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On the rational homotopy type of a moduli space of vector bundles over a curve

Biswas, Indrani and Muñoz, Vicente (2008) On the rational homotopy type of a moduli space of vector bundles over a curve. Communications in Analysis and Geometry, 15 (1). pp. 183-215. ISSN 1019-8385

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Abstract

We study the rational homotopy of the moduli space N-X that parametrizes the isomorphism classes of all stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g, with g >= 2. The symplectic group Aut(H-1(X, Z)) congruent to Sp(2g, Z) has a natural action on the rational homotopy groups pi(n)(N-X)circle times(Z)Q. We prove that this action extends to an action of Sp(2g, C) on pi(n)(N-X)circle times C-Z. We also show that pi(n)(N-X)circle times C-Z is a non-trivial representation of Sp(2g, C) congruent to Aut (H-1(X, C)) for all n >= 2g - 1. In particular, N-X is a rationally hyperbolic space. In the special case where g = 2, for each n is an element of N, we compute the leading Sp(2g, C) representation occurring in pi(n)(N-X)circle times C-Z.


Item Type:Article
Uncontrolled Keywords:Vector bundles; Moduli space; Smooth projective curve;Riemann surface;Rational homotopy groups
Subjects:Sciences > Mathematics > Algebraic geometry
Sciences > Mathematics > Topology
ID Code:17141
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