic singularities, most notably the multiplier ideals used in analysis and algebraic
geometry. Schnell and Yang’s recent preprint \cite{arXiv:2309.16763} suggested geometry. Schnell and Yang’s recent preprint \cite{arXiv:2309.16763} suggested
an alternative approach toward similar ends. The first applications pertain to an alternative approach toward similar ends. The first applications pertain to
@ -314,7 +272,7 @@ program put forward by Friedman--Laza aims at understanding the Hodge structures
of degenerating Calabi--Yau varieties. of degenerating Calabi--Yau varieties.
\subsubsection{Lagrangian fibrations} \subsubsection{Lagrangian Fibrations}
A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map
$f : M \to B$ whose generic fibers are Langrangian. $f : M \to B$ whose generic fibers are Langrangian.
@ -322,31 +280,27 @@ $f : M \to B$ whose generic fibers are Langrangian.
\paragraph{Compact Setting} \paragraph{Compact Setting}
If $M$ is compact, a well-known conjecture in the field predicts that $B$ should be the If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
projective space. A strong evidence for this problem is due to Hwang: he established the projective space. In the case where $B$ is smooth, Hwang established the
conjecture more than 16 years ago in a celebrated paper, provided that the base $B$ is smooth. conjecture more than 16 years ago in a celebrated paper. There is new insight
There is new insight today, as Bakker--Schnell recently found a purely Hodge theoretic proof of
today, with two alternative arguments for the proof of this theorem. Hwang's result in \cite{arXiv:2311.08977}. Hopefully, these methods will give
Bakker--Schnell recently found a purely Hodge theoretic proof of insight into the singular setting, which remains open to date.
Hwang's result in \cite{arXiv:2311.08977}. On the other hand Tosatti--Li, cf. \cite{arXiv:2308.10553}
found a more differential-geometric argument, which relied heavily on a singular version of Mok's uniformisation theorem.
Even if both methods are using results about rational curves -which confines them from the start to
the smooth case-, we hope that they put Hwang's result in a new perspective, hopefully helpful to progress towards the general case.
\paragraph{Non-compact Setting} \paragraph{Non-compact Setting}
In the non-compact setting, geometers study Lagrangian fibrations in the Geometers study Lagrangian fibrations over non-compact bases in the framework of
framework of the ``$P=W$ conjecture,'' which Maulik–Shen and the ``$P=W$ conjecture,'' which Maulik--Shen and Hausel--Mellit--Minets--Schiffmann
Hausel–Mellit–Minets–Schiffmann have recently proved \cite{arXiv:2209.02568, have recently proved \cite{arXiv:2209.02568, arXiv:2209.05429}. In the same
arXiv:2209.05429}. In the same setting, Shen–Yin discovered a remarkable setting, Shen–Yin discovered a remarkable symmetry of certain pushforward
symmetry of certain pushforward sheaves and conjectured that more general sheaves and conjectured that more general symmetries exist. Schnell has recently
symmetries exist. Schnell has recently established these conjectures in established these conjectures in \cite{arXiv:2303.05364} and also proved two
\cite{arXiv:2303.05364} and also proved two conjectures of Maulik–Shen–Yin on conjectures of Maulik–Shen–Yin on the behavior of certain perverse sheaves near
the behavior of certain perverse sheaves near singular fibers. singular fibers.
\subsubsection{Singer--Hopf conjecture and fundamental groups of Kähler manifolds} \subsubsection{Singer--Hopf Conjecture and Fundamental Groups of Kähler Manifolds}
The Singer-Hopf conjecture asserts that a closed aspherical manifold of real The Singer-Hopf conjecture asserts that a closed aspherical manifold of real
dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot
@ -368,105 +322,6 @@ applications of the methods from one field to problems in the other.
\section{Suggested dates}
We would prefer if our workshop took place in mid of September or early to mid April.
%If this date is not available, early to mid-April would be an alternative.
%The workshop ``Komplexe Analysis'' traditionally takes place in the first week
%of September. We would like to follow this tradition. If the traditional date is
%not available, early to mid-April would be an alternative.
\section{Preliminary list of proposed participants}
Below is a preliminary list of people we would like to invite\footnote{We list
colleagues as ``young'' if they have no tenured job, or if their tenure is
less than about three years old.}.
{\small
\begin{longtable}[c]{lccccccc}
\rowcolor{lightgray} \multicolumn{2}{l}{\textbf{Name}} & \textbf{Location} & \multicolumn{3}{c}{\textbf{Subject}} & \textbf{Young} & \textbf{Woman} \\
\rowcolor{lightgray} &&& §5.1 & §5.2 & & & \\\hline \endhead
%Araujo&Carolina&Rio de Janeiro&&&&&1 \\
Bakker&Benjamin&Chicago&&&&& \\
Berndtsson&Bo&Göteborg&&&&& \\
%Bertini&Valeria&Chemnitz&&1&&1&1 \\
%Blum&Harold&Stony Brook&1&1&&1 \\
Boucksom&Sebastien&Paris&&&&& \\
Braun&Lukas&Innsbruck&&&&1& \\
Brotbek&Damian&Nancy&&&&& \\
Brunebarbe&Yohan&CNRS/Bordeaux&&&&& \\
Cadorel&Benoit&Nancy&&&&1& \\
Campana&Frédéric&Nancy&&&&& \\
Cao&Junyan&Nice&&&&& \\
Castravet&Ana-Maria&Versailles&&&&&1\\
%Chen&Jiaming&Nancy&1&1&&1& \\
Claudon&Benoit&Rennes&&&&& \\
%Commelin&Johan&Freiburg&&&1&1& \\
%Darvas&Tamás&Maryland&1&&&& \\
Das&Omprokash &TIFR Mumbai&&&&1&\\
Delcroix&Thibaut&Montpellier&&&&& \\
Deng&Ya&CNRS/Nancy&&&&& \\
%di Nezza&Eleonora&CNRS/Palaiseau&1&&&1&1 \\
Dutta&Yagna&Leiden&&&&&1\\
Eyssidieux&Philippe &Grenoble&&&&&\\
Gachet&C\'ecile & Berlin&&&&&1\\
Graf&Patrick&Bayreuth&&&&1&\\
Greb&Daniel&Essen&&&&& \\
%Grossi&Annalisa&Chemnitz&&1&&1&1 \\
Guenancia&Henri&CNRS/Toulouse&&&&& \\
Hao&Feng& Shandong University&&&&&\\
Hein&Hans-Joachim&Münster&&&&& \\
%Huang&Xiaojun&Rutgers&1&&&& \\
Hulek&Klaus&Hannover&&&&& \\
Hoskins&Victoria&Essen&&&&&1\\
Höring&Andreas&Nice&&&&& \\
Hwang&Jun-Muk &Daejeon&&&&&\\
Javanpeykar&Ariyan &Nijmegen&&&&&\\
Kirwan&Frances&Oxford&&&&&1\\
Klingler&Bruno&Berlin&&&&& \\
%Koike&Takayuki&Osaka&1&1&&1& \\
Lehn&Christian&Chemnitz&&&&& \\
%Li&Chi&Rutgers&1&&&& \\
Lin&Hsueh-Yung&Taiwan&&&&&\\
Llosa-Isenrich& Claudio& Karlsruhe&&&&1&\\
%Lu&Hoang-Chinh&Orsay&1&&&1& \\
%Martinelli&Diletta&Amsterdam&&1&&1&1 \\
%Matsumura&Shin-Ichi&Tohoku&&1&&1& \\
Mauri&Mirko&Paris&&&&1& \\
Moraga&Joaquín&UCLA&&&&1& \\
M\"uller&Niklas&Essen&&&&1&\\
%Olano&Sebastián&Northwestern&&1&&1& \\
Ortega&Angela&Berlin&&&&&1\\
Ou&Wenhao&AMSS, China&&&&1&\\
Park&Sung Gi &Harvard&&&&&\\
Paulsen&Matthias&Marburg& &&& 1 & \\
%Paul&Sean T.&Wisconsin&1&&&& \\
Peternell&Thomas&Bayreuth&&&&& \\
Py& Pierre& Grenoble&&&&&\\
Rousseau& Erwan& Brest&&&&&\\
%Saccá&Giulia&NYU&1&1&&&1 \\
Schnell&Christian&Stony Brook&&&&& \\
%Shentu&Junchan&Heifei&&1&&1 \\
%Siarhei&Finski&Grenoble&&&&1& \\
Spelta&Irene&Barcelona&&&&&1\\
Stenger&Isabel&Hannover&&&&&1\\
Tasin&Luca&Mailand&&&&1&\\
%Takayama&Shigeharu&Tokyo&&1&&& \\
Tosatti&Valentino&Northwestern&&&&& \\
%Ungureanu&Mara&Freiburg&&1&&1&1 \\
Wang&Botong&University of Wisconsin&&&&&\\
%Wang&Juanyong&Beijing&&1&&1& \\
Witt-Nyström&David&Göteborg&&&&& \\
%Wu&Xiaojun&Bayreuth&&&&1& \\
%Xiao&Ming&UCSD&1&&&1& \\
%Xu&Chenyang&Princeton&&1&&& \\
Yang&Ruijie&Humboldt&&&&1&
\end{longtable}
} % \scriptsize
\bibstyle{alpha} \bibstyle{alpha}

79
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@ -1,3 +1,82 @@
@misc{arXiv:2207.03283,
title={Hyperbolicity in presence of a large local system},
author={Yohan Brunebarbe},
year={2022},
eprint={2207.03283},
archivePrefix={arXiv},
primaryClass={math.AG},
url={https://arxiv.org/abs/2207.03283},
}
@misc{arXiv:2305.09613,
title={The relative Green-Griffiths-Lang conjecture for families of varieties of maximal Albanese dimension},
author={Yohan Brunebarbe},
year={2023},
eprint={2305.09613},
archivePrefix={arXiv},
primaryClass={math.AG},
url={https://arxiv.org/abs/2305.09613},
}
@misc{arXiv:2007.12957,
title={Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications},
author={Damian Brotbek and Yohan Brunebarbe},
year={2020},
eprint={2007.12957},
archivePrefix={arXiv},
primaryClass={math.AG},
url={https://arxiv.org/abs/2007.12957},
}
@misc{arXiv:2112.13961,
title={Infinite energy maps and rigidity},
author={Daskalopoulos, Georgios and Mese, Chikako},
year={2021},
month={December},
url={https://arxiv.org/abs/2112.13961}
}
@Article{zbMATH07858206,
Author = {Conlon, Ronan J. and Hein, Hans-Joachim},
Title = {Classification of asymptotically conical {Calabi}-{Yau} manifolds},
FJournal = {Duke Mathematical Journal},
Journal = {Duke Math. J.},
ISSN = {0012-7094},
Volume = {173},
Number = {5},
Pages = {947--1015},
Year = {2024},
Language = {English},
DOI = {10.1215/00127094-2023-0030},
Keywords = {53C25,14J32},
zbMATH = {7858206}
}
@Article{zbMATH07810677,
Author = {Chiu, Shih-Kai and Sz{\'e}kelyhidi, G{\'a}bor},
Title = {Higher regularity for singular {K{\"a}hler}-{Einstein} metrics},
FJournal = {Duke Mathematical Journal},
Journal = {Duke Math. J.},
ISSN = {0012-7094},
Volume = {172},
Number = {18},
Pages = {3521--3558},
Year = {2023},
Language = {English},
DOI = {10.1215/00127094-2022-0107},
Keywords = {32Q20,32Q25,53C25},
URL = {projecteuclid.org/journals/duke-mathematical-journal/volume-172/issue-18/Higher-regularity-for-singular-K%c3%a4hlerEinstein-metrics/10.1215/00127094-2022-0107.full},
zbMATH = {7810677}
}
@misc{arXiv:2407.01150,
title={Degenerating conic Kähler-Einstein metrics to the normal cone},
author={Biquard, Olivier and Guenancia, Henri},
year={2024},
month={July},
url={https://arxiv.org/abs/2407.01150},
}
@Article{zbMATH07782497, @Article{zbMATH07782497,
Author = {Fu, Xin and Hein, Hans-Joachim and Jiang, Xumin}, Author = {Fu, Xin and Hein, Hans-Joachim and Jiang, Xumin},
Title = {Asymptotics of {K{\"a}hler}-{Einstein} metrics on complex hyperbolic cusps}, Title = {Asymptotics of {K{\"a}hler}-{Einstein} metrics on complex hyperbolic cusps},