Galois coverings and endomorphisms of projective varieties

We prove that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This results is applied to the study of ramified endomorphisms of Fano manifolds with b ₂ = 1. It is conjectured that is the only Fano manifold admitting an endomorphism of degree d ≥ 2, and we verify this conjecture in several cases. An important ingredient is a generalization of a theorem of Andreatta–Wisniewski, characterizing projective space via the existence of an ample subsheaf in the tangent bundle. Marian Aprodu was supported in part by a Humboldt Research Fellowship and a Humboldt Return Fellowship. He expresses his special thanks to the Mathematical Institute of Bayreuth University for hospitality during the first stage of this work. Stefan Kebekus and Thomas Peternell were supported by the DFG-Schwerpunkt “Globale Methoden in der komplexen Geometrie” and the DFG-Forschergruppe “Classification of Algebraic Surfaces and Compact Complex Manifolds”. A part of this paper was worked out while Stefan Kebekus visited the Korea Institute for Advanced Study. He would like to thank Jun-Muk Hwang for the invitation.