\documentclass[a4paper, british]{scrartcl} % % Local font definitions -- need to come first % \usepackage{amsthm} \usepackage{libertine} \usepackage[libertine]{newtxmath} \usepackage{xcolor} \usepackage{longtable} %\usepackage{ccfonts,color,comment} \usepackage[T1]{fontenc} \usepackage{hyperref} \usepackage[utf8]{inputenc} \newcounter{saveenum} \newenvironment{itemize-compact}{\begin{itemize}\itemsep -2pt}{\end{itemize}} \usepackage{colortbl} \usepackage{pdflscape} \sloppy \newtheorem*{q}{Question} % Colours for hyperlinks \definecolor{lightgray}{RGB}{220,220,220} \definecolor{gray}{RGB}{180,180,180} \definecolor{linkred}{rgb}{0.7,0.2,0.2} \definecolor{linkblue}{rgb}{0,0.2,0.6} % Limit table of contents to section titles \setcounter{tocdepth}{1} \title{Application for a Workshop on Complex Analysis} \author{Eleonora Di Nezza, Stefan Kebekus, Mihai Păun, Stefan Schreieder} \makeatletter \hypersetup{ pdfauthor={Eleonora Di Nezza, Stefan Kebekus, Mihai Păun, Stefan Schreieder}, pdftitle={\@title}, pdfstartview={Fit}, pdfpagelayout={TwoColumnRight}, pdfpagemode={UseOutlines}, colorlinks, linkcolor=linkblue, citecolor=linkred, urlcolor=linkred} \makeatother \newcommand\young[1]{{\textbf{#1}}} \begin{document} \maketitle \section{Workshop Title} Komplexe Analysis --- Differential and Algebraic Methods in the Theory of Kähler Spaces \section{Proposed Organisers} \begin{tabular}{ll} \parbox[t]{7cm}{ Eleonora Di Nezza\\ IMJ-PRG, Sorbonne Université,\\ 4 Place Jussieu\\ 75005 Paris\\ France\\[2mm] \href{mailto:eleonora.dinezza@imj-prg.fr}{eleonora.dinezza@imj-prg.fr}} & \parbox[t]{7cm}{ Stefan Kebekus\\ Albert-Ludwigs-Universität Freiburg\\ Ernst-Zermelo-Straße 1\\ 79104 Freiburg\\ Germany\\[2mm] \href{mailto:stefan.kebekus@math.uni-freiburg.de}{stefan.kebekus@math.uni-freiburg.de}} \\ \ \\ \ \\ \parbox[t]{6cm}{ Mihai Păun \\ Universität Bayreuth \\ Universitätsstraße 30\\ 95447 Bayreuth\\ Germany\\[2mm] \href{mailto:mihai.paun@uni-bayreuth.de}{mihai.paun@uni-bayreuth.de}} & \parbox[t]{6cm}{ Stefan Schreieder\\ Leibniz Universit\"at Hannover \\ Welfengarten 1\\ 30167 Hannover\\ Germany\\[2mm] \href{mailto:schreieder@math.uni-hannover.de}{schreieder@math.uni-hannover.de}} \end{tabular} \section{Abstract} Complex Analysis is a very active branch of mathematics with applications in many other fields. The proposed workshop presents recent results in complex analysis and especially the analytic and algebraic study of Kähler spaces, and surveys progress in topics that link the field to other branches of mathematics. This application highlights canonical metrics and their limits, hyperbolicity properties of complex algebraic varieties and the topology and Hodge theory of Kähler spaces. An important aspect of our workshop are its close ties to other branches of mathematics. Our aim is to invite a few experts from neighboring fields where we expect fruitful interactions in the future. For instance, we will include a small number of geometric group theorists, including Py and Llosa-Isenrich, that have recently applied methods from complex geometry and Hodge theory to solve longstanding open problems in geometric group theory. %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. %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! \section{Mathematics Subject Classification} \subsubsection*{2020 Mathematics Subject Classification} \begin{tabular}{llll} Primary & 32 &--& Several complex variables and analytic spaces\\ Secondary & 14 &--& Algebraic geometry \\ & 53 &--& Differential geometry \\ & 58 &--& Global analysis, analysis on manifolds \end{tabular} \section{Description of the Workshop} The proposed workshop presents recent results in Complex Geometry and surveys relations to other fields. For 2026, we would like to emphasize the fields described below. Each relates to complex analysis differently. Each has seen substantial progress recently, producing results that will be of importance for years to come. We will account for new developments that arise between the time of submission of this proposal and the time of the workshop. Following good Oberwolfach tradition, we will keep the number of talks small to provide ample opportunity for informal discussions. \subsection{Canonical Metrics and Hyperbolicity} \subsubsection{Kähler--Einstein Metrics with Conic Singularities and Their Limits} In the proof of the Donaldson--Tian--Yau conjecture, which Chen--Donaldson--Sun gave in a series of papers around 2015, Kähler--Einstein metrics with conic singularities along a smooth divisor emerged to play a vital role. The work of Brendle, Donaldson, Guenancia, Rubinstein, and many others provides a complete package of results that generalize Yau's celebrated solution of the Calabi conjecture to the conic setting. Since then, these metrics have become an object of study in their own right. Today, many exciting recent developments in this field gravitate around the following general question. \begin{q} Let $X$ be a projective manifold, and let $D \subsetneq X$ be a non-singular divisor. Assume that for every sufficiently small angle $0 < \beta \ll 1$, there exists a unique Kähler--Einstein metric $\omega_\beta$ with conic singularities of angle $2\pi\beta$ along $D$. In other words, assume that \[ Ricci_{\omega_\beta}= \lambda \cdot \omega_{\beta}+ (1-\beta)\cdot [D], \quad \text{where } \lambda \in \{ \pm 1\}. \] Is there a meaningful limit of $\omega_\beta$ as $\beta\to 0$, perhaps after rescaling? \end{q} In \cite{zbMATH07615186} and the very recent preprint \cite{arXiv:2407.01150}, Biquard--Guenancia begin settling relevant (and technically challenging!) particular cases of this question. \begin{itemize} \item If $(X,D)$ is the toroidal compactification of a ball quotient, then the limit of the metric exists and equals the hyperbolic metric. \item If $X$ is Fano and $D$ is a divisor whose class is proportional to the anticanonical class, then the limit of the rescaled metric exists and equals the Tian--Yau metric. \end{itemize} More work is ongoing, and we expect to report on substantial progress by the time our workshop takes place. \subsubsection{Kähler--Einstein Metrics on Singular Spaces} Motivated by progress in the Minimal Model Program, there has been increasing interest in Kähler--Einstein metrics on singular spaces. While one of the first results in this direction dates back to the early 1970s when Kobayashi constructed orbifold Kähler--Einstein metrics, a definitive existence result for a relevant class of singularities was obtained by Eyssidieux--Guedj--Zeriahi about 15 years ago in \cite{zbMATH05859416}, by combining Yau's technique with Kolodziej's $\mathcal C^0$ estimates. Much more recently, Li--Tian--Wang extended Chen-Donaldson-Sun's solution of the Yau--Tian--Donaldson conjecture to general $\mathbb Q$-Fano varieties \cite{zbMATH07382001, zbMATH07597119}. For most applications, it is essential to control the geometry of these metrics near the singularities. Despite the problem's obvious importance, little is known so far. The continuity of the metric's potential has been established quite recently in the preprint \cite{arXiv:2401.03935} of Cho--Choi. Beyond that, the main progress in this direction is due to Hein--Sun \cite{zbMATH06827885}, who showed that near a large class of smoothable isolated singularities that are locally isomorphic to a Calabi-Yau cone, the singular Calabi-Yau metric must be asymptotic in a strong sense to the Calabi-Yau cone metric. Using the bounded geometry method, Datar--Fu--Song recently showed an analogous result in the case of isolated log canonical singularities \cite{zbMATH07669617}. Fu–Hein–Jiang obtained precise asymptotics shortly after, \cite{zbMATH07782497}. Essential contributions directly connected to these topics are due to Chiu--Székelyhidi \cite{zbMATH07810677}, Delcroix, Conlon--Hein \cite{zbMATH07858206}, C.~Li, Y.~Li, Tosatti and Zhang. \subsubsection{Complex Hyperbolicity} The 1979 Green-Griffiths-Lang conjecture asserts that every complex-projective variety $X$ of general type contains a proper subvariety $Y \subsetneq X$, such that every non-constant entire holomorphic curve $\mathbb C \to X$ takes its values in $Y$. Its beginnings date back to 1926, when André Bloch showed that the Zariski closure of entire holomorphic curve $\varphi: \mathbb C \to A$ to a complex torus $A$ is the translate of a sub-torus. Today, the conjecture still drives substantial research in complex geometry. Several authors, including Brotbek, Brunebarbe, Deng, Cadorel, and Javanpeykar, opened a new research direction with relation to arithmetic, by combining techniques from Hodge theory with Nevanlinna theory and jet differentials, \cite{arXiv:2007.12957, arXiv:2207.03283, arXiv:2305.09613}. Besides, we highlight two additional advances that will be relevant for our workshop. \paragraph{Hypersurfaces in Projective Space} A remarkable paper of Bérczi and Kirwan, \cite{MR4688701} published in September last year, establishes hyperbolicity and proves the Green-Griffiths-Lang for generic hypersurfaces of the projective space, $X \subsetneq \mathbb P^n$, provided that the degree of $X$ is larger than an explicit polynomial in $n$. These are significant improvements of earlier degree bounds, which involve non-polynomial bounds of order $(\sqrt{n} \log n)^n$ or worse. The proof builds on a strategy of Diverio-Merker-Rousseau and combines non-reductive geometric invariant theory with the ``Grassmannian techniques'' of Riedl-Yang. A very recent preprint of Cadorel simplifies the proof Bérczi-Kirwan substantially, but still needs to undergo a peer review, \cite{arXiv:2406.19003}. \paragraph{Representations of Fundamental Groups} Using recent advances in the theory of harmonic maps due to Daskalopoulos--Mese \cite{arXiv:2112.13961}, Deng--Yamanoi were able to confirm the Green--Griffiths conjecture for manifolds whose fundamental group admits a representation having certain natural properties, in direct analogy to the case of general-type curves. \subsection{Topology and Hodge Theory of Kähler spaces} Ever since its invention, Hodge theory has been one of the most powerful tools in studying the geometry and topology of Kähler spaces. More recent developments connect the theory to singularity theory and commutative algebra. The following topics in this area will particularly interest our workshop. \subsubsection{Singularities and Hodge Ideals} In a series of influential papers starting with \cite{MR4081135}, Mustaţă and Popa used Hodge modules to refine and generalize well-known invariants of singularities, most notably the multiplier ideals used in analysis and algebraic geometry. Schnell and Yang’s recent preprint \cite{arXiv:2309.16763} suggested an alternative approach toward similar ends. The first applications pertain to Bernstein--Sato polynomials and their zero sets; these are essential invariants of singularities originating from commutative algebra that are hard to compute. Schnell and Yang apply their results to conjectures of Debarre--Casalaina-Martin--Grushevsky concerning the Riemann--Schottky problem and the singularities of Theta divisors of principally polarized Abelian varieties. Park and Popa recently applied perverse sheaves and D-module theory to improve Goresky--MacPherson's classic Lefschetz theorems in the singular setting. A program put forward by Friedman--Laza aims at understanding the Hodge structures of degenerating Calabi--Yau varieties. \subsubsection{Lagrangian Fibrations} A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map $f : M \to B$ whose generic fibers are Langrangian. \paragraph{Compact Setting} If $M$ is compact, a well-known conjecture in the field predicts that $B$ is projective space. In the case where $B$ is smooth, Hwang established the conjecture more than 16 years ago in a celebrated paper. There is new insight today, as Bakker--Schnell recently found a purely Hodge theoretic proof of Hwang's result in \cite{arXiv:2311.08977}. Hopefully, these methods will give insight into the singular setting, which remains open to date. \paragraph{Non-compact Setting} Geometers study Lagrangian fibrations over non-compact bases in the framework of the ``$P=W$ conjecture,'' which Maulik--Shen and Hausel--Mellit--Minets--Schiffmann have recently proved \cite{arXiv:2209.02568, arXiv:2209.05429}. In the same setting, Shen–Yin discovered a remarkable symmetry of certain pushforward sheaves and conjectured that more general symmetries exist. Schnell has recently established these conjectures in \cite{arXiv:2303.05364} and also proved two conjectures of Maulik–Shen–Yin on the behavior of certain perverse sheaves near singular fibers. \subsubsection{Singer--Hopf Conjecture and Fundamental Groups of Kähler Manifolds} The Singer-Hopf conjecture asserts that a closed aspherical manifold of real dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot \chi(X)\geq 0$. This conjecture goes back to 1931 when Hopf formulated a related version for Riemannian manifolds. Recently, Arapura–Maxim–Wang suggested Hodge-theoretic refinements of this conjecture for Kähler manifolds in \cite{arXiv:2310.14131}. While the methods of \cite{arXiv:2310.14131} suffice to show particular cases, the statement remains open in full generality. In a related direction, Llosa-Isenrich--Py found an application of complex geometry and Hodge theory to geometric group theory, settling an old question of Brady on the finiteness properties of groups \cite{zbMATH07790946}. As a byproduct, the authors also obtain a proof of the classical Singer conjecture in an essential particular case in the realm of Kähler manifolds. Our goal in this workshop is to bring together several experts in geometric group theory with experts on Hodge theory and to explore further potential applications of the methods from one field to problems in the other. \section{Suggested and Excluded Dates} We would prefer if our workshop took place in mid of September or early to mid of April. \bibstyle{alpha} \bibliographystyle{alpha} \bibliography{general} \end{document}