Geometry Seminar

 

University of Tor Vergata, Department of Mathematics

21^th of November 2023, 14:30-16:00, room “D’Antoni”

 

 

 

 

 

 

 

The geometric Riemann-Schottky problem and Hodge theory

 

 

 

Ruijie Yang

Humboldt-Universitä

 

 

 

It is a classical problem in algebraic geometry, dated back to Riemann, to characterize Jacobians of smooth projective curves among all principally polarized abelian varieties. In 2008, Casalaina-Martin proposed a conjecture in terms of singularities of theta divisors. In this talk, I will present a partial solution of this conjecture using Hodge theory and D-modules. We also show that this conjecture can be deduced from a conjecture of Pareschi and Popa on GV sheaves and minimal cohomology classes.

To achieve this, we develop a theory of higher multiplier ideals, which is a Hodge-theoretic refinement of the theory of multiplier ideals in birational geometry. This new theory relies on the Hodge theory of Kashiwara-Malgrange V-filtration, Sabbah-Schnell’s theory of complex mixed Hodge modules and the language of twisted D-modules by Beilinson and Bernstein.

 

 

 

 

 

 

 

 

 

 

 

 

 

This talk is part of the activity of the MIUR Excellence Department Projects MathMod@TOV, and the PRIN 2022 Moduli Spaces and Birational Geometry