\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{Description of the Workshop} % Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules % % %- Singularities and Hodge ideals etc: Mustata-Popa, Park, Ruijie Yang, Schnell % %- Lagrangian fibrations: Maulik-Shen, Shen-Yin, Schnell, Bakker % %- Singer-Hopf conjecture and fundamental groups of Kaehler manifolds: Arapura, Botong Wang, Maxim, Llosa-Isenrich—Py. \subsection{Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules} Over the last decade, Saito's theory of Hodge modules has seen spectacular applications in birational geometry. More recent developments, which are of significant importance, connect the theory to singularity theory, commutative algebra, and the topology of algebraic varieties. The following topics in this area will particularly interest our workshop. \subsubsection{Singularities and Hodge Ideals} 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. \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. Since then, these metrics have become an object of study in their own right. The work of Brendle, Donaldson, Guenancia, Rubinstein, and many others provides a complete package of results that generalize Yau's celebrated solution of the Calabi conjecture to the conic setting. 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\subset $ be a non-singular divisor. Assume that for every sufficiently small angle $0< \beta << 1$, 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} Starting with \cite{zbMATH07615186}, a series of articles by Biquard--Guenancia settles many 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, Delcroix, Hein, C.~Li, Y.~Li, Sun, Székelyhidi, Tosatti, and Zhang. \bigskip {\color{red}\textbf --- DO NOT READ ANYTHING BELOW THIS LINE ---} \subsubsection{Complex hyperbolicity} The Green-Griffiths conjecture (concerning the Zariski closure of holomorphic entire curves or more generally, of families of holomorphic disks on varieties of general type) continues to keep busy many complex geometers. Probably the most complete result in this field is due to A. Bloch (more than 100 years ago), who -in modern language- showed that the Zariski closure of a map $\varphi: \mathbb C \to A$ to a complex tori $A$ is the translate of a sub-tori. A decade ago, K.~Yamanoi established the Green-Griffiths conjecture for projective manifolds general type, which admit a generically finite map into an Abelian variety. This represents a very nice generalization of Bloch's theorem. In the last couple of years the field is taking a very interesting direction, by combining techniques from Hodge theory with the familiar Nevanlinna theory and jet differentials, cf. the articles by D. Brotbek, Y. Deng, Y. Brunebarbe, B. Cadorel and A. Javanpeykar. Using recent advances in the theory of harmonic maps (due to Daskalopoulos-Mese), Y. Deng and K. Yamanoi were able to confirm the Green-Griffiths conjecture for manifolds whose fundamental group admits a representation having certain natural properties (echoing the case of curves of genus at least two). Techniques from birational geometry, in connection with the work of F.~Campana are also present in the field via the -long awaited- work of E. Rousseau and its collaborators. \subsubsection{Complex hyperbolicity. Mark II} The 1979 Green-Griffiths-Lang conjecture asserts that every complex-projective 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 much of the research in complex geometry. We highlight several 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 peer review, \cite{arXiv:2406.19003}. \paragraph{Hyperbolicity and representations of fundamental groups} Using recent advances in the theory of harmonic maps (due to Daskalopoulos-Mese), Y. Deng and K. Yamanoi were able to confirm the Green-Griffiths conjecture for manifolds whose fundamental group admits a representation having certain natural properties (echoing the case of curves of genus at least two). \paragraph{Material collections} In the last couple of years the field is taking a very interesting direction, by combining techniques from Hodge theory with the familiar Nevanlinna theory and jet differentials, cf. the articles by D. Brotbek, Y. Deng, Y. Brunebarbe, B. Cadorel and A. Javanpeykar. \bibstyle{alpha} \bibliographystyle{alpha} \bibliography{general} \end{document}