Endomorphisms of singular del Pezzo surfaces

Abstract

A natural problem of algebraic dynamics is to classify the complex projective varieties that admit an endomorphism of degree greater than 1. Joshi solved the problem for all canonical del Pezzo surfaces with Picard number 1 except one, a surface with a du Val singularity of type $E_8$. The method of Bott vanishing does not resolve this case. We show here that the $E_8$ surface has no endomorphism of degree greater than 1. For the proof, we extend the method of Amerik-Rovinsky-Van de Ven, involving Chern number inequalities, from varieties to Deligne-Mumford stacks. This approach should be useful for other hard cases in the classification of varieties with endomorphisms.