Curvature Formulas Related to a Family of Stable Higgs Bundles

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