Hodge sheaves underlying flat projective families

TL;DR

This paper constructs a natural system of Hodge sheaves with pole-free Higgs fields from flat projective families, extending classical results and applying positivity to characterize base spaces of certain families.

Contribution

It introduces a new framework for Hodge sheaves in flat families, generalizing classical results and linking positivity to the geometry of base spaces.

Findings

Hodge sheaves with pole-free Higgs fields exist for flat projective families.

Positivity of direct image sheaves implies base spaces are of general type.

Construction of derived categorical objects generalizing relative logarithmic forms.

Abstract

We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms for smooth maps and their functorial properties.