Higher multiplier ideals

TL;DR

This paper introduces higher multiplier ideals associated with divisors on complex manifolds, exploring their properties, vanishing theorems, and applications to conjectures on theta divisors.

Contribution

It develops a new family of ideals using mixed Hodge modules, generalizing classical multiplier ideals and applying them to complex geometry problems.

Findings

Proved vanishing and restriction theorems for higher multiplier ideals.

Established criteria for nontriviality of these ideals.

Applied the theory to prove new cases of conjectures on theta divisors.

Abstract

We associate a family of ideal sheaves to any Q-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and V-filtrations. This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration. When the Hodge level is zero, they recover the usual multiplier ideals. We study the local and global properties of higher multiplier ideals systematically. In particular, we prove vanishing theorems and restriction theorems, provide criteria for the nontriviality, and introduce the center of minimal exponent (generalizing the notion of minimal log canonical center). The main idea is to exploit the global structure of the V-filtration along an effective divisor using the notion of twisted Hodge modules. As applications, we prove new cases of conjectures by Debarre, Casalaina-Martin and Grushevsky on singularities of theta divisors on principally polarized abelian varieties.