Differential Geometry of the Mixed Hodge Metric

TL;DR

This paper studies the geometric properties of the Hodge metric on mixed period domains, including curvature calculations and conditions for the pullback metric to be Kähler, with applications to normal functions and biextension bundles.

Contribution

It provides new curvature formulas and criteria for Kählerness of the pullback metric in the context of mixed Hodge structures.

Findings

Curvature of the Hodge metric is explicitly calculated.

Conditions for the pullback metric to be Kähler are established.

Applications to normal functions and biextension bundles are demonstrated.

Abstract

We investigate properties of the Hodge metric of a mixed period domain. In particular, we calculate its curvature and the curvature of the Hodge bundles. We also consider when the pull back metric via a period map is K\"ahler. Several applications in cases of geometric interest are given, such as for normal functions and biextension bundles.