Asian Journal of Mathematics

Volume 21 (2017)

Number 5

Curvatures of moduli space of curves and applications

Pages: 841 – 854

DOI: https://dx.doi.org/10.4310/AJM.2017.v21.n5.a3

Authors

Kefeng Liu (Department of Mathematics, University of California at Los Angeles)

Xiaofeng Sun (Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, U.S.A.)

Xiaokui Yang (Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, C.A.S., Beijing, China)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(\mathcal{M}_g , \omega_{WP})$ of curves with genus $g \gt 1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positive Riemannian curvature operator and also non-positive complex sectional curvature. We also prove that any submanifold in $\mathcal{M}_g$ which is totally geodesic in $\mathcal{A}_g$ with finite volume must be a ball quotient.

Keywords

curvature, moduli space, Weil–Petersson metric

2010 Mathematics Subject Classification

14K10, 32Cxx, 53C55

Received 7 November 2015

Accepted 21 March 2016

Published 9 February 2018