Publication: Moduli spaces of connections on a Riemann surface.
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2010
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International Press
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Abstract
Let E be a holomorphic vector bundle over a compact connected Riemann surface X. The vector bundle E admits a holomorphic projective connection if and only if for every holomorphic direct summand F of E of positive rank, the equality degree(E)=rank(E) = degree(F)=rank(F) holds. Fix a point x0 in X. There is a logarithmic connection on E, singular over x0 with residue ¡d n IdEx0 if and only if the
equality degree(E)=rank(E) = degree(F)=rank(F) holds. Fix an integer n ¸ 2, and also ¯x an integer d coprime to n. Let M(n; d) denote the moduli space of logarithmic
SL(n;C){connections on X singular of x0 with residue ¡ d n
Id. The isomorphism class of the variety M(n; d) determines the isomorphism class of the Riemann surface X.
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Proceedings of the U.S.-India workshop in Teichmüller theory and moduli problems,Harish-Chandra Research Institute, Allahabad, January 2006
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