# Spaces of abelian differentials and Hitchin's spectral covers

## TL;DR

This paper explores the relationship between Hitchin's spectral covers and meromorphic abelian differentials, deriving new residue formulas and connecting them to topological recursion for period matrix variations.

## Contribution

It introduces a novel embedding of Hitchin's spectral covers into the moduli space of abelian differentials, leading to generalized residue formulas for period matrix variations.

## Key findings

- Derived generalized residue formulas extending Donagi-Markman
- Reproduced second derivative formulas via topological recursion
- Connected spectral cover theory with abelian differential moduli

## Abstract

Using the embedding of the moduli space of generalized GL(n) Hitchin's spectral covers to the moduli space of meromorphic abelian differentials we study the variational formulae of the period matrix, the canonical bidifferential, the prime form and the Bergman tau function. This leads to residue formulae which generalize the Donagi-Markman formula for variations of the period matrix. Computation of second derivatives of the period matrix reproduces the formula derived by Baraglia and Zhenxi Huang using the framework of topological recursion.