In-Person Talk 

For any effective 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 applications in algebraic geometry. In this talk, I will discuss the construction of a family of ideal sheaves associated to (X,D), indexed by an integer indicating the Hodge level, such that the lowest level recovers the usual multiplier ideals. Their local and global properties are established using various types of D-modules. One key mechanism is the gluing of local nearby cycles along the divisor as twisted complex Hodge modules using the language of D-algebra by Bernstein and Beilinson and the theory of complex Hodge modules by Sabbah and Schnell. Some application to the Riemann-Schottky problem via the singularity of theta divisors on principally polarized abelian varieties will be given.

This is based on the joint work with Christian Schnell.