Moduli spaces of vector bundles on a curve and opers

2023/03/17 by Biswas, Indranil, Hurtubise, Jacques, Roubtsov, Vladimir
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2303.09701

Abstract

Let X be a compact connected Riemann surface of genus g, with g ≥ 2, and let ξ be a holomorphic line bundle on X with ξ^{⊗ 2} = \mathcal O_X. Fix a theta characteristic \mathbb L on X. Let \mathcal M_X(r,ξ) be the moduli space of stable vector bundles E on X of rank r such that \bigwedge^r E = ξ and H⁰(X, E⊗\mathbb L) = 0. Consider the quotient of \mathcal M_X(r,ξ) by the involution given by E \longmapsto E^*. We construct an algebraic morphism from this quotient to the moduli space of \rm SL(r,\mathbb C) opers on X. Since dim \mathcal M_X(r,ξ) coincides with the dimension of the moduli space of \rm SL(r,\mathbb C) opers, it is natural to ask about the injectivity and surjectivity of this map.

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