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 Kähler 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.

在本文中，我们研究了固定紧凑 Kähler 流形上稳定希格斯丛的有效参数化全纯族的基本复流形的几何形状。我们研究的起点是 Schumacher-Toma/Biswas-Schumacher 的 Weil-Petersson 型度量的曲率公式，在第 3 节。2，我们给出了他们的公式在基流形的几何性质上的一些应用。昆虫。3，我们计算更高直接图像束的曲率，它恢复了 Biswas-Schumacher 的曲率公式。昆虫。在图 4 中，我们为基础流形构建了一个平滑且强伪凸的复 Finsler 度量，明确计算了相应的全纯截面曲率。