Augmented Weil-Petersson metrics on moduli spaces of polarized Ricci-flat Kähler manifolds and orbifolds
type: Item
instance of: scholarly article
title: Augmented Weil-Petersson metrics on moduli spaces of polarized Ricci-flat Kähler manifolds and orbifolds
zbMATH Open document ID: 1411.32014
author: Wing-Keung To
author: Sai-Kee Yeung
published in: The Asian Journal of Mathematics
publication date: 2018-10-04T00:00:00Z
review text: 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)].
reviewed by: Antonella Nannicini
Mathematics Subject Classification ID: 32G13
Mathematics Subject Classification ID: 32Q25
Mathematics Subject Classification ID: 32Q45
zbMATH DE Number: 6947086
zbMATH Keywords: moduli space
zbMATH Keywords: Ricci-flat Kähler manifolds
zbMATH Keywords: Kähler orbifolds
zbMATH Keywords: augmented Weil-Petersson metric
MaRDI profile type: Publication
DOI: 10.4310/AJM.2018.V22.N4.A6
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description: scientific article; zbMATH DE number 6947086