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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 $M_\tau(n,d)$ be the moduli space of$\tau$-semistable pairs of rank $n$ and degree $d$ over $X$. We prove that thecohomology groups of $M_\tau(n,d)$ are Hodge structures isomorphic to directsummands of tensor products of the Hodge structure $H^1(X)$. This implies asimilar result for the moduli spaces of stable vector bundles over $X$.

Hodge structures of the moduli spaces of pairs