# Hodge structures of the moduli spaces of pairs - Mathematics > Algebraic Geometry

Hodge structures of the moduli spaces of pairs - Mathematics > Algebraic Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complexnumbers. Fix $n\geq 2$, and an integer $d$. A pair $E,\phi$ over $X$ consistsof an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $X$ and asection $\phi$. There is a concept of stability for pairs which depends on areal parameter $\tau$. Let $M \taun,d$ be the moduli space of$\tau$-semistable pairs of rank $n$ and degree $d$ over $X$. We prove that thecohomology groups of $M \taun,d$ are Hodge structures isomorphic to directsummands of tensor products of the Hodge structure $H^1X$. This implies asimilar result for the moduli spaces of stable vector bundles over $X$.

Author: ** Vicente MuĂ±oz**

Source: https://arxiv.org/