Boundedness of finite morphisms onto Fano manifolds with large Fano index

Abstract

Let $f:Y\to X$ 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 $X\cong {\mathbb{P}}^4$; 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.

30 pages, minor revisions, Journal of Algebra (to appear), comments are welcome!

References in corpus (6)

• Questions on self maps of algebraic varieties
• Rigidity of rationally connected smooth projective varieties from dynamical viewpoints
• Non-isomorphic endomorphisms of Fano threefolds
• Amplified endomorphisms of Fano fourfolds
• Totally invariant divisors of int-amplified endomorphisms of normal projective varieties
• Double lines in the quintic del Pezzo fourfold