Motives and the Hodge Conjecture for moduli spaces of pairs

TL;DR

This paper proves that for generic curves, the moduli spaces of stable pairs satisfy the Hodge Conjecture for ranks up to 4, using their motivation by the underlying curve.

Contribution

It establishes the Hodge Conjecture for moduli spaces of pairs on generic curves and shows these spaces are motivated by the curve, for ranks up to 4.

Findings

Hodge Conjecture holds for moduli spaces of pairs with rank ≤ 4.

Moduli spaces of pairs are motivated by the underlying curve.

Results apply to generic curves of genus ≥ 2.

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 $C$.