Siegel metric and curvature of the moduli space of curves

TL;DR

This paper investigates the curvature properties of the moduli space of curves using the Siegel metric, providing explicit curvature formulas and extending the metric's Kaehler form to the compactification.

Contribution

It derives an explicit formula for the holomorphic sectional curvature of M_g and extends the Siegel metric's Kaehler form to the Deligne-Mumford compactification.

Findings

Explicit formula for holomorphic sectional curvature along Schiffer variations

Extension of the Kaehler form as a closed current on the compactification

Cohomology class of the extended form as a multiple of the first Chern class

Abstract

We study the curvature of the moduli space M_g of curves of genus g with the Siegel metric induced by the period map. We give an explicit formula for the holomorphic sectional curvature of M_g along a Schiffer variation at a point P on the curve X, in terms of the holomorphic sectional curvature of A_g and the second Gaussian map. Finally we extend the Kaehler form of the Siegel metric as a closed current on the Deligne-Mumford compatification of M_g and we determine its cohomology class as a multiple of the first Chern class of the Hodge bundle.