Saving work

.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

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
+Geometers study Lagrangian fibrations over non-compact bases in the framework of
-framework of the ``$P=W$ conjecture,'' which Maulik–Shen and
+the ``$P=W$ conjecture,'' which Maulik--Shen and Hausel--Mellit--Minets--Schiffmann
-Hausel–Mellit–Minets–Schiffmann have recently proved \cite{arXiv:2209.02568,
+have recently proved \cite{arXiv:2209.02568, arXiv:2209.05429}.  In the same
-arXiv:2209.05429}.  In the same setting, Shen–Yin discovered a remarkable
+setting, Shen–Yin discovered a remarkable symmetry of certain pushforward
-symmetry of certain pushforward sheaves and conjectured that more general
+sheaves and conjectured that more general symmetries exist. Schnell has recently
-symmetries exist. Schnell has recently established these conjectures in
+established these conjectures in \cite{arXiv:2303.05364} and also proved two
-\cite{arXiv:2303.05364} and also proved two conjectures of Maulik–Shen–Yin on
+conjectures of Maulik–Shen–Yin on the behavior of certain perverse sheaves near
-the behavior of certain perverse sheaves near singular fibers.
+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
+In the proof of the Donaldson--Tian--Yau conjecture, which Chen--Donaldson--Sun
-metrics with conic singularities along a smooth divisor are playing a key role.
+gave in a series of papers around 2015, Kähler--Einstein metrics with conic
-Since then, they have become an object of study in its own right. For example,
+singularities along a smooth divisor emerged to play a vital role. Since then,
-we currently dispose of results which are completely analog to Yau’s celebrated
+these metrics have become an object of study in their own right.  The work of
-solution of Calabi conjecture in conic setting, by the work of S. Brendle, S.
+Brendle, Donaldson, Guenancia, Rubinstein, and many others provides a complete
-Donaldson, H. Guenancia, Y. Rubinstein,  among many others. 
+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
+\begin{q}
-gravitating around the following general question: \emph{let $X$ be a projective
+  Let $X$ be a projective manifold, and let $D\subset $ be a non-singular
-manifold, and let $D\subset $ be a non-singular divisor. We assume that for each
+  divisor.  Assume that for every sufficiently small angle $0< \beta << 1$,
-angle $0< \beta<< 1$ small enough, there exists a unique KE metric
+  there exists a unique Kähler--Einstein metric $\omega_\beta$ with conic
-$\omega_\beta$ with conic singularities of angle $2\pi\beta$ along $D$, i.e.
+  singularities of angle $2\pi\beta$ along $D$.  In other words, assume that
-$$Ricci_{\omega_\beta}= \lambda \omega_{\beta}+ (1-\beta)[D],$$
+  \[
-where $\lambda$ is equal to -1 or 1. Can one extract a limit of $(\omega_\beta)$
+    Ricci_{\omega_\beta}= \lambda \cdot \omega_{\beta}+ (1-\beta)\cdot [D],
-as $\beta\to 0$, eventually after rescaling}?
+    \quad
+    \text{where } \lambda \in \{ \pm 1\}.
+  \]
+  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
+Starting with \cite{zbMATH07615186}, a series of articles by Biquard--Guenancia
-interesting and technically challenging particular casses of this question:
+settles many relevant (and technically challenging!) particular cases of this
-toroidal compactifications of ball quotients -in which the limit mentioned above
+question.
-is the hyperbolic metric- and the case of a Fano manifold together with a
+\begin{itemize}
-divisor $D$ proportional to the anticanonical class -the limit of the rescaled
+  \item If $(X,D)$ is the toroidal compactification of a ball quotient, then the
-metric is the Tian-Yau metric. 
+  limit of the metric exists and equals the hyperbolic metric.
-\smallskip
 
-On the other hand, there has been increasing interest in the understanding of
+  \item If $X$ is Fano and $D$ is a divisor whose class is proportional to the
-Kähler-Einstein metrics on singular spaces. Perhaps one of the first result in
+  anticanonical class, then the limit of the rescaled metric exists and equals
-this direction is due to S. Kobayashi (construction of orbifold Kähler-Einstein
+  the Tian--Yau metric. 
-metrics), while a definitive existence result for a large class of singularities
+\end{itemize}
-was obtained by Eyssidieux-Guedj-Zeriahi by combining Yau's technique with S.
+More work is ongoing, and we expect to report on substantial progress by the
-Kolodziej's $\mathcal C^0$ estimates. Recently Li-Tian-Wang extended
+time our workshop takes place.
-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.
 
-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
+\subsubsection{Kähler--Einstein Metrics on Singular Spaces}
-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
+Motivated by progress in the Minimal Model Program, there has been increasing
-direction is due to Hein-Sun, who showed that near a large class of smoothable
+interest in Kähler--Einstein metrics on singular spaces.  While one of the first
-isolated singularities that are locally isomorphic to a Calabi-Yau cone, the
+results in this direction dates back to the early 1970s when Kobayashi
-singular Calabi-Yau metric must be asymptotic in a strong sense to the
+constructed orbifold Kähler--Einstein metrics, a definitive existence result for
-Calabi-Yau cone metric. Recently an analogous result was shown by Datar-Fu-Song
+a relevant class of singularities was obtained by Eyssidieux--Guedj--Zeriahi
-in the case of isolated log canonical singularities using the bounded geometry
+about 15 years ago in \cite{zbMATH05859416}, by combining Yau's technique with
-method, and precise asymptotics were obtained shortly after by Fu-Hein-Jiang.
+Kolodziej's $\mathcal C^0$ estimates.  Much more recently, Li--Tian--Wang
-Important contributions in direct connection with these topics are due to S.-K.
+extended Chen-Donaldson-Sun's solution of the Yau--Tian--Donaldson conjecture to
-Chiu,T. Delcroix, H.-J. Hein, C. Li, Y. Li, S. Sun, G. Székelyhidi, V. Tosatti
+general $\mathbb Q$-Fano varieties \cite{zbMATH07382001, zbMATH07597119}.
-and K. Zhang.
+
+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}

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},