diff --git a/MFO26.tex b/MFO26.tex index e41d1ae..ad6a83e 100644 --- a/MFO26.tex +++ b/MFO26.tex @@ -72,30 +72,30 @@ \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 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. +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. -\subsubsection{Singularities and Hodge ideals} +\subsubsection{Singularities and Hodge Ideals} 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 +geometry. Schnell and Yang’s recent preprint \cite{arXiv:2309.16763} suggested +an alternative approach toward similar ends. The first applications pertain to +Bernstein--Sato polynomials and their zero sets; these are essential 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. -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. +Park and Popa recently 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} @@ -103,41 +103,47 @@ structures of degenerating Calabi--Yau varieties. 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--Shen and Hausel--Mellit--Minets--Schiffmann, \cite{arXiv:2209.02568, -arXiv:2209.05429}. In the same setting, Shen--Yin discovered a remarkable +\paragraph{Compact Setting} + +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. These conjectures have recently been established by Schnell, -\cite{arXiv:2303.05364}, who also proved two conjectures of Maulik--Shen--Yin on +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} -The Singer--Hopf conjecture asserts 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, $(-1)^n \cdot -\chi(X)\geq 0$. 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. +\chi(X)\geq 0$. This conjecture goes back to 1931 when Hopf formulated a related +version for Riemannian manifolds. Recently, Arapura–Maxim–Wang suggested +Hodge-theoretic refinements of this conjecture for Kähler manifolds in +\cite{arXiv:2310.14131}. While the methods of \cite{arXiv:2310.14131} suffice to +show particular cases, the statement remains open in full generality. 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 Kähler manifolds. +Brady on the finiteness properties of groups \cite{zbMATH07790946}. As a +byproduct, the authors also obtain a proof of the classical Singer conjecture in +an essential particular 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 +group theory with experts on Hodge theory and to explore further potential applications of the methods from one field to problems in the other.