Comparison of Poisson structures on moduli spaces

TL;DR

This paper proves that an isomorphism between moduli spaces of Hitchin pairs and spectral data preserves their algebraic Poisson structures, highlighting a deep geometric relationship.

Contribution

It demonstrates that the spectral data isomorphism between moduli spaces preserves the algebraic Poisson structures, extending understanding of Hitchin systems.

Findings

The isomorphism between moduli spaces preserves Poisson structures.

Spectral data provides a Poisson-preserving correspondence.

The result applies to moduli spaces on complex curves with line bundles.

Abstract

Let $X$ be a complex irreducible smooth projective curve, and let ${\mathbb L}$ be an algebraic line bundle on $X$ with a nonzero section $\sigma_0$. Let $\mathcal{M}$ denote the moduli space of stable Hitchin pairs $(E,\, \theta)$, where $E$ is an algebraic vector bundle on $X$ of fixed rank $r$ and degree $\delta$, and $\theta\, \in\, H^0(X,\, End(E)\otimes K_X\otimes{\mathbb L})$. Associating to every stable Hitchin pair its spectral data, an isomorphism of $\mathcal{M}$ with a moduli space $\mathcal{P}$ of stable sheaves of pure dimension one on the total space of $K_X\otimes{\mathbb L}$ is obtained. Both the moduli spaces $\mathcal{P}$ and $\mathcal{M}$ are equipped with algebraic Poisson structures, which are constructed using $\sigma_0$. Here we prove that the above isomorphism between $\mathcal{P}$ and $\mathcal{M}$ preserves the Poisson structures.