On families of holomorphic differentials on degenerating annuli

Details On families of holomorphic differentials on degenerating annuli On families of holomorphic differentials on degenerating annuli

2011-08-16

arxiv.org/abs/1108.3227v2

Mathematics

Geometric Topology

Version 2 includes small edits and a general discussion of divisors of families in a neighborhood of a node

Scientific paper

We consider the local analytic behavior for a family of holomorphic differentials on a family of degenerating annuli. Three results and discussion are presented. The first is the normal families Lemma 1. The second is an isomorphism of sheaves, formula (3), giving a direct description of families of regular $k$-differentials (sections of powers of the relative dualizing sheaf) in terms of $k$-canonical forms on the total space of the family. The third is a general holomorphic extension property, Lemma 3, for families given on smooth Riemann surfaces/curves to extend to the limiting nodal Riemann surfaces/curves. Divisors of families of holomorphic differentials are also discussed.

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