A universal construction for moduli spaces of decorated vector bundles over curves

Abstract

Let $X$ be a smooth projective curve over the complex numbers. To every representation $ρ\colon \GL(r)\lra \GL(V)$ of the complex general linear group on the finite dimensional complex vector space $V$ which satisfies the assumption that there be an integer $α$ with $ρ(z \id_{\C^r})=z^α\id_V$ for all $z\in\C^*$ we associate the problem of classifying triples $(E,L,ϕ)$ where $E$ is a vector bundle of rank $r$ on $X$, $L$ is a line bundle on $X$, and $ϕ\colon E_ρ\lra L$ is a non trivial homomorphism. Here, $E_ρ$ is the vector bundle of rank $\dim V$ associated to $E$ via $ρ$. If we take, for example, the standard representation of $\GL(r)$ on $\C^r$ we have to classify triples $(E,L,ϕ)$ consisting of $E$ as before and a non-zero homomorphism $ϕ\colon E\lra L$ which includes the so-called Bradlow pairs. For the representation of $\GL(r)$ on $S^2\C^3$ we find the conic bundles of Gomez and Sols. In the present paper, we will formulate a general semistability concept for the above triples which depends on a rational parameter $δ$ and establish the existence of moduli spaces of $δ$-(semi)stable triples of fixed topological type. The notion of semistability mimics the Hilbert-Mumford criterion for $SL(r)$ which is the main reason that such a general approach becomes feasible. In the known examples (the above, Higgs bundles, extension pairs, oriented framed bundles) we show how to recover the "usual" semistability concept. This process of simplification can also be formalized. Altogether, our results provide a unifying construction for the moduli spaces of most decorated vector bundle problems together with an automatism for finding the right notion of semistability and should therefore be of some interest.