439 lines
18 KiB
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439 lines
18 KiB
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\title{Application for a Workshop on Complex Analysis}
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\author{Eleonora Di Nezza, Stefan Kebekus, Mihai Păun, Stefan Schreieder}
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\makeatletter
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\hypersetup{
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pdfauthor={Eleonora Di Nezza, Stefan Kebekus, Mihai Păun, Stefan Schreieder},
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\newcommand\young[1]{{\textbf{#1}}}
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\begin{document}
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\maketitle
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\section{Title and proposed organizers}
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\subsection{Workshop Title}
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Komplexe Analysis --- Analytic and Algebraic Methods in the Theory of Kähler Spaces
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\subsection{Proposed Organizers}
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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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\clearpage
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\section{Abstract}
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The proposed workshop will present recent advances in the analytic and algebraic
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study of Kähler spaces. Key topics to be covered include:
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\begin{itemize}
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\item Canonical metrics and their limits,
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\item Hyperbolicity properties of complex algebraic varieties,
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\item The topology and Hodge theory of Kähler spaces.
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\end{itemize}
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While these topics are classical, various breakthroughs were achieved only
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recently. Moreover, each is closely linked to various other branches of
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mathematics. For example, geometric group theorists have recently applied
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methods from complex geometry and Hodge theory to address long-standing open
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problems in geometric group theory. Similarly, concepts used in the framework
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of hyperbolicity questions, such as entire curves, jet differentials and
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Nevanlinna theory have recently seen important applications in the study of
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rational and integral points in number theory. To foster further
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interdisciplinary collaboration, we will invite several experts from related
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fields to participate in the workshop.
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The workshop has a distinguished history, originating with Grauert and Remmert.
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For the 2026 edition, it will feature 50\% new organizers and participants,
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ensuring fresh perspectives and innovative contributions.
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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 Kähler
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spaces, focusing on a combination of analytic and algebraic methods. We aim to
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emphasize the fields described below, each rooted in complex analysis and
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interconnected with various other branches of mathematics.
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An important goal of our workshop is to foster collaborations between
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mathematicians from different communities, with diverse backgrounds and
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perspectives. We have invited experts from related fields, and we will ask them
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to give survey talks on their work early in the week. This will allow them to
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introduce themselves to the complex geometers attending the workshop and provide
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ample opportunities for discussions throughout the rest of the week.
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Following the Oberwolfach guidelines, we will keep the number of talks
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comparatively small (no more than 25) to allow for plenty of informal
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discussions.
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One evening during the workshop, we will hold a special session where junior
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participants who are not selected to give a 60-minute talk can give a 5-10
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minute pitch on their work to introduce themselves to the community.
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\subsection{Canonical Metrics and Hyperbolicity}
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\subsubsection{Kähler--Einstein Metrics with Conic Singularities and Their Limits}
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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. 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. 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 $D \subsetneq X$ 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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\[
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Ricci_{\omega_\beta}= \lambda \cdot \omega_{\beta}+ (1-\beta)\cdot [D],
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\quad
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\text{where } \lambda \in \{ \pm 1\}.
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\]
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Is there a meaningful limit of $\omega_\beta$ as $\beta\to 0$, perhaps after
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rescaling?
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\end{q}
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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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\item If $X$ is Fano and $D$ is a divisor whose class is proportional to the
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anticanonical class, then the limit of the rescaled metric exists and equals
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the Tian--Yau metric.
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\end{itemize}
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More work is ongoing, and we expect to report on substantial progress by the
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time our workshop takes place.
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\subsubsection{Kähler--Einstein Metrics on Singular Spaces}
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Motivated by progress in the Minimal Model Program, there has been increasing
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interest in Kähler--Einstein metrics on singular spaces. While one of the first
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results in this direction dates back to the early 1970s when Kobayashi
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constructed orbifold Kähler--Einstein metrics, a definitive existence result for
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a relevant class of singularities was obtained by Eyssidieux--Guedj--Zeriahi
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about 15 years ago in \cite{zbMATH05859416}, by combining Yau's technique with
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Kolodziej's $\mathcal C^0$ estimates. Much more recently, Li--Tian--Wang
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extended Chen-Donaldson-Sun's solution of the Yau--Tian--Donaldson conjecture to
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general $\mathbb Q$-Fano varieties \cite{zbMATH07382001, zbMATH07597119}.
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For most applications, it is essential to control the geometry of these metrics
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near the singularities. Despite the problem's obvious importance, little is
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known so far. The continuity of the metric's potential has been established
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quite recently in the preprint \cite{arXiv:2401.03935} of Cho--Choi. Beyond
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that, the main progress in this direction is due to Hein--Sun
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\cite{zbMATH06827885}, who showed that near a large class of smoothable isolated
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singularities that are locally isomorphic to a Calabi-Yau cone, the singular
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Calabi-Yau metric must be asymptotic in a strong sense to the Calabi-Yau cone
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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--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 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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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 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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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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generic hypersurfaces of the projective space, $X \subsetneq \mathbb P^n$,
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provided that the degree of $X$ is larger than an explicit polynomial in $n$.
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These are significant improvements of earlier degree bounds, which involve
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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 a peer review, \cite{arXiv:2406.19003}.
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\paragraph{Representations of Fundamental Groups}
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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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\subsection{Topology and Hodge Theory of Kähler spaces}
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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 and commutative algebra.
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The following topics in this 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}. On the other hand Li--Tosatti found
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a more differential-geometric argument \cite{zbMATH07863260}, which relied
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heavily on a singular version of Mok's uniformization theorem. Even if both
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methods use results about rational curves, which confines them from the start to
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the smooth case, there is hope that they will give insight into the singular
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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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\section{Suggested and Excluded Dates}
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We would prefer if our workshop took place in mid of September or early to mid
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of April.
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\section{Preliminary list of proposed participants}
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Below is a preliminary list of 55 people (including organizers) we would like to
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invite\footnote{We list colleagues as ``young'' if they have no tenured job, or
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if their tenure is less than about three years old.}. The list meets or exceeds
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the quota on diversity and regionality laid out in the ``Proposal Guidelines for
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Workshops''.
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{\small
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\begin{longtable}[c]{lccccccc}
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\rowcolor{lightgray} \textbf{Name} & \textbf{Location} & \textbf{German} & \textbf{Young} & \textbf{Woman} & \textbf{New to conference} \\
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di Nezza, Eleonora&Sorbonne&&&1 \\
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Paun, Mihai&Bayreuth&1&& \\
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Kebekus, Stefan&Freiburg&1&& \\
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Schreieder, Stefan&Hannover&1&& \\
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&&&& \\
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Arapura, Donu&Purdue&&&&1 \\
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Bakker, Ben&Chicago&&&& \\
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Bérczi, Gergely&Aarhus&&&&1 \\
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Biquard, Olivier&Sorbonne&&&&1 \\
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Boucksom, Sébastien&Paris&&&& \\
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Braun, Lukas&Innsbruck&1&1&& \\
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Brotbek, Damian&Nancy&&&&1 \\
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Brunebarbe, Yohan&Bordeaux&&&& \\
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Cadorel, Benoît&Nancy&&&& \\
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Chiu, Shih-Kai&Vanderbilt&&1&&1 \\
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Conlon, Ronan&Dallas&&&&1 \\
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Delcroix, Thibault&Montpellier&&&& \\
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Deng, Ya&Nancy&&&& \\
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Dutta, Yajnaseni&Leiden&&&1&1 \\
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Engel, Phil&Bonn&1&1&&1 \\
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Eyssidieux, Phillippe&Grenoble&&&& \\
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Floris, Enrica&Poitiers&&&1& \\
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Friedman, Robert&Columbia&&&&1 \\
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Gachet, Cécile&Bochum&1&1&1&1 \\
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Graf, Patrick&Bayreuth&1&&& \\
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Greb, Daniel&Essen&1&&& \\
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Guedj, Vincent&Toulouse&&&& \\
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Guenancia, Henri&Toulouse&&&& \\
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Hausel, Tamas&IST Austria&&&&1 \\
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Hein, Hans-Joachim&Münster&1&&& \\
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Höring, Andreas&Nice&&&& \\
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Hoskins, Victoria&Nijmegen&&&1&1 \\
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Hwang, Jun-Muk&Daejon, Korea&&&& \\
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Javanpeykar, Arian&Nijmegen&&&& \\
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Kirwan, Frances&Oxford&&&1&1 \\
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Klingler, Bruno&Berlin&1&&& \\
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Li, Chi&Purdue&&&& \\
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Llosa Isenrich, Claudio&Karlsruhe&1&&&1 \\
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Mauri, Mirko&Paris&&&& \\
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Maxim, Laurenţiu&Wisconsin, Madison&&&&1 \\
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Mustață, Mircea&Ann Arbor&&&& \\
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Park, Sung Gi&Harvard&&1&&1 \\
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Peternell, Thomas&Bayreuth/Hannover&1&&& \\
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Pieropan, Marta&Utrecht&&&1&1 \\
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Popa, Mihnea&Harvard&&&& \\
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Pozzetti, Beatrice&Heidelberg&1&&1&1 \\
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Py, Pierre&Strasbourg&&&&1 \\
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Rousseau, Erwan&Brest&&&& \\
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Schnell, Christian&Stony Brook&&&& \\
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Spelta, Irene&Berlin&1&&1&1 \\
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Stenger, Isabel&Hannover&1&1&1&1 \\
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Sun, Song&Berkely&&&&1 \\
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Székelyhidi, Gabor&Northwestern&&&&1 \\
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Wang, Botong&Wisconsin&&&&1 \\
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Wang, Julie Tzu-Yueh&Taiwan&&&1&1 \\
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Xie, Zhixin&Nancy&&1&1&1
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\end{longtable}
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} % \scriptsize
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\bibstyle{alpha}
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\bibliographystyle{alpha}
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\bibliography{general}
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\end{document}
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