Torelli theorem for the moduli spaces of connections on a Riemann surface

2005/12/12 by Indranil Biswas, Biswas, Indranil, Vicente Munoz +1
Mathematics · #14C34 #14D20 #14H60 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #math.AG #math.DG #msc:14C34 #msc:14D20 #msc:14H60

paper · pdf · doi:10.48550/arxiv.math/0512236

25 pages, no figures; v2. Revised version. To appear in Topology

arxiv created 2007/02/05 · arxiv updated 2009/12/01

Abstract

Let (X,x₀) be any one--pointed compact connected Riemann surface of genus g, with g≥ 3. Fix two mutually coprime integers r>1 and d. Let \mathcal M_X denote the moduli space parametrizing all logarithmic SL(r,\mathbb C)--connections, singular over x₀, on vector bundles over X of degree d. We prove that the isomorphism class of the variety \mathcal M_X determines the Riemann surface X uniquely up to an isomorphism, although the biholomorphism class of \mathcal M_X is known to be independent of the complex structure of X. The isomorphism class of the variety \mathcal M_X is independent of the point x₀ ∈ X. A similar result is proved for the moduli space parametrizing logarithmic GL(r,\mathbb C)--connections, singular over x₀, on vector bundles over X of degree d.

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