Going through section Hodge theory

.vscode/ltex.dictionary.en-US.txt

@@ -0,0 +1,60 @@
+Zariski
+holomorphic
+geometers
+Yamanoi
+Brotbek
+Brunebarbe
+Cadorel
+Javanpeykar
+Campana
+hyperbolicity
+subvariety
+Bérczi
+Kirwan
+hypersurfaces
+Diverio-Merker-Rousseau
+Grassmannian
+Riedl-Yang
+Bérczi-Kirwan
+Nezza
+Kebekus
+Mihai
+Păun
+Schreieder
+Kähler
+Saito
+Mustaţă
+Popa
+Schnell
+Bernstein-Sato
+Debarre
+Casalaina-Martin
+Grushevsky
+Riemann-Schottky
+Lefschetz
+Goresky-MacPherson
+Laza
+Calabi-Yau
+fibration
+fibrations
+hyperkähler
+Langrangian
+Bakker
+Shen
+Maulik
+Hausel
+Mellit
+Minets
+Schiffmann
+pushforward
+Maulik-Shen-Yin
+Singer-Hopf
+Hopf
+Arapura
+Sato
+Llosa-Isenrich
+Py
+Goresky
+Schottky
+Calabi
+Yau

.vscode/ltex.hiddenFalsePositives.en-US.txt

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+{"rule":"PREPOSITION_VERB","sentence":"^\\QProbably 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 \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q to a complex torus \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is the translate of a sub-tori.\\E$"}
+{"rule":"PREPOSITION_VERB","sentence":"^\\QIts beginnings date back to 1926, when André Bloch showed that the Zariski closure of entire holomorphic curve \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q to a complex torus \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is the translate of a sub-torus.\\E$"}
+{"rule":"MISSING_GENITIVE","sentence":"^\\QHodge modules are used to define generalizations of well-known ideals of singularities, such as multiplier ideals from analysis and algebraic geometry.\\E$"}

MFO26.tex

@@ -71,69 +71,70 @@
 
 \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. Over the last few years the theory has been
-further developed and branched out to yield exciting applications to the
-topology of algebraic varieties, singularity theory and commutative algebra.  
-The following topics in this area will be of particular interest to our workshop.
+Over the last decade, Saito's theory of Hodge modules has seen spectacular
+applications in birational geometry.  More recent developments connect the
+theory to singularity theory, commutative algebra, and the topology of algebraic
+varieties.  The following topics in this area will be of particular interest to
+our workshop.
+
 
 \subsubsection{Singularities and Hodge ideals}
 
-Hodge modules are used to define generalizations of well-known ideals of
-singularities, such as multiplier ideals from analysis and algebraic geometry.
-This theory has been put forward by Mustata and Popa, an alternative approach
-was suggested by Schnell and Yang. These generalizations allow to study for
-instance Bernstein-Sato polynomials, which are important commutative algebra
-invariants of singularities that are typically hard to compute. Geometric
-applications are given by the study of singularities of Theta divisors of
-principally polarized abelian varieties, as pursued by Schnell and Yang. 
+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. An alternative approach towards similar ends was recently suggested in
+the preprint \cite{arXiv:2309.16763} of Schnell and Yang.  First applications
+pertain to Bernstein--Sato polynomials and their zero sets; these are important
+invariants of singularities originating from commutative algebra that are hard
+to compute.  Schnell and Yang apply their results to conjectures of
+Debarre--Casalaina-Martin--Grushevsky concerning the Riemann--Schottky problem
+and the singularities of Theta divisors of principally polarized Abelian
+varieties. 
 
-In most recent developments by Park and Popa, related methods have been used to
-improve classical Lefschetz theorems for singular varieties due to Goresky-Mac
-Pherson. Originally, Lefschetz theorems for singular varieties have been proven
-via stratified Morse theory, while the recent improvements rely on perverse
-sheaves and D-module theory. 
-
-A related program put forward by Friedman and Laza aims at understanding the
-Hodge structures of degenerating Calabi-Yau varieties. This led to the notions
-of higher Du Bois and higher rational singularities which can be understood via
-Hodge modules and will. 
+Very recently, Park and Popa applied perverse sheaves and D-module theory to
+improve Goresky--MacPherson's classic Lefschetz theorems in the singular setting.
+A program put forward by Friedman--Laza aims at understanding the Hodge
+structures of degenerating Calabi--Yau varieties. 
 
 
 \subsubsection{Lagrangian fibrations}
 
-A Lagrangian fibration of a (not necessarily compact) hyperkaehler manifold $M$
-is a proper holomorphic map $f:M\to B$ whose fibres are Langrangian
-submanifolds. If $M$ is compact, then a well-known conjecture in the field
-predicts that $B$ is projective space. This is known if $B$ is smooth by
-celebrated work of Hwang. A Hodge theoretic proof of Hwang‘s result has recently
-been found by Bakker and Schnell; the case where $B$ is allowed to be singular
-remains open. 
+A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map
+$f : M \to B$ whose generic fibers are Langrangian. 
+
+If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
+projective space.  In case where $B$ is smooth, the conjecture has been
+established more than 16 years ago in a celebrated work of Hwang.  Today, there
+is new insight, as Bakker--Schnell recently found a purely Hodge theoretic proof
+of Hwang's result, \cite{arXiv:2311.08977}.  There is hope that these methods
+give insight into the singular setting, which remains open to date. 
 
 In the non-compact setting, Lagrangian fibrations have been studied in the
-framework of the so called P=W conjecture, which has recently been proven by
-Maulik and Shen for the Hitchin fibration associated to the general linear group
-and curves of arbitrary genus $g\geq 2$. Shen and Yin discovered a remarkable
-symmetry of certain pushforward sheaves in the case of Lagrangian fibrations
-over possibly non-compact bases. Recently, Schnell used Saito‘s theory of Hodge
-modules to prove the conjecture of Shen and Yin in full generality.
+framework of the so called $P=W$ conjecture, which has recently been proven by
+Maulik--Shen and Hausel--Mellit--Minets--Schiffmann, \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.  These conjectures have recently been established by Schnell,
+\cite{arXiv:2303.05364}, who 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 K\"ahler manifolds}
+\subsubsection{Singer--Hopf conjecture and fundamental groups of Kähler manifolds}
 
-The Singer-Hopf conjecture says that a closed aspherical manifold of real
-dimension $2n$ has positive signed Euler characteristic $(-1)^n\chi(X)\geq 0$.
-This conjecture goes back to 1931, when Hopf formulated a related version for
-Riemannian manifolds. Recently, Hodge-theoretic refinements of this conjecture
-for K\"ahler manifolds have been put forward by Arapura, Maxim and Wang. Special
-cases of these conjecture have recently been proven, but the statement remains
-open in full generality.
+The Singer--Hopf conjecture asserts that a closed aspherical manifold of real
+dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot
+\chi(X)\geq 0$. This conjecture goes back to 1931, when Hopf formulated a
+related version for Riemannian manifolds. Recently, Hodge-theoretic refinements
+of this conjecture for Kähler manifolds have been put forward by
+Arapura--Maxim--Wang, \cite{arXiv:2310.14131}. Special cases of these conjecture
+have been proven, but the statement remains open in full generality.
 
-In a related direction, Llosa-Isenrich and Py found recently an application of
-complex geometry and Hodge theory to geometric group theory, thereby settling an
-old question of of Brady  on finiteness properties of groups. As a byproduct,
+In a related direction, Llosa-Isenrich--Py found an application of complex
+geometry and Hodge theory to geometric group theory, settling an old question of
+Brady on finiteness properties of groups, \cite{zbMATH07790946}. As a byproduct,
 the authors also obtain a proof of the classical Singer conjecture in an
-important special case in the realm of K\"ahler manifolds.
+important special case in the realm of Kähler manifolds.
 
 Our goal in this workshop is to bring together several experts in geometric
 group theory with experts on Hodge theory, and to explore further potential

general.bib

@@ -1,3 +1,98 @@
+@Article{zbMATH07790946,
+ Author = {Llosa Isenrich, Claudio and Py, Pierre},
+ Title = {Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices},
+ FJournal = {Inventiones Mathematicae},
+ Journal = {Invent. Math.},
+ ISSN = {0020-9910},
+ Volume = {235},
+ Number = {1},
+ Pages = {233--254},
+ Year = {2024},
+ Language = {English},
+ DOI = {10.1007/s00222-023-01223-3},
+ Keywords = {20F65,20F67,57M07,32J27},
+ zbMATH = {7790946},
+ Zbl = {1530.20138}
+}
+
+
+@misc{arXiv:2310.14131,
+      title={Hodge-theoretic variants of the Hopf and Singer Conjectures}, 
+      author={Donu Arapura and Laurentiu Maxim and Botong Wang},
+      year={2024},
+      eprint={2310.14131},
+      archivePrefix={arXiv},
+      primaryClass={math.AG},
+      url={https://arxiv.org/abs/2310.14131}, 
+}
+
+@misc{arXiv:2303.05364,
+      title={Hodge theory and Lagrangian fibrations on holomorphic symplectic manifolds}, 
+      author={Christian Schnell},
+      year={2023},
+      eprint={2303.05364},
+      archivePrefix={arXiv},
+      primaryClass={math.AG},
+      url={https://arxiv.org/abs/2303.05364}, 
+}
+
+@misc{arXiv:2209.05429,
+      title={$P=W$ via $H_2$}, 
+      author={Tamas Hausel and Anton Mellit and Alexandre Minets and Olivier Schiffmann},
+      year={2022},
+      eprint={2209.05429},
+      archivePrefix={arXiv},
+      primaryClass={math.AG},
+      url={https://arxiv.org/abs/2209.05429}, 
+}
+
+@misc{arXiv:2209.02568,
+      title={The $P=W$ conjecture for $\mathrm{GL}_n$}, 
+      author={Davesh Maulik and Junliang Shen},
+      year={2024},
+      eprint={2209.02568},
+      archivePrefix={arXiv},
+      primaryClass={math.AG},
+      url={https://arxiv.org/abs/2209.02568}, 
+}
+
+@misc{arXiv:2311.08977,
+      title={A Hodge-theoretic proof of Hwang's theorem on base manifolds of Lagrangian fibrations}, 
+      author={Benjamin Bakker and Christian Schnell},
+      year={2023},
+      eprint={2311.08977},
+      archivePrefix={arXiv},
+      primaryClass={math.AG},
+      url={https://arxiv.org/abs/2311.08977}, 
+}
+
+@misc{arXiv:2309.16763,
+      title={Higher multiplier ideals}, 
+      author={Christian Schnell and Ruijie Yang},
+      month={September},
+      year={2023},
+      eprint={2309.16763},
+      archivePrefix={arXiv},
+      primaryClass={math.AG},
+      url={https://arxiv.org/abs/2309.16763}, 
+}
+
+@article {MR4081135,
+    AUTHOR = {Mustaţă, Mircea and Popa, Mihnea},
+     TITLE = {Hodge filtration, minimal exponent, and local vanishing},
+   JOURNAL = {Invent. Math.},
+  FJOURNAL = {Inventiones Mathematicae},
+    VOLUME = {220},
+      YEAR = {2020},
+    NUMBER = {2},
+     PAGES = {453--478},
+      ISSN = {0020-9910,1432-1297},
+   MRCLASS = {14F10 (14F17 14J17 32S25)},
+  MRNUMBER = {4081135},
+MRREVIEWER = {Zhi\ Jiang},
+       DOI = {10.1007/s00222-019-00933-x},
+       URL = {https://doi.org/10.1007/s00222-019-00933-x},
+}
 
 @misc{arXiv:2406.19003,
       title={Hyperbolicity of generic hypersurfaces of polynomial degree via Green-Griffiths jet differentials},