Real structures on rational surfaces and automorphisms acting trivially on Picard groups (Q269913)

Recall that a \textit{real structure} on a complex analytic variety is an anti-holomorphic involution. Most algebraic surfaces are known to admit but finitely many \textit{real forms}, i.e., isomorphism classes of real structures. Surprisingly, this finiteness question is still open for rational surfaces. The author shows that, if a rational surface~\(X\) admits infinitely many real forms, then \(X\) is a blow-up of \(\mathbb P^2\) at at least \(10\) points and has an automorphism~\(\varphi\) inducing an automorphism of positive topological entropy on \(\text{Pic}X\otimes_{\mathbb Z}{\mathbb R}\). The proof uses an algebraic restatement of the latter condition, case-by-case analysis, and a few previously known finiteness results. In the last section, the main theorem is extended to a few other classes of rational surfaces.