ncts

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 2012, 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 is based on the joint work with Christian Schnell.