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Stefan Kebekus
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@ -79,3 +79,9 @@ Delcroix
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Székelyhidi
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Tosatti
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Chiu
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Oberwolfach
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Conlon
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Daskalopoulos
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Mese
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Nevanlinna
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arithmetics
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257
MFO26.tex
257
MFO26.tex
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@ -54,9 +54,6 @@
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\maketitle
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\section{Workshop Title}
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Komplexe Analysis --- Differential and Algebraic methods in Kähler spaces
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@ -122,15 +119,18 @@ Secondary & 14 &--& Algebraic geometry \\
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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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The proposed workshop presents recent results in Complex Geometry and surveys
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relations to other fields. For 2026, we would like to emphasize the fields
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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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%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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We will account for new developments that arise between the time of submission
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of this proposal and the time of the workshop. Following good Oberwolfach
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tradition, we will keep the number of talks small to provide ample opportunity
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for informal discussions.
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%After so many months of the pandemic, this will be more than welcome!
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@ -140,16 +140,16 @@ Following good Oberwolfach tradition, we will keep the number of talks small to
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In the proof of the Donaldson--Tian--Yau conjecture, which Chen--Donaldson--Sun
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gave in a series of papers around 2015, Kähler--Einstein metrics with conic
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singularities along a smooth divisor emerged to play a vital role. Since then,
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these metrics have become an object of study in their own right. The work of
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singularities along a smooth divisor emerged to play a vital role. The work of
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Brendle, Donaldson, Guenancia, Rubinstein, and many others provides a complete
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package of results that generalize Yau's celebrated solution of the Calabi
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conjecture to the conic setting. Today, many exciting recent developments in
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conjecture to the conic setting. Since then, these metrics have become an
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object of study in their own right. Today, many exciting recent developments in
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this field gravitate around the following general question.
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\begin{q}
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Let $X$ be a projective manifold, and let $D\subset $ be a non-singular
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divisor. Assume that for every sufficiently small angle $0< \beta << 1$,
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Let $X$ be a projective manifold, and let $D \subsetneq X$ be a non-singular
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divisor. Assume that for every sufficiently small angle $0 < \beta \ll 1$,
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there exists a unique Kähler--Einstein metric $\omega_\beta$ with conic
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singularities of angle $2\pi\beta$ along $D$. In other words, assume that
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\[
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@ -161,9 +161,9 @@ this field gravitate around the following general question.
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rescaling?
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\end{q}
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Starting with \cite{zbMATH07615186}, a series of articles by Biquard--Guenancia
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settles many relevant (and technically challenging!) particular cases of this
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question.
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In \cite{zbMATH07615186} and the very recent preprint \cite{arXiv:2407.01150},
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Biquard--Guenancia begin settling relevant (and technically challenging!)
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particular cases of this question.
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\begin{itemize}
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\item If $(X,D)$ is the toroidal compactification of a ball quotient, then the
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limit of the metric exists and equals the hyperbolic metric.
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@ -200,43 +200,11 @@ metric. Using the bounded geometry method, Datar--Fu--Song recently showed an
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analogous result in the case of isolated log canonical singularities
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\cite{zbMATH07669617}. Fu–Hein–Jiang obtained precise asymptotics shortly
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after, \cite{zbMATH07782497}. Essential contributions directly connected to
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these topics are due to Chiu, Delcroix, Hein, C.~Li, Y.~Li, Sun, Székelyhidi,
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Tosatti, and Zhang.
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\bigskip
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{\color{red}\textbf --- DO NOT READ ANYTHING BELOW THIS LINE ---}
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these topics are due to Chiu--Székelyhidi \cite{zbMATH07810677}, Delcroix,
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Conlon--Hein \cite{zbMATH07858206}, C.~Li, Y.~Li, Tosatti and Zhang.
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\subsubsection{Complex hyperbolicity}
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The Green-Griffiths conjecture (concerning the Zariski closure of holomorphic
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entire curves or more generally, of families of holomorphic disks on varieties
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of general type) continues to keep busy many complex geometers. Probably the
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most complete result in this field is due to A. Bloch (more than 100 years ago),
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who -in modern language- showed that the Zariski closure of a map $\varphi:
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\mathbb C \to A$ to a complex tori $A$ is the translate of a sub-tori. A decade
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ago, K.~Yamanoi established the Green-Griffiths conjecture for projective
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manifolds general type, which admit a generically finite map into an Abelian
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variety. This represents a very nice generalization of Bloch's theorem.
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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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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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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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Techniques from birational geometry, in connection with the work of F.~Campana
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are also present in the field via the -long awaited- work of E. Rousseau and its
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collaborators.
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\subsubsection{Complex hyperbolicity. Mark II}
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\subsubsection{Complex Hyperbolicity}
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The 1979 Green-Griffiths-Lang conjecture asserts that every complex-projective
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variety $X$ of general type contains a proper subvariety $Y \subsetneq X$, such
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@ -244,10 +212,15 @@ that every non-constant entire holomorphic curve $\mathbb C \to X$ takes its
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values in $Y$. Its beginnings date back to 1926, when André Bloch showed that
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the Zariski closure of entire holomorphic curve $\varphi: \mathbb C \to A$ to a
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complex torus $A$ is the translate of a sub-torus. Today, the conjecture still
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drives much of the research in complex geometry. We highlight several advances
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that will be relevant for our workshop.
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drives substantial research in complex geometry. Several authors, including
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Brotbek, Brunebarbe, Deng, Cadorel, and Javanpeykar, opened a new research
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direction with relation to arithmetic, by combining techniques from Hodge theory
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with Nevanlinna theory and jet differentials, \cite{arXiv:2007.12957,
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arXiv:2207.03283, arXiv:2305.09613}. Besides, we highlight two additional
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advances that will be relevant for our workshop.
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\paragraph{Hypersurfaces in projective space}
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\paragraph{Hypersurfaces in Projective Space}
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A remarkable paper of Bérczi and Kirwan, \cite{MR4688701} published in September
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last year, establishes hyperbolicity and proves the Green-Griffiths-Lang for
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@ -261,43 +234,28 @@ recent preprint of Cadorel simplifies the proof Bérczi-Kirwan substantially, bu
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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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\paragraph{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, 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), cf. \cite{arXiv:2212.12225}.
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Using recent advances in the theory of harmonic maps due to Daskalopoulos--Mese
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\cite{arXiv:2112.13961}, Deng--Yamanoi were able to confirm the Green--Griffiths
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conjecture for manifolds whose fundamental group admits a representation having
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certain natural properties, in direct analogy to the case of general-type
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curves.
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\paragraph{Material collections}
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\subsection{Topology and Hodge Theory of Kähler spaces}
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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, 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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Ever since its invention, Hodge theory has been one of the most powerful tools
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in studying the geometry and topology of Kähler spaces. More recent
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developments connect the theory to singularity theory, commutative algebra, and
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the topology of algebraic varieties. The following topics in this area will
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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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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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@ -314,7 +272,7 @@ 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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\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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@ -322,31 +280,27 @@ $f : M \to B$ 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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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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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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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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\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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@ -368,105 +322,6 @@ 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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\bibstyle{alpha}
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79
general.bib
79
general.bib
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@ -1,3 +1,82 @@
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@misc{arXiv:2207.03283,
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title={Hyperbolicity in presence of a large local system},
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author={Yohan Brunebarbe},
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year={2022},
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eprint={2207.03283},
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archivePrefix={arXiv},
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primaryClass={math.AG},
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url={https://arxiv.org/abs/2207.03283},
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}
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@misc{arXiv:2305.09613,
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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,
|
||||
Author = {Fu, Xin and Hein, Hans-Joachim and Jiang, Xumin},
|
||||
Title = {Asymptotics of {K{\"a}hler}-{Einstein} metrics on complex hyperbolic cusps},
|
||||
|
|
|
|||
Loading…
Reference in New Issue