Curvatures of moduli space of curves and applications

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 $(M_g,{\omega }_{WP})$ 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.

References in corpus (4)

• The Torelli locus and special subvarieties
• The Riemannian Sectional Curvature Operator Of The Weil-Petersson Metric and Its Application
• Curvature of vector bundles associated to holomorphic fibrations
• On Shimura curves in the Torelli locus of curves

Cited by in corpus (6)

• RC-positivity and the generalized energy density I: Rigidity
• FQHE and $t{t}^{\ast }$ geometry
• Hermitian Harmonic maps and non-degenerate curvatures
• On the Weil-Petersson curvature of the moduli space of Riemann surfaces of large genus
• Effective vanishing theorems for ample and globally generated vector bundles
• RC-positivity and rigidity of harmonic maps into Riemannian manifolds