Minimal extension property of direct images

TL;DR

This paper proves Griffiths semi-positivity and minimal extension properties for direct image sheaves arising from projective morphisms, involving complex geometric structures like Hodge variations and harmonic bundles.

Contribution

It establishes new positivity and extension results for direct image sheaves in complex geometry, encompassing various sheaf types such as dualizing, multiplier ideal, and Hodge structures.

Findings

Proves Griffiths semi-positivity of direct image sheaves.

Establishes minimal extension property for these sheaves.

Applies to sheaves involving Hodge structures and harmonic bundles.

Abstract

Given a projective morphism $f:X\to Y$ from a complex space to a complex manifold, we prove the Griffiths semi-positivity and minimal extension property of the direct image sheaf $f_\ast(\mathscr{F})$. Here, $\mathscr{F}$ is a coherent sheaf on $X$, which consists of the Grauert-Riemenschneider dualizing sheaf, a multiplier ideal sheaf, and a variation of Hodge structure (or more generally, a tame harmonic bundle).