Asian Journal of Mathematics

Volume 22 (2018)

Number 4

Special issue in honor of Ngaiming Mok (3 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

Augmented Weil–Petersson metrics on moduli spaces of polarized Ricci-flat Kähler manifolds and orbifolds

Pages: 705 – 728



Wing-Keung To (Department of Mathematics, National University of Singapore)

Sai-Kee Yeung (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


We show that the base complex manifold of an effectively parametrized family of compact polarized Ricci-flat Kähler orbifolds, and in particular manifolds, admits a smooth augmented Weil–Petersson metric whose holomorphic sectional curvature is bounded above by a negative constant. As a consequence, we conclude that such base manifold is Kobayashi hyperbolic.


augmented Weil–Petersson metric, moduli space, Ricci-flat Kähler manifolds, Kähler orbifolds

2010 Mathematics Subject Classification

32G13, 32Q25, 32Q45

Wing-Keung To was partially supported by the research grant R-146-000-142-112 from the National University of Singapore and the Ministry of Education.

Sai-Kee Yeung was partially supported by a grant from the National Science Foundation.

Received 13 September 2016

Accepted 2 June 2017

Published 20 September 2018