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

TL;DR

This paper proves that complex smooth rational surfaces without automorphisms of positive entropy have finitely many real forms and that automorphisms acting trivially on the Picard group form a linear algebraic group over the reals.

Contribution

It establishes finiteness of real forms for certain rational surfaces and characterizes the automorphism group acting trivially on the Picard group as a real algebraic group.

Findings

Finite number of real forms for specific rational surfaces.

Automorphism group acting trivially on Picard group is a real algebraic group.

Results apply especially when the surface is not obtained by blowing up 10 or more points in P^2.

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 $\mathbb R$.