Curvatures of moduli space of curves and applications

Kefeng Liu, Xiaofeng Sun, Xiaokui Yang, Shing-Tung Yau

Published 2013-12-25, updated 2015-05-02Version 2

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 $(M_g, \omega_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positive Riemannain curvature operator and also non-positive complex sectional curvature. As applications, we prove that any submanifold in $M_g$ which is totally geodesic in $A_g$ with finite volume must be a ball quotient. We also show that, on a Kodaira (deformation) surface, there is no K\"ahler metric with non-positive Riemannain sectional curvature although it possesses K\"ahler metrics with strictly negative holomorphic bisectional curvature.