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

2006/05/19 by Indranil Biswas, Vicente Muñoz, Biswas, Indranil +1
Mathematics · #14H60 #55P62 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #math.AT #math.DG #msc:14H60 #msc:55P62

paper · pdf · doi:10.48550/arxiv.math/0605542

27 pages, no figures; v2. final version. To appear in Comm. Analysis and Geom

openalex publication_date 2006/05/19 · arxiv created 2007/10/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the rational homotopy of the moduli space \mathcal N_X of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g≥ 2. The symplectic group Aut(H₁(X,\mathbb Z))=Sp(2g,\mathbb Z) has a natural action on the rational homotopy groups π_n(\mathcal N_X) ⊗ \mathbb Q. We prove that this action extends to an action of Sp(2g,\mathbb C) on π_n(\mathcal N_X) ⊗ \mathbb C. We also show that π_n(\mathcal N_X) ⊗ \mathbb C is a non-trivial Sp(2g,\mathbb C)-representation for each n≥ 2g-1. In particular, \mathcal N_X is a rationally hyperbolic space. In the special case where g=2, we compute the leading Sp(2g,\mathbb C)-representation occurring in π_n(\mathcal N_X) ⊗ \mathbb C, for each n.

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