Real structures on rational surfaces and automorphisms acting trivially on Picard groups

Abstract

In this article, we prove that any complex smooth rational surface $X$ which has no automorphism of positive entropy has a finite number of real forms (this is especially the case if $X$ cannot be obtained by blowing up ${\mathbb{P}}_{\mathbb{C}}^2$ at $r\ge 10$ points). In particular, we prove that the group ${\mathrm{Aut}}^{\#}X$ of complex automorphisms of $X$ which act trivially on the Picard group of $X$ is a linear algebraic group defined over $\mathbb{R}$.

Minor changes. To appear in Mathematische Zeitschrift

Cited by in corpus (3)

• Finiteness of Klein actions and real structures on compact hyperkähler manifolds
• A surface with discrete and non-finitely generated automorphism group
• Smooth complex projective rational surfaces with infinitely many real forms