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 $\mathbb P^2_{\mathbb C}$ at $r\geq 10$ 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 $\mathbb R$.