  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.
