Smooth projective surfaces with infinitely many real forms

TL;DR

This paper investigates conditions under which smooth complex projective surfaces have finitely or infinitely many real forms, confirming criteria for finiteness and constructing examples with infinitely many real forms, especially among Enriques surfaces.

Contribution

It establishes criteria for the finiteness of real forms of smooth projective surfaces and constructs explicit examples of surfaces with infinitely many real forms.

Findings

Finiteness of real forms is linked to automorphism groups, cone conjecture, and entropy.

Most surfaces have finitely many real forms unless they are rational or certain birational types.

Constructed an Enriques surface with infinitely many real forms, answering Kondo's question.

Abstract

The aim of this paper is twofold. First of all, we confirm a few basic criteria of the finiteness of real forms of a given smooth complex projective variety, in terms of the Galois cohomology set of the discrete part of the automorphism group, the cone conjecture and the topological entropy. We then apply them to show that a smooth complex projective surface has at most finitely many non-isomorphic real forms unless it is either rational or a non-minimal surface birational to either a K3 surface or an Enriques surface. In the second part of the paper, we construct an Enriques surface whose blow-up at one point admits infinitely many non-isomorphic real forms. This answers a question of Kondo to us and also shows the three exceptional cases really occur.