Hodge structures of the moduli spaces of pairs

TL;DR

This paper proves that the cohomology of moduli spaces of pairs over a curve has a Hodge structure decomposing into tensor products of the curve's Hodge structure, revealing deep geometric properties.

Contribution

It establishes that the cohomology of moduli spaces of pairs and stable bundles decomposes into tensor products of the curve's Hodge structure, extending known results.

Findings

Cohomology groups are Hodge structures isomorphic to summands of tensor products of H^1(X)

Results apply to moduli spaces of pairs and stable vector bundles

Provides new insights into the geometric structure of these moduli spaces

Abstract

Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. Fix $n\geq 2$, and an integer $d$. A pair $(E,\phi)$ over $X$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $X$ and a section $\phi$. There is a concept of stability for pairs which depends on a real parameter $\tau$. Let $M_\tau(n,d)$ be the moduli space of $\tau$-semistable pairs of rank $n$ and degree $d$ over $X$. We prove that the cohomology groups of $M_\tau(n,d)$ are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure $H^1(X)$. This implies a similar result for the moduli spaces of stable vector bundles over $X$.