Positivity for Hodge modules and geometric applications

Abstract

This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry.

Hodge Ideals

“…For simplicity, we will call such objects Hodge D-modules. It is not the place here to give a detailed account of the theory of Hodge modules; for details we refer to the original [Sai88], [Sai90], for summaries of the results needed here to the surveys [Sai16a] and [Sch14a], and for a review of how they have been recently used for geometric applications to [Pop16]. We will however review properties of Hodge D-modules that make them special among all filtered D-modules, and that will be used here, as well as some results specific to ω X ( * D).…”

𝒟-Modules in Birational Geometry

“…[DMS11], [Sch12], [PS13], [Wan16], [PS14], [PS17], [PPS17], [Wei17]. The bulk of the recent survey [Pop16b] treats part of this body of work, so I have decided not to discuss it here again. In any event, the reader is advised to use [Pop16b] as a companion to this article, as introductory material on D-modules and Hodge modules together with a guide to technical literature can be found there (especially in Ch.…”

“…Saito's theory of mixed Hodge modules produces useful filtered D-modules of geometric and Hodge theoretic origin on complex varieties, which extend the notion of a variation of Hodge structure when singularities (of fibers of morphisms, of hypersurfaces, of ambient varieties, etc.) are involved; see for instance the examples in [Pop16b,§2]. Usually the D-module itself is quite complicated, but here we deal with one of the simplest ones.…”