Schiffer variations and Abelian differentials

TL;DR

This paper studies deformations of compact Riemann surfaces and Abelian differentials using cohomology, with explicit calculations and expansions for period matrices and differentials, advancing understanding of their deformation theory.

Contribution

It introduces a comprehensive deformation expansion for Abelian differentials and Riemann period matrices using cohomological and kernel function methods.

Findings

Calculated cocycles for conformal deformations at zeros of Abelian differentials

Presented second order deformation expansion for the Riemann period matrix

Developed a complete deformation expansion for Abelian differentials

Abstract

Deformations of compact Riemann surfaces are considered using a \v{C}ech 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.