# General Variational Formulas for Abelian Differentials

## TL;DR

This paper develops explicit variational formulas for stable differentials and their periods near degenerations, using jump problem techniques, with applications to period matrix degeneration and strata closure theorems.

## Contribution

It introduces a new method to compute variational formulas for stable differentials and their periods, providing explicit formulas and alternative proofs for existing theorems.

## Key findings

- Explicit variational formulas for stable differentials near degenerations.
- Reproof of Yamada's results on period matrix degeneration.
- Alternative proof of the closure theorem for strata of differentials.

## Abstract

We use the jump problem technique developed in a recent paper by Grushevsky, Krichever and the second author to compute the variational formula of any stable differential and its periods to arbitrary precision in plumbing coordinates. In particular, we give the explicit variational formula for the degeneration of the period matrices near an arbitrary stable curve, easily reproving the results of Yamada for nodal curves with one node. Concrete examples are included. We also apply the same technique to give an alternative proof of the sufficiency part of the theorem by Bainbridge, Chen, Gendron, Grushevsky and M\"oller on the closures of strata of differentials with prescribed multiplicities of zeroes and poles.