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-Schumachers 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-Schumachers 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.
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