Applications of the deformation formula of holomorphic one-forms

Quanting Zhao and Sheng Rao

Vol. 266 (2013), No. 1, 221–255

DOI: 10.2140/pjm.2013.266.221

Abstract

This paper studies some geometric aspects of moduli of curves ℳ_g, using as a tool the deformation formula of holomorphic one-forms. Quasi-isometry guarantees the L² convergence of deformation of holomorphic one-forms, which is a kind of global result. After giving the period map a full expansion, we can also write out the Siegel metric, curvature and second fundamental form of a nonhyperelliptic locus of ℳ_g in a quite detailed manner, while gaining some understanding of a totally geodesic manifold in a nonhyperelliptic locus.

Keywords

moduli of Riemann surfaces, Teichmüller theory, period matrices, variation of Hodge structures, differentials on Riemann surfaces

Mathematical Subject Classification 2010

Primary: 32G15

Secondary: 32G20, 30F30

Milestones

Received: 11 August 2012

Accepted: 5 April 2013

Published: 23 September 2013