diff --git a/.vscode/ltex.dictionary.en-US.txt b/.vscode/ltex.dictionary.en-US.txt index a1768c1..7c9b2e4 100644 --- a/.vscode/ltex.dictionary.en-US.txt +++ b/.vscode/ltex.dictionary.en-US.txt @@ -79,3 +79,9 @@ Delcroix Székelyhidi Tosatti Chiu +Oberwolfach +Conlon +Daskalopoulos +Mese +Nevanlinna +arithmetics diff --git a/MFO26.tex b/MFO26.tex index 86ffd4f..ac7d1f0 100644 --- a/MFO26.tex +++ b/MFO26.tex @@ -54,9 +54,6 @@ \maketitle - - - \section{Workshop Title} Komplexe Analysis --- Differential and Algebraic methods in Kähler spaces @@ -122,15 +119,18 @@ Secondary & 14 &--& Algebraic geometry \\ \section{Description of the Workshop} -The proposed workshop presents recent results in Complex Geometry and surveys relations to other fields. -For 2026, we would like to emphasize the fields described below. +The proposed workshop presents recent results in Complex Geometry and surveys +relations to other fields. For 2026, we would like to emphasize the fields +described below. Each relates to complex analysis differently. Each has seen substantial progress recently, producing results that will be of importance for years to come. %The bullet items list some of the latest developments that have attracted our attention. %We plan to include at least one broader overview talk for each of the three subjects, as well as more specialized presentations by senior experts and junior researchers. -We will account for new developments that arise between the time of submission of this proposal and the time of the workshop. -Following good Oberwolfach tradition, we will keep the number of talks small to provide ample opportunity for informal discussions. +We will account for new developments that arise between the time of submission +of this proposal and the time of the workshop. Following good Oberwolfach +tradition, we will keep the number of talks small to provide ample opportunity +for informal discussions. %After so many months of the pandemic, this will be more than welcome! @@ -140,16 +140,16 @@ Following good Oberwolfach tradition, we will keep the number of talks small to 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 +singularities along a smooth divisor emerged to play a vital role. 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 +conjecture to the conic setting. Since then, these metrics have become an +object of study in their own right. 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 $0< \beta << 1$, + Let $X$ be a projective manifold, and let $D \subsetneq X$ be a non-singular + divisor. Assume that for every sufficiently small angle $0 < \beta \ll 1$, there exists a unique Kähler--Einstein metric $\omega_\beta$ with conic singularities of angle $2\pi\beta$ along $D$. In other words, assume that \[ @@ -161,9 +161,9 @@ this field gravitate around the following general question. 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. +In \cite{zbMATH07615186} and the very recent preprint \cite{arXiv:2407.01150}, +Biquard--Guenancia begin settling 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. @@ -200,43 +200,11 @@ 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 ---} +these topics are due to Chiu--Székelyhidi \cite{zbMATH07810677}, Delcroix, +Conlon--Hein \cite{zbMATH07858206}, C.~Li, Y.~Li, Tosatti and Zhang. -\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} +\subsubsection{Complex Hyperbolicity} The 1979 Green-Griffiths-Lang conjecture asserts that every complex-projective variety $X$ of general type contains a proper subvariety $Y \subsetneq X$, such @@ -244,10 +212,15 @@ 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. +drives substantial research in complex geometry. Several authors, including +Brotbek, Brunebarbe, Deng, Cadorel, and Javanpeykar, opened a new research +direction with relation to arithmetic, by combining techniques from Hodge theory +with Nevanlinna theory and jet differentials, \cite{arXiv:2007.12957, +arXiv:2207.03283, arXiv:2305.09613}. Besides, we highlight two additional +advances that will be relevant for our workshop. -\paragraph{Hypersurfaces in projective space} + +\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 @@ -261,43 +234,28 @@ recent preprint of Cadorel simplifies the proof Bérczi-Kirwan substantially, bu still needs to undergo a peer review, \cite{arXiv:2406.19003}. -\paragraph{Hyperbolicity and representations of fundamental groups} +\paragraph{Representations of Fundamental Groups} -Using recent advances in the theory of harmonic maps (due to -Daskalopoulos-Mese, cf. \cite{arXiv:2112.13961}), B. Cadorel, Y. Deng 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), cf. \cite{arXiv:2212.12225}. +Using recent advances in the theory of harmonic maps due to Daskalopoulos--Mese +\cite{arXiv:2112.13961}, Deng--Yamanoi were able to confirm the Green--Griffiths +conjecture for manifolds whose fundamental group admits a representation having +certain natural properties, in direct analogy to the case of general-type +curves. -\paragraph{Material collections} +\subsection{Topology and Hodge Theory of Kähler spaces} - - -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, cf. \cite{arXiv:2007.12957}, \cite{arXiv:2305.09613}, \cite{arXiv:2207.03283}. - - - - - - - -\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. +Ever since its invention, Hodge theory has been one of the most powerful tools +in studying the geometry and topology of Kähler spaces. More recent +developments 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{MR4044463}, % \cite{MR4081135} is not the first one -Mustaţă and Popa used Hodge modules to refine and generalize well-known invariants of +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 @@ -314,7 +272,7 @@ program put forward by Friedman--Laza aims at understanding the Hodge structures of degenerating Calabi--Yau varieties. -\subsubsection{Lagrangian fibrations} +\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. @@ -322,31 +280,27 @@ $f : M \to B$ whose generic fibers are Langrangian. \paragraph{Compact Setting} -If $M$ is compact, a well-known conjecture in the field predicts that $B$ should be the -projective space. A strong evidence for this problem is due to Hwang: he established the -conjecture more than 16 years ago in a celebrated paper, provided that the base $B$ is smooth. -There is new insight -today, with two alternative arguments for the proof of this theorem. -Bakker--Schnell recently found a purely Hodge theoretic proof of -Hwang's result in \cite{arXiv:2311.08977}. On the other hand Tosatti--Li, cf. \cite{arXiv:2308.10553} -found a more differential-geometric argument, which relied heavily on a singular version of Mok's uniformisation theorem. -Even if both methods are using results about rational curves -which confines them from the start to -the smooth case-, we hope that they put Hwang's result in a new perspective, hopefully helpful to progress towards the general case. +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} -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} +\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 @@ -368,105 +322,6 @@ applications of the methods from one field to problems in the other. -\section{Suggested dates} - -We would prefer if our workshop took place in mid of September or early to mid April. -%If this date is not available, early to mid-April would be an alternative. -%The workshop ``Komplexe Analysis'' traditionally takes place in the first week -%of September. We would like to follow this tradition. If the traditional date is -%not available, early to mid-April would be an alternative. - - -\section{Preliminary list of proposed participants} - -Below is a preliminary list of people we would like to invite\footnote{We list - colleagues as ``young'' if they have no tenured job, or if their tenure is - less than about three years old.}. - -{\small -\begin{longtable}[c]{lccccccc} - - \rowcolor{lightgray} \multicolumn{2}{l}{\textbf{Name}} & \textbf{Location} & \multicolumn{3}{c}{\textbf{Subject}} & \textbf{Young} & \textbf{Woman} \\ - \rowcolor{lightgray} &&& §5.1 & §5.2 & & & \\\hline \endhead - -%Araujo&Carolina&Rio de Janeiro&&&&&1 \\ -Bakker&Benjamin&Chicago&&&&& \\ -Berndtsson&Bo&Göteborg&&&&& \\ -%Bertini&Valeria&Chemnitz&&1&&1&1 \\ -%Blum&Harold&Stony Brook&1&1&&1 \\ -Boucksom&Sebastien&Paris&&&&& \\ -Braun&Lukas&Innsbruck&&&&1& \\ -Brotbek&Damian&Nancy&&&&& \\ -Brunebarbe&Yohan&CNRS/Bordeaux&&&&& \\ -Cadorel&Benoit&Nancy&&&&1& \\ -Campana&Frédéric&Nancy&&&&& \\ -Cao&Junyan&Nice&&&&& \\ -Castravet&Ana-Maria&Versailles&&&&&1\\ -%Chen&Jiaming&Nancy&1&1&&1& \\ -Claudon&Benoit&Rennes&&&&& \\ -%Commelin&Johan&Freiburg&&&1&1& \\ -%Darvas&Tamás&Maryland&1&&&& \\ -Das&Omprokash &TIFR Mumbai&&&&1&\\ -Delcroix&Thibaut&Montpellier&&&&& \\ -Deng&Ya&CNRS/Nancy&&&&& \\ -%di Nezza&Eleonora&CNRS/Palaiseau&1&&&1&1 \\ -Dutta&Yagna&Leiden&&&&&1\\ -Eyssidieux&Philippe &Grenoble&&&&&\\ -Gachet&C\'ecile & Berlin&&&&&1\\ -Graf&Patrick&Bayreuth&&&&1&\\ -Greb&Daniel&Essen&&&&& \\ -%Grossi&Annalisa&Chemnitz&&1&&1&1 \\ -Guenancia&Henri&CNRS/Toulouse&&&&& \\ -Hao&Feng& Shandong University&&&&&\\ -Hein&Hans-Joachim&Münster&&&&& \\ -%Huang&Xiaojun&Rutgers&1&&&& \\ -Hulek&Klaus&Hannover&&&&& \\ -Hoskins&Victoria&Essen&&&&&1\\ -Höring&Andreas&Nice&&&&& \\ -Hwang&Jun-Muk &Daejeon&&&&&\\ -Javanpeykar&Ariyan &Nijmegen&&&&&\\ -Kirwan&Frances&Oxford&&&&&1\\ -Klingler&Bruno&Berlin&&&&& \\ -%Koike&Takayuki&Osaka&1&1&&1& \\ -Lehn&Christian&Chemnitz&&&&& \\ -%Li&Chi&Rutgers&1&&&& \\ -Lin&Hsueh-Yung&Taiwan&&&&&\\ -Llosa-Isenrich& Claudio& Karlsruhe&&&&1&\\ -%Lu&Hoang-Chinh&Orsay&1&&&1& \\ -%Martinelli&Diletta&Amsterdam&&1&&1&1 \\ -%Matsumura&Shin-Ichi&Tohoku&&1&&1& \\ -Mauri&Mirko&Paris&&&&1& \\ -Moraga&Joaquín&UCLA&&&&1& \\ -M\"uller&Niklas&Essen&&&&1&\\ -%Olano&Sebastián&Northwestern&&1&&1& \\ -Ortega&Angela&Berlin&&&&&1\\ -Ou&Wenhao&AMSS, China&&&&1&\\ -Park&Sung Gi &Harvard&&&&&\\ -Paulsen&Matthias&Marburg& &&& 1 & \\ -%Paul&Sean T.&Wisconsin&1&&&& \\ -Peternell&Thomas&Bayreuth&&&&& \\ -Py& Pierre& Grenoble&&&&&\\ -Rousseau& Erwan& Brest&&&&&\\ -%Saccá&Giulia&NYU&1&1&&&1 \\ -Schnell&Christian&Stony Brook&&&&& \\ -%Shentu&Junchan&Heifei&&1&&1 \\ -%Siarhei&Finski&Grenoble&&&&1& \\ -Spelta&Irene&Barcelona&&&&&1\\ -Stenger&Isabel&Hannover&&&&&1\\ -Tasin&Luca&Mailand&&&&1&\\ -%Takayama&Shigeharu&Tokyo&&1&&& \\ -Tosatti&Valentino&Northwestern&&&&& \\ -%Ungureanu&Mara&Freiburg&&1&&1&1 \\ -Wang&Botong&University of Wisconsin&&&&&\\ -%Wang&Juanyong&Beijing&&1&&1& \\ -Witt-Nyström&David&Göteborg&&&&& \\ -%Wu&Xiaojun&Bayreuth&&&&1& \\ -%Xiao&Ming&UCSD&1&&&1& \\ -%Xu&Chenyang&Princeton&&1&&& \\ -Yang&Ruijie&Humboldt&&&&1& -\end{longtable} -} % \scriptsize - \bibstyle{alpha} diff --git a/general.bib b/general.bib index 0ab15bf..2130bbd 100644 --- a/general.bib +++ b/general.bib @@ -1,3 +1,82 @@ +@misc{arXiv:2207.03283, + title={Hyperbolicity in presence of a large local system}, + author={Yohan Brunebarbe}, + year={2022}, + eprint={2207.03283}, + archivePrefix={arXiv}, + primaryClass={math.AG}, + url={https://arxiv.org/abs/2207.03283}, +} + +@misc{arXiv:2305.09613, + title={The relative Green-Griffiths-Lang conjecture for families of varieties of maximal Albanese dimension}, + author={Yohan Brunebarbe}, + year={2023}, + eprint={2305.09613}, + archivePrefix={arXiv}, + primaryClass={math.AG}, + url={https://arxiv.org/abs/2305.09613}, +} + +@misc{arXiv:2007.12957, + title={Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications}, + author={Damian Brotbek and Yohan Brunebarbe}, + year={2020}, + eprint={2007.12957}, + archivePrefix={arXiv}, + primaryClass={math.AG}, + url={https://arxiv.org/abs/2007.12957}, +} + +@misc{arXiv:2112.13961, + title={Infinite energy maps and rigidity}, + author={Daskalopoulos, Georgios and Mese, Chikako}, + year={2021}, + month={December}, + url={https://arxiv.org/abs/2112.13961} +} + +@Article{zbMATH07858206, + Author = {Conlon, Ronan J. and Hein, Hans-Joachim}, + Title = {Classification of asymptotically conical {Calabi}-{Yau} manifolds}, + FJournal = {Duke Mathematical Journal}, + Journal = {Duke Math. J.}, + ISSN = {0012-7094}, + Volume = {173}, + Number = {5}, + Pages = {947--1015}, + Year = {2024}, + Language = {English}, + DOI = {10.1215/00127094-2023-0030}, + Keywords = {53C25,14J32}, + zbMATH = {7858206} +} + +@Article{zbMATH07810677, + Author = {Chiu, Shih-Kai and Sz{\'e}kelyhidi, G{\'a}bor}, + Title = {Higher regularity for singular {K{\"a}hler}-{Einstein} metrics}, + FJournal = {Duke Mathematical Journal}, + Journal = {Duke Math. J.}, + ISSN = {0012-7094}, + Volume = {172}, + Number = {18}, + Pages = {3521--3558}, + Year = {2023}, + Language = {English}, + DOI = {10.1215/00127094-2022-0107}, + Keywords = {32Q20,32Q25,53C25}, + URL = {projecteuclid.org/journals/duke-mathematical-journal/volume-172/issue-18/Higher-regularity-for-singular-K%c3%a4hlerEinstein-metrics/10.1215/00127094-2022-0107.full}, + zbMATH = {7810677} +} + +@misc{arXiv:2407.01150, + title={Degenerating conic Kähler-Einstein metrics to the normal cone}, + author={Biquard, Olivier and Guenancia, Henri}, + year={2024}, + month={July}, + url={https://arxiv.org/abs/2407.01150}, +} + @Article{zbMATH07782497, Author = {Fu, Xin and Hein, Hans-Joachim and Jiang, Xumin}, Title = {Asymptotics of {K{\"a}hler}-{Einstein} metrics on complex hyperbolic cusps},