Wronskian curvature positivity is equivalent to the existence of a holomorphic projective connection

Abstract

Noguchi introduced the notion of Wronskian curvature positivity in his Second Main Theorem and asked for further examples to which his theorem applies. We give a complete answer: a complex manifold admits a smooth connection satisfying this condition if and only if it admits a holomorphic projective connection. For a fixed torsion-free connection, the condition is equivalent to the holomorphicity of its projective class.