We prove a dichotomy theorem about compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields, which has the following consequence: either such a manifold satisfies an additional integrability condition, or through every zero of the real-holomorphic geodesic gradient there passes an uncountable family of totally geodesic, holomorphically immersed complex projective spaces, each carrying a fixed multiple of the Fubini–Study metric. We also obtain a classification result for the case where the integrability condition holds, implying that the manifold must then be biholomorphically isometric to a bundle of complex projective spaces with a bundle-like metric.
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Andrzej Derdzinski, Paolo Piccione, Kähler manifolds with geodesic holomorphic gradients. Rev. Mat. Iberoam. 36 (2020), no. 5, pp. 1489–1526DOI 10.4171/RMI/1173