Kähler manifolds with geodesic holomorphic gradients

Abstract

A vector field on a Riemannian manifold is called geodesic if its integral curves are reparametrized geodesics. We classify compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields that satisfy an additional integrability condition. They are all biholomorphic to bundles of complex projective spaces.

52 pages