Extension of twisted Hodge metrics for K\"ahler morphisms

TL;DR

This paper extends the understanding of Hodge metrics for Kähler morphisms, showing how to extend induced metrics as semi-positive singular Hermitian metrics, with implications for weak positivity in algebraic geometry.

Contribution

It introduces an extension of the Hodge metric on direct image sheaves as a semi-positive singular Hermitian metric over the base.

Findings

Extended the Hodge metric to singular Hermitian metrics.

Proved weak positivity of direct image sheaves over the base.

Applied results to projective base cases.

Abstract

Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \otimes E) for q \geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over Y-Z'' as a singular Hermitian metric with semi-positive curvature ``over Y''. As a particular consequence, if Y is projective, R^qf_*(K_{X/Y} \otimes E) is weakly positive over Y-Z in the sense of Viehweg.