Merging changes, part I
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MFO26.tex
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MFO26.tex
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@ -57,97 +57,81 @@
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\section{Workshop Title}
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Komplexe Analysis --- Differential and Algebraic methods in Kähler spaces
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\section{Proposed Organisers}
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\begin{tabular}{ll}
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\parbox[t]{7cm}{
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Eleonora Di Nezza\\
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IMJ-PRG, Sorbonne Université,\\
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4 Place Jussieu\\
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75005 Paris\\
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France\\[2mm]
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\href{mailto:eleonora.dinezza@imj-prg.fr}{eleonora.dinezza@imj-prg.fr}} &
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\parbox[t]{7cm}{
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Stefan Kebekus\\
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Albert-Ludwigs-Universität Freiburg\\
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Ernst-Zermelo-Straße 1\\
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79104 Freiburg\\
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Germany\\[2mm]
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\href{mailto:stefan.kebekus@math.uni-freiburg.de}{stefan.kebekus@math.uni-freiburg.de}}
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\\
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\ \\
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\ \\
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\parbox[t]{6cm}{
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Mihai Păun \\
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Universität Bayreuth \\
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Universitätsstraße 30\\
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95447 Bayreuth\\
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Germany\\[2mm]
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\href{mailto:mihai.paun@uni-bayreuth.de}{mihai.paun@uni-bayreuth.de}}
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&
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\parbox[t]{6cm}{
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Stefan Schreieder\\
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Leibniz Universit\"at Hannover \\
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Welfengarten 1\\
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30167 Hannover\\
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Germany\\[2mm]
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\href{mailto:schreieder@math.uni-hannover.de}{schreieder@math.uni-hannover.de}}
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\end{tabular}
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\section{Abstract}
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Complex Analysis is a very active branch of mathematics with applications in many other fields.
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The proposed workshop presents recent results in complex
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analysis and surveys progress in topics that link the field to other branches of mathematics.
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%This application highlights differential-geometric methods in the study of singular spaces, the interplay between analytic and algebraic methods, and the relation between complex analysis and Scholze-Clausen's condensed mathematics.
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%The meeting has always been a venue where confirmed researchers from different backgrounds meet and where young mathematicians are giving their first talks at an international conference. While we are happy to see a growing number of talented, young researchers, we feel that this age group suffers the most from the ongoing COVID crisis and the resulting lack of exchange and interaction. We would therefore like to emphasize the contributions of younger researchers and invite a relatively higher number of them. We are looking forward to welcoming them to Oberwolfach, rediscover the pleasure of meeting in person, and exchange points of view!
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\section{Mathematics Subject Classification}
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\subsubsection*{2020 Mathematics Subject Classification}
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\begin{tabular}{llll}
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Primary & 32 &--& Several complex variables and analytic spaces\\
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Secondary & 14 &--& Algebraic geometry \\
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& 53 &--& Differential geometry \\
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& 58 &--& Global analysis, analysis on manifolds
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\end{tabular}
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\section{Description of the Workshop}
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The proposed workshop presents recent results in Complex Geometry and surveys relations to other fields.
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For 2026, we would like to emphasize the fields described below.
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Each relates to complex analysis differently.
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Each has seen substantial progress recently, producing results that will be of importance for years to come.
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%The bullet items list some of the latest developments that have attracted our attention.
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% Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules
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%
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%
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%- Singularities and Hodge ideals etc: Mustata-Popa, Park, Ruijie Yang, Schnell
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%
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%- Lagrangian fibrations: Maulik-Shen, Shen-Yin, Schnell, Bakker
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%
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%- Singer-Hopf conjecture and fundamental groups of Kaehler manifolds: Arapura, Botong Wang, Maxim, Llosa-Isenrich—Py.
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\subsection{Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules}
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Over the last decade, Saito's theory of Hodge modules has seen spectacular
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applications in birational geometry. More recent developments, which are of
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significant importance, connect the theory to singularity theory, commutative
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algebra, and the topology of algebraic varieties. The following topics in this
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area will particularly interest our workshop.
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\subsubsection{Singularities and Hodge Ideals}
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In a series of influential papers starting with \cite{MR4081135}, Mustaţă and
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Popa used Hodge modules to refine and generalize well-known invariants of
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singularities, most notably the multiplier ideals used in analysis and algebraic
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geometry. Schnell and Yang’s recent preprint \cite{arXiv:2309.16763} suggested
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an alternative approach toward similar ends. The first applications pertain to
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Bernstein--Sato polynomials and their zero sets; these are essential invariants
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of singularities originating from commutative algebra that are hard to compute.
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Schnell and Yang apply their results to conjectures of
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Debarre--Casalaina-Martin--Grushevsky concerning the Riemann--Schottky problem
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and the singularities of Theta divisors of principally polarized Abelian
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varieties.
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Park and Popa recently applied perverse sheaves and D-module theory to improve
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Goresky--MacPherson's classic Lefschetz theorems in the singular setting. A
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program put forward by Friedman--Laza aims at understanding the Hodge structures
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of degenerating Calabi--Yau varieties.
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\subsubsection{Lagrangian fibrations}
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A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map
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$f : M \to B$ whose generic fibers are Langrangian.
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\paragraph{Compact Setting}
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If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
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projective space. In the case where $B$ is smooth, Hwang established the
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conjecture more than 16 years ago in a celebrated paper. There is new insight
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today, as Bakker--Schnell recently found a purely Hodge theoretic proof of
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Hwang's result in \cite{arXiv:2311.08977}. Hopefully, these methods will give
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insight into the singular setting, which remains open to date.
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\paragraph{Non-compact Setting}
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Geometers study Lagrangian fibrations over non-compact bases in the framework of
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the ``$P=W$ conjecture,'' which Maulik--Shen and Hausel--Mellit--Minets--Schiffmann
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have recently proved \cite{arXiv:2209.02568, arXiv:2209.05429}. In the same
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setting, Shen–Yin discovered a remarkable symmetry of certain pushforward
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sheaves and conjectured that more general symmetries exist. Schnell has recently
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established these conjectures in \cite{arXiv:2303.05364} and also proved two
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conjectures of Maulik–Shen–Yin on the behavior of certain perverse sheaves near
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singular fibers.
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\subsubsection{Singer--Hopf conjecture and fundamental groups of Kähler manifolds}
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The Singer-Hopf conjecture asserts that a closed aspherical manifold of real
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dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot
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\chi(X)\geq 0$. This conjecture goes back to 1931 when Hopf formulated a related
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version for Riemannian manifolds. Recently, Arapura–Maxim–Wang suggested
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Hodge-theoretic refinements of this conjecture for Kähler manifolds in
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\cite{arXiv:2310.14131}. While the methods of \cite{arXiv:2310.14131} suffice to
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show particular cases, the statement remains open in full generality.
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In a related direction, Llosa-Isenrich--Py found an application of complex
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geometry and Hodge theory to geometric group theory, settling an old question of
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Brady on the finiteness properties of groups \cite{zbMATH07790946}. As a
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byproduct, the authors also obtain a proof of the classical Singer conjecture in
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an essential particular case in the realm of Kähler manifolds.
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Our goal in this workshop is to bring together several experts in geometric
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group theory with experts on Hodge theory and to explore further potential
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applications of the methods from one field to problems in the other.
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%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.
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We will account for new developments that arise between the time of submission of this proposal and the time of the workshop.
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Following good Oberwolfach tradition, we will keep the number of talks small to provide ample opportunity for informal discussions.
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%After so many months of the pandemic, this will be more than welcome!
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\subsection{Canonical Metrics and Hyperbolicity}
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@ -274,16 +258,16 @@ non-polynomial bounds of order $(\sqrt{n} \log n)^n$ or worse. The proof builds
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on a strategy of Diverio-Merker-Rousseau and combines non-reductive geometric
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invariant theory with the ``Grassmannian techniques'' of Riedl-Yang. A very
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recent preprint of Cadorel simplifies the proof Bérczi-Kirwan substantially, but
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still needs to undergo peer review, \cite{arXiv:2406.19003}.
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still needs to undergo a peer review, \cite{arXiv:2406.19003}.
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\paragraph{Hyperbolicity and representations of fundamental groups}
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Using recent advances in the theory of harmonic maps (due to
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Daskalopoulos-Mese), Y. Deng and K. Yamanoi were able to confirm the
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Daskalopoulos-Mese, cf. \cite{arXiv:2112.13961}), B. Cadorel, Y. Deng K. Yamanoi were able to confirm the
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Green-Griffiths conjecture for manifolds whose fundamental group admits a
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representation having certain natural properties (echoing the case of curves of
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genus at least two).
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genus at least two), cf. \cite{arXiv:2212.12225}.
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\paragraph{Material collections}
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@ -293,7 +277,195 @@ genus at least two).
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In the last couple of years the field is taking a very interesting direction, by
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combining techniques from Hodge theory with the familiar Nevanlinna theory and
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jet differentials, cf. the articles by D. Brotbek, Y. Deng, Y. Brunebarbe, B.
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Cadorel and A. Javanpeykar.
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Cadorel and A. Javanpeykar, cf. \cite{arXiv:2007.12957}, \cite{arXiv:2305.09613}, \cite{arXiv:2207.03283}.
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\subsection{Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules}
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Over the last decade, Saito's theory of Hodge modules has seen spectacular
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applications in birational geometry. More recent developments, which are of
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significant importance, connect the theory to singularity theory, commutative
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algebra, and the topology of algebraic varieties. The following topics in this
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area will particularly interest our workshop.
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\subsubsection{Singularities and Hodge Ideals}
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In a series of influential papers starting with \cite{MR4044463}, % \cite{MR4081135} is not the first one
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Mustaţă and Popa used Hodge modules to refine and generalize well-known invariants of
|
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singularities, most notably the multiplier ideals used in analysis and algebraic
|
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geometry. Schnell and Yang’s recent preprint \cite{arXiv:2309.16763} suggested
|
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an alternative approach toward similar ends. The first applications pertain to
|
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Bernstein--Sato polynomials and their zero sets; these are essential invariants
|
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of singularities originating from commutative algebra that are hard to compute.
|
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Schnell and Yang apply their results to conjectures of
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Debarre--Casalaina-Martin--Grushevsky concerning the Riemann--Schottky problem
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and the singularities of Theta divisors of principally polarized Abelian
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varieties.
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Park and Popa recently applied perverse sheaves and D-module theory to improve
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Goresky--MacPherson's classic Lefschetz theorems in the singular setting. A
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program put forward by Friedman--Laza aims at understanding the Hodge structures
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of degenerating Calabi--Yau varieties.
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\subsubsection{Lagrangian fibrations}
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A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map
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$f : M \to B$ whose generic fibers are Langrangian.
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\paragraph{Compact Setting}
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If $M$ is compact, a well-known conjecture in the field predicts that $B$ should be the
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projective space. A strong evidence for this problem is due to Hwang: he established the
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conjecture more than 16 years ago in a celebrated paper, provided that the base $B$ is smooth.
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There is new insight
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today, with two alternative arguments for the proof of this theorem.
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Bakker--Schnell recently found a purely Hodge theoretic proof of
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Hwang's result in \cite{arXiv:2311.08977}. On the other hand Tosatti--Li, cf. \cite{arXiv:2308.10553}
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found a more differential-geometric argument, which relied heavily on a singular version of Mok's uniformisation theorem.
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Even if both methods are using results about rational curves -which confines them from the start to
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the smooth case-, we hope that they put Hwang's result in a new perspective, hopefully helpful to progress towards the general case.
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\paragraph{Non-compact Setting}
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In the non-compact setting, geometers study Lagrangian fibrations in the
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framework of the ``$P=W$ conjecture,'' which Maulik–Shen and
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Hausel–Mellit–Minets–Schiffmann have recently proved \cite{arXiv:2209.02568,
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arXiv:2209.05429}. In the same setting, Shen–Yin discovered a remarkable
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symmetry of certain pushforward sheaves and conjectured that more general
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symmetries exist. Schnell has recently established these conjectures in
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\cite{arXiv:2303.05364} and also proved two conjectures of Maulik–Shen–Yin on
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the behavior of certain perverse sheaves near singular fibers.
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\subsubsection{Singer--Hopf conjecture and fundamental groups of Kähler manifolds}
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The Singer-Hopf conjecture asserts that a closed aspherical manifold of real
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dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot
|
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\chi(X)\geq 0$. This conjecture goes back to 1931 when Hopf formulated a related
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version for Riemannian manifolds. Recently, Arapura–Maxim–Wang suggested
|
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Hodge-theoretic refinements of this conjecture for Kähler manifolds in
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\cite{arXiv:2310.14131}. While the methods of \cite{arXiv:2310.14131} suffice to
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show particular cases, the statement remains open in full generality.
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In a related direction, Llosa-Isenrich--Py found an application of complex
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geometry and Hodge theory to geometric group theory, settling an old question of
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Brady on the finiteness properties of groups \cite{zbMATH07790946}. As a
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byproduct, the authors also obtain a proof of the classical Singer conjecture in
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an essential particular case in the realm of Kähler manifolds.
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Our goal in this workshop is to bring together several experts in geometric
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group theory with experts on Hodge theory and to explore further potential
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applications of the methods from one field to problems in the other.
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\section{Suggested dates}
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We would prefer if our workshop took place in mid of September or early to mid April.
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%If this date is not available, early to mid-April would be an alternative.
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%The workshop ``Komplexe Analysis'' traditionally takes place in the first week
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%of September. We would like to follow this tradition. If the traditional date is
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%not available, early to mid-April would be an alternative.
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\section{Preliminary list of proposed participants}
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Below is a preliminary list of people we would like to invite\footnote{We list
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colleagues as ``young'' if they have no tenured job, or if their tenure is
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less than about three years old.}.
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{\small
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\begin{longtable}[c]{lccccccc}
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\rowcolor{lightgray} \multicolumn{2}{l}{\textbf{Name}} & \textbf{Location} & \multicolumn{3}{c}{\textbf{Subject}} & \textbf{Young} & \textbf{Woman} \\
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\rowcolor{lightgray} &&& §5.1 & §5.2 & & & \\\hline \endhead
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%Araujo&Carolina&Rio de Janeiro&&&&&1 \\
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Bakker&Benjamin&Chicago&&&&& \\
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Berndtsson&Bo&Göteborg&&&&& \\
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%Bertini&Valeria&Chemnitz&&1&&1&1 \\
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%Blum&Harold&Stony Brook&1&1&&1 \\
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Boucksom&Sebastien&Paris&&&&& \\
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Braun&Lukas&Innsbruck&&&&1& \\
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Brotbek&Damian&Nancy&&&&& \\
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Brunebarbe&Yohan&CNRS/Bordeaux&&&&& \\
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Cadorel&Benoit&Nancy&&&&1& \\
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Campana&Frédéric&Nancy&&&&& \\
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Cao&Junyan&Nice&&&&& \\
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Castravet&Ana-Maria&Versailles&&&&&1\\
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%Chen&Jiaming&Nancy&1&1&&1& \\
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Claudon&Benoit&Rennes&&&&& \\
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%Commelin&Johan&Freiburg&&&1&1& \\
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%Darvas&Tamás&Maryland&1&&&& \\
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Das&Omprokash &TIFR Mumbai&&&&1&\\
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Delcroix&Thibaut&Montpellier&&&&& \\
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Deng&Ya&CNRS/Nancy&&&&& \\
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%di Nezza&Eleonora&CNRS/Palaiseau&1&&&1&1 \\
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Dutta&Yagna&Leiden&&&&&1\\
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Eyssidieux&Philippe &Grenoble&&&&&\\
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Gachet&C\'ecile & Berlin&&&&&1\\
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Graf&Patrick&Bayreuth&&&&1&\\
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Greb&Daniel&Essen&&&&& \\
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%Grossi&Annalisa&Chemnitz&&1&&1&1 \\
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Guenancia&Henri&CNRS/Toulouse&&&&& \\
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Hao&Feng& Shandong University&&&&&\\
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Hein&Hans-Joachim&Münster&&&&& \\
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%Huang&Xiaojun&Rutgers&1&&&& \\
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Hulek&Klaus&Hannover&&&&& \\
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Hoskins&Victoria&Essen&&&&&1\\
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Höring&Andreas&Nice&&&&& \\
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Hwang&Jun-Muk &Daejeon&&&&&\\
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Javanpeykar&Ariyan &Nijmegen&&&&&\\
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Kirwan&Frances&Oxford&&&&&1\\
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Klingler&Bruno&Berlin&&&&& \\
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%Koike&Takayuki&Osaka&1&1&&1& \\
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Lehn&Christian&Chemnitz&&&&& \\
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%Li&Chi&Rutgers&1&&&& \\
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Lin&Hsueh-Yung&Taiwan&&&&&\\
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Llosa-Isenrich& Claudio& Karlsruhe&&&&1&\\
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%Lu&Hoang-Chinh&Orsay&1&&&1& \\
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%Martinelli&Diletta&Amsterdam&&1&&1&1 \\
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%Matsumura&Shin-Ichi&Tohoku&&1&&1& \\
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Mauri&Mirko&Paris&&&&1& \\
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Moraga&Joaquín&UCLA&&&&1& \\
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M\"uller&Niklas&Essen&&&&1&\\
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%Olano&Sebastián&Northwestern&&1&&1& \\
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Ortega&Angela&Berlin&&&&&1\\
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Ou&Wenhao&AMSS, China&&&&1&\\
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Park&Sung Gi &Harvard&&&&&\\
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Paulsen&Matthias&Marburg& &&& 1 & \\
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%Paul&Sean T.&Wisconsin&1&&&& \\
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Peternell&Thomas&Bayreuth&&&&& \\
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Py& Pierre& Grenoble&&&&&\\
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Rousseau& Erwan& Brest&&&&&\\
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%Saccá&Giulia&NYU&1&1&&&1 \\
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Schnell&Christian&Stony Brook&&&&& \\
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%Shentu&Junchan&Heifei&&1&&1 \\
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%Siarhei&Finski&Grenoble&&&&1& \\
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Spelta&Irene&Barcelona&&&&&1\\
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Stenger&Isabel&Hannover&&&&&1\\
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Tasin&Luca&Mailand&&&&1&\\
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%Takayama&Shigeharu&Tokyo&&1&&& \\
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Tosatti&Valentino&Northwestern&&&&& \\
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%Ungureanu&Mara&Freiburg&&1&&1&1 \\
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Wang&Botong&University of Wisconsin&&&&&\\
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%Wang&Juanyong&Beijing&&1&&1& \\
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Witt-Nyström&David&Göteborg&&&&& \\
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%Wu&Xiaojun&Bayreuth&&&&1& \\
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%Xiao&Ming&UCSD&1&&&1& \\
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%Xu&Chenyang&Princeton&&1&&& \\
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Yang&Ruijie&Humboldt&&&&1&
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\end{longtable}
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} % \scriptsize
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