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Going through section Hodge theory

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

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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$"}

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MFO26.tex
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@ -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

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general.bib
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@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},