Augmented Weil-Petersson metrics on moduli spaces of polarized Ricci-flat Kähler manifolds and orbifolds
The moduli space of compact polarized Ricci-flat Kähler manifolds and orbifolds are investigated. By modifying the Weil-Petersson metric the authors construct a Finsler metric, called augmented Weil-Petersson metric, whose holomorphic sectional curvature is bounded above by a negative constant. Precisely they prove that given an effectively parametrized holomorphic family of compact polarized Kähler manifolds of zero first Chern class \(\pi : \chi \rightarrow S\) over a complex manifold, \(S\) admits a \(C^{\infty}\) \(\mathrm{Aut}(\pi)\)-invariant augmented Weil-Petersson metric whose holomorphic sectional curvature is bounded above by a negative constant. Moreover, for an effectively parametrized holomorphic family of compact polarized Ricci-flat Kähler orbifolds, \(S\) admits a \(C^{\infty}\) augmented Weil-Petersson metric whose holomorphic sectional curvature is bounded above by a negative constant. As an application, in both cases, they obtain that \(S\) is Kobayashi hyperbolic. The augmented Weil-Petersson metric is constructed starting from the curvature expression of the usual Weil-Petersson metric computed by \textit{A. Nannicini} [Manuscr. Math. 54, 405--438 (1986; Zbl 0589.32041)].
scientific article; zbMATH DE number 6947086