Endomorphisms of projective varieties

TL;DR

This paper investigates the structure of complex projective manifolds with surjective endomorphisms of degree at least two, providing classification results, especially for uniruled threefolds and Fano manifolds, and proposing conjectures about projective space.

Contribution

It offers new structure theorems for endomorphisms on projective varieties, characterizes projective space via tangent bundle properties, and advances the understanding of ramified endomorphisms on Fano manifolds.

Findings

Structure theorems for etale endomorphisms of projective manifolds

Ample vector bundles associated with Galois coverings

Partial proof of the conjecture that only projective space admits endomorphisms of degree >1

Abstract

We study complex projective manifolds X that admit surjective endomorphisms f:X->X of degree at least two. In case f is etale, we prove structure theorems that describe X. In particular, a rather detailed description is given if X is a uniruled threefold. As to the ramified case, we first prove a general theorem stating that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This is applied to the study of ramified endomorphisms of Fano manifolds with second Betti number one. It is conjectured that the projective space is the only Fano manifold admitting admitting an endomorphism of degree d>1, and we prove that in several cases. A part of the argumentation is based on a new characterization of the projective space as the only manifold that admits an ample subsheaf in its tangent bundle.