Applications of the deformation formula of holomorphic one-forms

Abstract

This paper studies some geometric aspects of moduli of curves Mg, using as a tool the deformation formula of holomorphic one-forms. Quasi-isometry guarantees the L 2 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 Mg in a quite detailed manner, while gaining some understanding of a totally geodesic manifold in a nonhyperelliptic locus. This paper is a complement to our joint paper [Liu et al. 2012b] with Kefeng Liu, and explores more applications of the deformation formula of holomorphic oneforms to some problems related to moduli spaces of Riemann surfaces, including the full expansion of the period map, the Siegel metric and its curvature formulae, the second fundamental form of Torelli space’s nonhyperelliptic locus, and also a global result about the deformation of holomorphic one-forms. We start with the Kuranishi coordinate of the Teichmuller space Tg of Riemann surfaces of genus g and the deformation formula of holomorphic one-forms. t/, whose construction is contained in Section 2. The key points of the deformation formula lie in Theorem 2.1. To be more precise, on the Kuranishi family$VX!1 with a Riemann surface $ 1 .0/D X0 as its central fiber and a global holomorphic one-form of the central fiber 2 H 0 .X0; 1 X0 /, the deformation formula of holomorphic one-forms emerges as