Boundedness of finite morphisms onto Fano manifolds with large Fano index

Abstract

Let f:Y→X be a finite morphism between Fano manifolds Y and X such that the Fano index of X is greater than 1. On the one hand, when both X and Y are fourfolds of Picard number 1, we show that the degree of f is bounded in terms of X and Y unless X≅P4; hence, such X does not admit any non-isomorphic surjective endomorphism. On the other hand, when X=Y is either a fourfold or a del Pezzo manifold, we prove that, if f is an int-amplified endomorphism, then X is toric. Moreover, we classify all the singular quadrics admitting non-isomorphic endomorphisms.