Curvature formulas related to a family of stable Higgs bundles

TL;DR

This paper explores the geometry of the base manifold of stable Higgs bundles, deriving curvature formulas and constructing a Finsler metric to understand its geometric properties.

Contribution

It extends curvature formulas to the setting of Higgs bundles and constructs a new Finsler metric with explicit curvature calculations.

Findings

Derived curvature formulas for the base manifold of Higgs bundles.

Calculated the curvature on higher direct image bundles.

Constructed a strongly pseudo-convex Finsler metric with explicit curvature.

Abstract

In this paper, we investigate the geometry of the base complex manifold of an effectively parametrized holomorphic family of stable Higgs bundles over a fixed compact K\"{a}hler manifold. The starting point of our study is Schumacher-Toma/Biswas-Schumacher's curvature formulas for Weil-Petersson-type metrics, in Sect. 2, we give some applications of their formulas on the geometric properties of the base manifold. In Sect. 3, we calculate the curvature on the higher direct image bundle, which recovers Biswas-Schumacher's curvature formula. In Sect. 4, we construct a smooth and strongly pseudo-convex complex Finsler metric for the base manifold, the corresponding holomorphic sectional curvature is calculated explicitly.