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 $\alpha$ with $ρ(z \id_{\C^r})=z^α\id_V$ for all $z\in\C^*$ we associate the problem of classifying triples $(E,L,\varphi )$ 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, $E_{\rho }$ is the vector bundle of rank $\dim V$ associated to $E$ via $\rho$. If we take, for example, the standard representation of $\GL(r)$ on $\C^r$ we have to classify triples $(E,L,\varphi )$ 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 $\delta$ and establish the existence of moduli spaces of $\delta$-(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.

Final Version (To appear in Transformation Groups); V2: Example 3.7 corrected, other minor modifications; V3: Notion of polystability corrected, other minor modifications

References in corpus (4)

• Stable pairs on curves and surfaces
• A Hitchin-Kobayashi correspondence for Kaehler fibrations
• Singular principal bundles over higher dimensional manifolds and their moduli spaces
• Framed Hitchin Pairs

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• Torelli theorem for the moduli spaces of pairs
• Global Boundedness for Decorated Sheaves
• Enumerative invariants and wall-crossing formulae in abelian categories
• Quantization of Hitchin's equations for Higgs Bundles I
• Stability and Hermitian-Einstein metrics for vector bundles on framed manifolds
• Moduli problems of sheaves associated with oriented trees
• Rationality and Poincare families for vector bundles with extra structure on a curve
• Moduli for decorated tuples of sheaves and representation spaces for quivers