Higher multiplier ideals

For any Q-divisor D on a complex manifold X, there is a multiplier ideal associated to the pair (X,D), which is an ideal sheaf measuring the singularity of the pair and has many important applications in algebraic geometry and commutative algebra. It turns out that this is only a small piece of a larger picture. In this talk, I will discuss the construction of a family of ideal sheaves attached to (X,D) indexed by an integer indicating the Hodge level, such that the lowest level recovers the usual multiplier ideals. We describe their local and global properties: the local properties rely on Saito's theory of rational mixed Hodge modules and a small technical result from Sabbah's theory of twistor D-modules; while the global properties need Sabbah-Schnell's theory of complex mixed Hodge modules and  Beilinson-Bernstein’s theory of twisted D-modules from geometric representation theory. I will also compare this with the theory of (weighted) Hodge ideals recently developed by Mustata, Popa and Olano. If time permits, I will discuss some application to the singularity of theta divisors on principally polarized abelian varieties. This is joint work with Christian Schnell.