Hodge sheaves underlying flat projective families

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.

arXiv commentMore streamlined exposition following referee's suggestions. Corrected some minor errors in Sections 4 and 5. Main results and statements unchanged