Criterion for logarithmic connections with prescribed residues

2017/03/29 by Biswas, Indranil, Dan, Ananyo, Paul, Arjun · 1 citation
#14H60 #53B15 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1703.09864

Abstract

A theorem of Weil and Atiyah says that a holomorphic vector bundle E on a compact Riemann surface X admits a holomorphic connection if and only if the degree of every direct summand of E is zero. Fix a finite subset S of X, and fix an endomorphism A(x) ∈ End(E_x) for every x ∈ S. It is natural to ask when there is a logarithmic connection on E singular over S with residue A(x) at every x ∈ S. We give a necessary and sufficient condition for it under the assumption that the residues A(x) are rigid.

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