Going through section Hodge theory

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

@@ -16,3 +16,45 @@ 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

@@ -1,2 +1,3 @@
 {"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
+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
+applications in birational geometry.  More recent developments connect the
-further developed and branched out to yield exciting applications to the
+theory to singularity theory, commutative algebra, and the topology of algebraic
-topology of algebraic varieties, singularity theory and commutative algebra.  
+varieties.  The following topics in this area will be of particular interest to
-The following topics in this area will be of particular interest to our workshop.
+our workshop.
+
 
 \subsubsection{Singularities and Hodge ideals}
 
-Hodge modules are used to define generalizations of well-known ideals of
+In a series of influential papers starting with \cite{MR4081135}, Mustaţă and
-singularities, such as multiplier ideals from analysis and algebraic geometry.
+Popa used Hodge modules to refine and generalize well-known invariants of
-This theory has been put forward by Mustata and Popa, an alternative approach
+singularities, most notably the multiplier ideals used in analysis and algebraic
-was suggested by Schnell and Yang. These generalizations allow to study for
+geometry. An alternative approach towards similar ends was recently suggested in
-instance Bernstein-Sato polynomials, which are important commutative algebra
+the preprint \cite{arXiv:2309.16763} of Schnell and Yang.  First applications
-invariants of singularities that are typically hard to compute. Geometric
+pertain to Bernstein--Sato polynomials and their zero sets; these are important
-applications are given by the study of singularities of Theta divisors of
+invariants of singularities originating from commutative algebra that are hard
-principally polarized abelian varieties, as pursued by Schnell and Yang. 
+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
+Very recently, Park and Popa applied perverse sheaves and D-module theory to
-improve classical Lefschetz theorems for singular varieties due to Goresky-Mac
+improve Goresky--MacPherson's classic Lefschetz theorems in the singular setting.
-Pherson. Originally, Lefschetz theorems for singular varieties have been proven
+A program put forward by Friedman--Laza aims at understanding the Hodge
-via stratified Morse theory, while the recent improvements rely on perverse
+structures of degenerating Calabi--Yau varieties. 
-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. 
 
 
 \subsubsection{Lagrangian fibrations}
 
-A Lagrangian fibration of a (not necessarily compact) hyperkaehler manifold $M$
+A Lagrangian fibration of a hyperkähler manifold $M$ is a proper holomorphic map
-is a proper holomorphic map $f:M\to B$ whose fibres are Langrangian
+$f : M \to B$ whose generic fibers 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
+If $M$ is compact, a well-known conjecture in the field predicts that $B$ is
-celebrated work of Hwang. A Hodge theoretic proof of Hwang‘s result has recently
+projective space.  In case where $B$ is smooth, the conjecture has been
-been found by Bakker and Schnell; the case where $B$ is allowed to be singular
+established more than 16 years ago in a celebrated work of Hwang.  Today, there
-remains open. 
+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
+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
+Maulik--Shen and Hausel--Mellit--Minets--Schiffmann, \cite{arXiv:2209.02568,
-and curves of arbitrary genus $g\geq 2$. Shen and Yin discovered a remarkable
+arXiv:2209.05429}.  In the same setting, Shen--Yin discovered a remarkable
-symmetry of certain pushforward sheaves in the case of Lagrangian fibrations
+symmetry of certain pushforward sheaves and conjectured that more general
-over possibly non-compact bases. Recently, Schnell used Saito‘s theory of Hodge
+symmetries exist.  These conjectures have recently been established by Schnell,
-modules to prove the conjecture of Shen and Yin in full generality.
+\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
+The Singer--Hopf conjecture asserts that a closed aspherical manifold of real
-dimension $2n$ has positive signed Euler characteristic $(-1)^n\chi(X)\geq 0$.
+dimension $2n$ has positive signed Euler characteristic, $(-1)^n \cdot
-This conjecture goes back to 1931, when Hopf formulated a related version for
+\chi(X)\geq 0$. This conjecture goes back to 1931, when Hopf formulated a
-Riemannian manifolds. Recently, Hodge-theoretic refinements of this conjecture
+related version for Riemannian manifolds. Recently, Hodge-theoretic refinements
-for K\"ahler manifolds have been put forward by Arapura, Maxim and Wang. Special
+of this conjecture for Kähler manifolds have been put forward by
-cases of these conjecture have recently been proven, but the statement remains
+Arapura--Maxim--Wang, \cite{arXiv:2310.14131}. Special cases of these conjecture
-open in full generality.
+have been proven, but the statement remains open in full generality.
 
-In a related direction, Llosa-Isenrich and Py found recently an application of
+In a related direction, Llosa-Isenrich--Py found an application of complex
-complex geometry and Hodge theory to geometric group theory, thereby settling an
+geometry and Hodge theory to geometric group theory, settling an old question of
-old question of of Brady  on finiteness properties of groups. As a byproduct,
+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},