Boundedness of finite morphisms onto Fano manifolds with large Fano index

TL;DR

This paper investigates the boundedness and classification of finite morphisms and endomorphisms on Fano manifolds with large Fano index, revealing conditions under which these morphisms are limited or lead to toric structures.

Contribution

It establishes bounds on the degree of finite morphisms between certain Fano fourfolds and classifies singular quadrics with non-isomorphic endomorphisms, advancing understanding of morphisms on Fano manifolds.

Findings

Degree of morphisms bounded unless X is projective space

X admits no non-isomorphic surjective endomorphisms when Picard number is 1

X is toric if endomorphism is int-amplified on certain Fano manifolds

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.