Smooth projective surfaces with infinitely many real forms

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.
Comment: v2: the structure of the paper has been changed to improve the exposition; the blow-up at one point of an Enriques surface with infinitely many real forms that we construct is now proved to have non-finitely generated automorphism group