diff --git a/.vscode/ltex.dictionary.en-US.txt b/.vscode/ltex.dictionary.en-US.txt index c61cc82..a1768c1 100644 --- a/.vscode/ltex.dictionary.en-US.txt +++ b/.vscode/ltex.dictionary.en-US.txt @@ -58,3 +58,24 @@ Goresky Schottky Calabi Yau +Tian +Brendle +Guenancia +Biquard +Fano +anticanonical +orbifold +Eyssidieux +Guedj +Zeriahi +Kolodziej +Cho +Choi +Hein +smoothable +Datar +Fu +Delcroix +Székelyhidi +Tosatti +Chiu diff --git a/MFO26.tex b/MFO26.tex index ad6a83e..d3b82c3 100644 --- a/MFO26.tex +++ b/MFO26.tex @@ -3,6 +3,7 @@ % % Local font definitions -- need to come first % +\usepackage{amsthm} \usepackage{libertine} \usepackage[libertine]{newtxmath} @@ -20,6 +21,8 @@ \sloppy +\newtheorem*{q}{Question} + % Colours for hyperlinks \definecolor{lightgray}{RGB}{220,220,220} \definecolor{gray}{RGB}{180,180,180} @@ -116,14 +119,14 @@ insight into the singular setting, which remains open to date. \paragraph{Non-compact Setting} -In the non-compact setting, geometers study Lagrangian fibrations 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. +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} @@ -147,57 +150,78 @@ 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 Kobayashi hyperbolicity} +\subsection{Canonical Metrics and Hyperbolicity} -\subsubsection{Kähler-Einstein metrics with conic singularities and their limits} +\subsubsection{Kähler--Einstein Metrics with Conic Singularities and Their Limits} -In the proof of Donaldson-Tian-Yau conjecture -around 2015-, the Kähler-Einstein -metrics with conic singularities along a smooth divisor are playing a key role. -Since then, they have become an object of study in its own right. For example, -we currently dispose of results which are completely analog to Yau’s celebrated -solution of Calabi conjecture in conic setting, by the work of S. Brendle, S. -Donaldson, H. Guenancia, Y. Rubinstein, among many others. +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. -An important number of the exciting recent developments in this field are -gravitating around the following general question: \emph{let $X$ be a projective -manifold, and let $D\subset $ be a non-singular divisor. We assume that for each -angle $0< \beta<< 1$ small enough, there exists a unique KE metric -$\omega_\beta$ with conic singularities of angle $2\pi\beta$ along $D$, i.e. -$$Ricci_{\omega_\beta}= \lambda \omega_{\beta}+ (1-\beta)[D],$$ -where $\lambda$ is equal to -1 or 1. Can one extract a limit of $(\omega_\beta)$ -as $\beta\to 0$, eventually after rescaling}? +\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} -The series of articles by Biquard-Guenancia —2022 and 2024-- settle many -interesting and technically challenging particular casses of this question: -toroidal compactifications of ball quotients -in which the limit mentioned above -is the hyperbolic metric- and the case of a Fano manifold together with a -divisor $D$ proportional to the anticanonical class -the limit of the rescaled -metric is the Tian-Yau metric. -\smallskip +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. -On the other hand, there has been increasing interest in the understanding of -Kähler-Einstein metrics on singular spaces. Perhaps one of the first result in -this direction is due to S. Kobayashi (construction of orbifold Kähler-Einstein -metrics), while a definitive existence result for a large class of singularities -was obtained by Eyssidieux-Guedj-Zeriahi by combining Yau's technique with S. -Kolodziej's $\mathcal C^0$ estimates. Recently Li-Tian-Wang extended -Chen-Donaldson-Sun’s solution of the Yau-Tian-Donaldson conjecture to general -$\mathbb Q$-Fano varieties. Thus, we now have several sources/motivations for -studying singular Kähler-Einstein metrics on normal varieties. + \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. -For applications it is desirable to have control of the geometry of these -metrics near the singularities, but so far little is known in general. The -continuity of their potential has only been established very recently (beginning -of 2024) by Y.-W- Luke and Y.-J. Choi. Beyond that, the main progress in this -direction is due to Hein-Sun, 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. Recently an analogous result was shown by Datar-Fu-Song -in the case of isolated log canonical singularities using the bounded geometry -method, and precise asymptotics were obtained shortly after by Fu-Hein-Jiang. -Important contributions in direct connection with these topics are due to S.-K. -Chiu,T. Delcroix, H.-J. Hein, C. Li, Y. Li, S. Sun, G. Székelyhidi, V. Tosatti -and K. Zhang. + +\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} diff --git a/general.bib b/general.bib index 7a7e591..8b2221a 100644 --- a/general.bib +++ b/general.bib @@ -1,3 +1,132 @@ +@Article{zbMATH07782497, + Author = {Fu, Xin and Hein, Hans-Joachim and Jiang, Xumin}, + Title = {Asymptotics of {K{\"a}hler}-{Einstein} metrics on complex hyperbolic cusps}, + FJournal = {Calculus of Variations and Partial Differential Equations}, + Journal = {Calc. Var. Partial Differ. Equ.}, + ISSN = {0944-2669}, + Volume = {63}, + Number = {1}, + Pages = {34}, + Note = {Id/No 6}, + Year = {2024}, + Language = {English}, + DOI = {10.1007/s00526-023-02613-4}, + Keywords = {32Q20,35J96,53C55}, + zbMATH = {7782497}, + Zbl = {1535.32023} +} + +@Article{zbMATH07669617, + Author = {Datar, Ved and Fu, Xin and Song, Jian}, + Title = {K{\"a}hler-{Einstein} metrics near an isolated log-canonical singularity}, + FJournal = {Journal f{\"u}r die Reine und Angewandte Mathematik}, + Journal = {J. Reine Angew. Math.}, + ISSN = {0075-4102}, + Volume = {797}, + Pages = {79--116}, + Year = {2023}, + Language = {English}, + DOI = {10.1515/crelle-2022-0095}, + Keywords = {83C75,53C21,32Q20,03C80,35B35,83C30}, + zbMATH = {7669617}, + Zbl = {1521.83164} +} + +@Article{zbMATH06827885, + Author = {Hein, Hans-Joachim and Sun, Song}, + Title = {Calabi-{Yau} manifolds with isolated conical singularities}, + FJournal = {Publications Math{\'e}matiques}, + Journal = {Publ. Math., Inst. Hautes {\'E}tud. Sci.}, + ISSN = {0073-8301}, + Volume = {126}, + Pages = {73--130}, + Year = {2017}, + Language = {English}, + DOI = {10.1007/s10240-017-0092-1}, + Keywords = {32Q25,32Q20,14J32}, + URL = {www.numdam.org/articles/10.1007/s10240-017-0092-1/}, + zbMATH = {6827885}, + Zbl = {1397.32009} +} + +@misc{arXiv:2401.03935, + title={Continuity of solutions to complex Monge-Amp\`{e}re equations on compact K\"{a}hler spaces}, + author={Ye-Won Luke Cho and Young-Jun Choi}, + year={2024}, + eprint={2401.03935}, + archivePrefix={arXiv}, + primaryClass={math.DG}, + url={https://arxiv.org/abs/2401.03935}, +} + +@Article{zbMATH07597119, + Author = {Li, Chi and Tian, Gang and Wang, Feng}, + Title = {The uniform version of {Yau}-{Tian}-{Donaldson} conjecture for singular {Fano} varieties}, + FJournal = {Peking Mathematical Journal}, + Journal = {Peking Math. J.}, + ISSN = {2096-6075}, + Volume = {5}, + Number = {2}, + Pages = {383--426}, + Year = {2022}, + Language = {English}, + DOI = {10.1007/s42543-021-00039-5}, + Keywords = {32Q20,32Q26,14J45}, + zbMATH = {7597119}, + Zbl = {1504.32068} +} + +@Article{zbMATH07382001, + Author = {Li, Chi and Tian, Gang and Wang, Feng}, + Title = {On the {Yau}-{Tian}-{Donaldson} conjecture for singular {Fano} varieties}, + FJournal = {Communications on Pure and Applied Mathematics}, + Journal = {Commun. Pure Appl. Math.}, + ISSN = {0010-3640}, + Volume = {74}, + Number = {8}, + Pages = {1748--1800}, + Year = {2021}, + Language = {English}, + DOI = {10.1002/cpa.21936}, + Keywords = {32Q20,14J45,53C55}, + zbMATH = {7382001}, + Zbl = {1484.32041} +} + +@Article{zbMATH05859416, + Author = {Eyssidieux, Philippe and Guedj, Vincent and Zeriahi, Ahmed}, + Title = {Singular {K{\"a}hler}-{Einstein} metrics}, + FJournal = {Journal of the American Mathematical Society}, + Journal = {J. Am. Math. Soc.}, + ISSN = {0894-0347}, + Volume = {22}, + Number = {3}, + Pages = {607--639}, + Year = {2009}, + Language = {English}, + DOI = {10.1090/S0894-0347-09-00629-8}, + Keywords = {32W20,32Q20,32J27,14J17}, + zbMATH = {5859416}, + Zbl = {1215.32017} +} + +@Article{zbMATH07615186, + Author = {Biquard, Olivier and Guenancia, Henri}, + Title = {Degenerating {K{\"a}hler}-{Einstein} cones, locally symmetric cusps, and the {Tian}-{Yau} metric}, + FJournal = {Inventiones Mathematicae}, + Journal = {Invent. Math.}, + ISSN = {0020-9910}, + Volume = {230}, + Number = {3}, + Pages = {1101--1163}, + Year = {2022}, + Language = {English}, + DOI = {10.1007/s00222-022-01138-5}, + Keywords = {32Q20,53C55,35J99}, + zbMATH = {7615186}, + Zbl = {1510.32057} +} + @Article{zbMATH07790946, Author = {Llosa Isenrich, Claudio and Py, Pierre}, Title = {Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices},