Asian Journal of Mathematics

Volume 19 (2015)

Number 2

Motives and the Hodge conjecture for moduli spaces of pairs

Pages: 281 – 306

DOI: http://dx.doi.org/10.4310/AJM.2015.v19.n2.a5

Authors

Vicente Muñoz (Facultad de Matemáticas, Universidad Complutense de Madrid, Spain)

André G. Oliveira (Departamento de Matemática, Escola de Ciências e Tecnologia, Universidade de Trás-os-Montes e Alto Douro, Quinta dos Prados, Vila Real, Portugal)

Jonathan Sánchez (Facultad de Matemáticas, Universidad Complutense de Madrid, Spain)

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 $C$. 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 $C$.

Keywords

moduli space, complex curve, vector bundle, motives, Hodge conjecture

2010 Mathematics Subject Classification

Primary 14F45. Secondary 14D20, 14H60.

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