This is a joint work with Professors Tien-Cuong Dinh and Xun Yu. The real form problem asks how many different ways one can describe a given complex variety by polynomial equations with real coefficients up to isomorphisms over the real number field. For instance, the complex projective line has exactly two real forms up to isomorphisms. This problem is in the limelight again after a breakthrough work due to Lesieutre in 2018.

In this talk, among other relevant things, we would like to show that in each dimension greater than or equal to two, there is a smooth complex projective rational variety with infinitely many real forms. This answers a question of Kharlamov in 1999.