A smooth complex rational affine surface with uncountably many nonisomorphic real forms

A real form of a complex algebraic variety $X$ is a real algebraic variety whose complexification is isomorphic to $X$. Up until recently, it was known that many families of complex varieties have a finite number of nonisomorphic real forms. In 2019, Lesieutre constructed an example of a projective variety of dimension six with infinitely many, and now, Dinh, Oguiso and Yu found a projective rational surface with infinitely many as well. In this talk, I’ll present the first example of a rational affine surface having uncountably many nonisomorphic real forms. The first example with infinitely countably many real forms on an affine rational variety is due to Dubouloz, Freudenberg and Moser-Jauslin.