Schiffer variations and Abelian differentials

Abstract

Deformations of compact Riemann surfaces are considered using a Čech cohomology sliding overlaps approach. Cocycles are calculated for conformal cutting and regluing deformations at zeros of Abelian differentials. A second order deformation expansion is presented for the Riemann period matrix. A complete deformation expansion is presented for Abelian differentials. Schiffer's kernel function approach for deformations of a Green's function is followed.

26 pages, 3 figures; version 2 includes discussion of the relation to the Kontsevich-Zorich breaking up a zero deformation and of breaking up higher order zeros