  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.
