Motives and the Hodge Conjecture for moduli spaces of pairs

Abstract

Let $C$ be a smooth projective curve of genus $g\geq 2$ over $\mathbb C$. Fix $n\geq 1$, $d\in {\mathbb Z}$. A pair $(E,\phi)$ over $C$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $C$ and a section $\phi \in H^0(E)$. There is a concept of stability for pairs which depends on a real parameter $\tau$. Let ${\mathfrak M}_\tau(n,d)$ be the moduli space of $\tau$-polystable pairs of rank $n$ and degree $d$ over $C$. Here we prove that for a generic curve $C$, the moduli space ${\mathfrak M}_\tau(n,d)$ satisfies the Hodge Conjecture for $n \leq 4$. For obtaining this, we prove first that ${\mathfrak M}_\tau(n,d)$ is motivated by $C$.