229 lines
10 KiB
TeX
229 lines
10 KiB
TeX
\documentclass[a4paper, british]{scrartcl}
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% Limit table of contents to section titles
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\setcounter{tocdepth}{1}
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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={Philippe Eyssidieux, Jun-Muk Hwang, Stefan Kebekus, Mihai Paun},
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pdftitle={\@title},
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\makeatother
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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{Description of the Workshop}
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% Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules
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%- Singularities and Hodge ideals etc: Mustata-Popa, Park, Ruijie Yang, Schnell
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%- Lagrangian fibrations: Maulik-Shen, Shen-Yin, Schnell, Bakker
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%- Singer-Hopf conjecture and fundamental groups of Kaehler manifolds: Arapura, Botong Wang, Maxim, Llosa-Isenrich—Py.
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\subsection{Topology of Kähler spaces: D-modules, perverse sheaves, Hodge modules}
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Over the last decade, Saito’s theory of Hodge modules has seen spectacular
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applications in birational geometry. Over the last few years the theory has been
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further developed and branched out to yield exciting applications to the
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topology of algebraic varieties, singularity theory and commutative algebra.
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The following topics in this area will be of particular interest to our workshop.
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\subsubsection{Singularities and Hodge ideals}
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Hodge modules are used to define generalizations of well-known ideals of
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singularities, such as multiplier ideals from analysis and algebraic geometry.
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This theory has been put forward by Mustata and Popa, an alternative approach
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was suggested by Schnell and Yang. These generalizations allow to study for
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instance Bernstein-Sato polynomials, which are important commutative algebra
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invariants of singularities that are typically hard to compute. Geometric
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applications are given by the study of singularities of Theta divisors of
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principally polarized abelian varieties, as pursued by Schnell and Yang.
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In most recent developments by Park and Popa, related methods have been used to
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improve classical Lefschetz theorems for singular varieties due to Goresky-Mac
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Pherson. Originally, Lefschetz theorems for singular varieties have been proven
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via stratified Morse theory, while the recent improvements rely on perverse
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sheaves and D-module theory.
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A related program put forward by Friedman and Laza aims at understanding the
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Hodge structures of degenerating Calabi-Yau varieties. This led to the notions
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of higher Du Bois and higher rational singularities which can be understood via
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Hodge modules and will.
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\subsubsection{Lagrangian fibrations}
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A Lagrangian fibration of a (not necessarily compact) hyperkaehler manifold $M$
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is a proper holomorphic map $f:M\to B$ whose fibres are Langrangian
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submanifolds. If $M$ is compact, then a well-known conjecture in the field
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predicts that $B$ is projective space. This is known if $B$ is smooth by
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celebrated work of Hwang. A Hodge theoretic proof of Hwang‘s result has recently
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been found by Bakker and Schnell; the case where $B$ is allowed to be singular
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remains open.
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In the non-compact setting, Lagrangian fibrations have been studied in the
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framework of the so called P=W conjecture, which has recently been proven by
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Maulik and Shen for the Hitchin fibration associated to the general linear group
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and curves of arbitrary genus $g\geq 2$. Shen and Yin discovered a remarkable
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symmetry of certain pushforward sheaves in the case of Lagrangian fibrations
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over possibly non-compact bases. Recently, Schnell used Saito‘s theory of Hodge
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modules to prove the conjecture of Shen and Yin in full generality.
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\subsubsection{Singer-Hopf conjecture and fundamental groups of K\"ahler manifolds}
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The Singer-Hopf conjecture says that a closed aspherical manifold of real
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dimension $2n$ has positive signed Euler characteristic $(-1)^n\chi(X)\geq 0$.
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This conjecture goes back to 1931, when Hopf formulated a related version for
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Riemannian manifolds. Recently, Hodge-theoretic refinements of this conjecture
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for K\"ahler manifolds have been put forward by Arapura, Maxim and Wang. Special
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cases of these conjecture have recently been proven, but the statement remains
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open in full generality.
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In a related direction, Llosa-Isenrich and Py found recently an application of
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complex geometry and Hodge theory to geometric group theory, thereby settling an
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old question of of Brady on finiteness properties of groups. As a byproduct,
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the authors also obtain a proof of the classical Singer conjecture in an
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important special case in the realm of K\"ahler 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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\subsection{Canonical metrics and Kobayashi hyperbolicity}
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\subsubsection{Kähler-Einstein metrics with conic singularities and their limits}
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In the proof of Donaldson-Tian-Yau conjecture -around 2015-, the Kähler-Einstein
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metrics with conic singularities along a smooth divisor are playing a key role.
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Since then, they have become an object of study in its own right. For example,
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we currently dispose of results which are completely analog to Yau’s celebrated
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solution of Calabi conjecture in conic setting, by the work of S. Brendle, S.
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Donaldson, H. Guenancia, Y. Rubinstein, among many others.
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An important number of the exciting recent developments in this field are
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gravitating around the following general question: \emph{let $X$ be a projective
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manifold, and let $D\subset $ be a non-singular divisor. We assume that for each
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angle $0< \beta<< 1$ small enough, there exists a unique KE metric
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$\omega_\beta$ with conic singularities of angle $2\pi\beta$ along $D$, i.e.
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$$Ricci_{\omega_\beta}= \lambda \omega_{\beta}+ (1-\beta)[D],$$
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where $\lambda$ is equal to -1 or 1. Can one extract a limit of $(\omega_\beta)$
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as $\beta\to 0$, eventually after rescaling}?
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The series of articles by Biquard-Guenancia —2022 and 2024-- settle many
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interesting and technically challenging particular casses of this question:
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toroidal compactifications of ball quotients -in which the limit mentioned above
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is the hyperbolic metric- and the case of a Fano manifold together with a
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divisor $D$ proportional to the anticanonical class -the limit of the rescaled
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metric is the Tian-Yau metric.
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\smallskip
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On the other hand, there has been increasing interest in the understanding of
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Kähler-Einstein metrics on singular spaces. Perhaps one of the first result in
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this direction is due to S. Kobayashi (construction of orbifold Kähler-Einstein
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metrics), while a definitive existence result for a large class of singularities
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was obtained by Eyssidieux-Guedj-Zeriahi by combining Yau's technique with S.
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Kolodziej's $\mathcal C^0$ estimates. Recently Li-Tian-Wang extended
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Chen-Donaldson-Sun’s solution of the Yau-Tian-Donaldson conjecture to general
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$\mathbb Q$-Fano varieties. Thus, we now have several sources/motivations for
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studying singular Kähler-Einstein metrics on normal varieties.
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For applications it is desirable to have control of the geometry of these
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metrics near the singularities, but so far little is known in general. The
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continuity of their potential has only been established very recently (beginning
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of 2024) by Y.-W- Luke and Y.-J. Choi. Beyond that, the main progress in this
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direction is due to Hein-Sun, who showed that near a large class of smoothable
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isolated singularities that are locally isomorphic to a Calabi-Yau cone, the
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singular Calabi-Yau metric must be asymptotic in a strong sense to the
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Calabi-Yau cone metric. Recently an analogous result was shown by Datar-Fu-Song
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in the case of isolated log canonical singularities using the bounded geometry
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method, and precise asymptotics were obtained shortly after by Fu-Hein-Jiang.
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Important contributions in direct connection with these topics are due to S.-K.
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Chiu,T. Delcroix, H.-J. Hein, C. Li, Y. Li, S. Sun, G. Székelyhidi, V. Tosatti
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and K. 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
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$\varphi:\mathbb C\to A$ to a complex tori $A$ is the translate of a sub-tori. A
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decade 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 generalisation 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 Nevalinna 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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\end{document}
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