Kähler manifolds with geodesic holomorphic gradients

• Andrzej Derdzinski ^[1] ; Paolo Piccione ^[2]
  1. [1] Ohio State University

     Ohio State University

     City of Columbus, Estados Unidos

     
  2. [2] Universidade de São Paulo

     Universidade de São Paulo

     Brasil

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 5, 2020, págs. 1489-1526
• Idioma: inglés
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• Resumen

  • 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.