Saving work
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Stefan Kebekus
parent
28adc27616
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e91c315a85
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@ -58,3 +58,24 @@ Goresky
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Schottky
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Calabi
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Yau
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Tian
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Brendle
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Guenancia
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Biquard
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Fano
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anticanonical
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orbifold
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Eyssidieux
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Guedj
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Zeriahi
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Kolodziej
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Cho
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Choi
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Hein
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smoothable
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Datar
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Fu
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Delcroix
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Székelyhidi
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Tosatti
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Chiu
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130
MFO26.tex
130
MFO26.tex
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@ -3,6 +3,7 @@
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%
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% Local font definitions -- need to come first
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%
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\usepackage{amsthm}
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\usepackage{libertine}
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\usepackage[libertine]{newtxmath}
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@ -20,6 +21,8 @@
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\sloppy
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\newtheorem*{q}{Question}
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% Colours for hyperlinks
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\definecolor{lightgray}{RGB}{220,220,220}
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\definecolor{gray}{RGB}{180,180,180}
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@ -116,14 +119,14 @@ 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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@ -147,57 +150,78 @@ 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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\subsection{Canonical Metrics and Hyperbolicity}
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\subsubsection{Kähler-Einstein metrics with conic singularities and their limits}
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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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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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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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this field gravitate around the following general question.
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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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\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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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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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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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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\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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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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\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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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{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, 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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\subsubsection{Complex hyperbolicity}
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129
general.bib
129
general.bib
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@ -1,3 +1,132 @@
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@Article{zbMATH07782497,
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Author = {Fu, Xin and Hein, Hans-Joachim and Jiang, Xumin},
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Title = {Asymptotics of {K{\"a}hler}-{Einstein} metrics on complex hyperbolic cusps},
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FJournal = {Calculus of Variations and Partial Differential Equations},
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Journal = {Calc. Var. Partial Differ. Equ.},
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ISSN = {0944-2669},
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Volume = {63},
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Number = {1},
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Pages = {34},
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Note = {Id/No 6},
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Year = {2024},
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Language = {English},
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DOI = {10.1007/s00526-023-02613-4},
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Keywords = {32Q20,35J96,53C55},
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zbMATH = {7782497},
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Zbl = {1535.32023}
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}
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@Article{zbMATH07669617,
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Author = {Datar, Ved and Fu, Xin and Song, Jian},
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Title = {K{\"a}hler-{Einstein} metrics near an isolated log-canonical singularity},
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FJournal = {Journal f{\"u}r die Reine und Angewandte Mathematik},
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Journal = {J. Reine Angew. Math.},
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ISSN = {0075-4102},
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Volume = {797},
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Year = {2023},
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Language = {English},
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DOI = {10.1515/crelle-2022-0095},
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Keywords = {83C75,53C21,32Q20,03C80,35B35,83C30},
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zbMATH = {7669617},
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Zbl = {1521.83164}
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}
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@Article{zbMATH06827885,
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Author = {Hein, Hans-Joachim and Sun, Song},
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Title = {Calabi-{Yau} manifolds with isolated conical singularities},
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FJournal = {Publications Math{\'e}matiques},
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Journal = {Publ. Math., Inst. Hautes {\'E}tud. Sci.},
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Year = {2017},
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DOI = {10.1007/s10240-017-0092-1},
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Keywords = {32Q25,32Q20,14J32},
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URL = {www.numdam.org/articles/10.1007/s10240-017-0092-1/},
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zbMATH = {6827885},
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Zbl = {1397.32009}
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}
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@misc{arXiv:2401.03935,
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title={Continuity of solutions to complex Monge-Amp\`{e}re equations on compact K\"{a}hler spaces},
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author={Ye-Won Luke Cho and Young-Jun Choi},
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year={2024},
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eprint={2401.03935},
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archivePrefix={arXiv},
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primaryClass={math.DG},
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url={https://arxiv.org/abs/2401.03935},
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}
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@Article{zbMATH07597119,
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Author = {Li, Chi and Tian, Gang and Wang, Feng},
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Title = {The uniform version of {Yau}-{Tian}-{Donaldson} conjecture for singular {Fano} varieties},
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FJournal = {Peking Mathematical Journal},
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Journal = {Peking Math. J.},
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ISSN = {2096-6075},
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Volume = {5},
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Number = {2},
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Year = {2022},
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Language = {English},
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DOI = {10.1007/s42543-021-00039-5},
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Keywords = {32Q20,32Q26,14J45},
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zbMATH = {7597119},
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Zbl = {1504.32068}
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}
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@Article{zbMATH07382001,
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Author = {Li, Chi and Tian, Gang and Wang, Feng},
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Title = {On the {Yau}-{Tian}-{Donaldson} conjecture for singular {Fano} varieties},
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FJournal = {Communications on Pure and Applied Mathematics},
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DOI = {10.1002/cpa.21936},
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Keywords = {32Q20,14J45,53C55},
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zbMATH = {7382001},
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Zbl = {1484.32041}
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}
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@Article{zbMATH05859416,
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Author = {Eyssidieux, Philippe and Guedj, Vincent and Zeriahi, Ahmed},
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Title = {Singular {K{\"a}hler}-{Einstein} metrics},
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FJournal = {Journal of the American Mathematical Society},
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Journal = {J. Am. Math. Soc.},
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ISSN = {0894-0347},
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Volume = {22},
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Pages = {607--639},
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Year = {2009},
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Language = {English},
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DOI = {10.1090/S0894-0347-09-00629-8},
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Keywords = {32W20,32Q20,32J27,14J17},
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zbMATH = {5859416},
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Zbl = {1215.32017}
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}
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@Article{zbMATH07615186,
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Author = {Biquard, Olivier and Guenancia, Henri},
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Title = {Degenerating {K{\"a}hler}-{Einstein} cones, locally symmetric cusps, and the {Tian}-{Yau} metric},
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FJournal = {Inventiones Mathematicae},
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Journal = {Invent. Math.},
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ISSN = {0020-9910},
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Volume = {230},
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Number = {3},
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Pages = {1101--1163},
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Year = {2022},
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Language = {English},
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DOI = {10.1007/s00222-022-01138-5},
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Keywords = {32Q20,53C55,35J99},
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zbMATH = {7615186},
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Zbl = {1510.32057}
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}
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@Article{zbMATH07790946,
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Author = {Llosa Isenrich, Claudio and Py, Pierre},
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Title = {Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices},
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