Smooth rational projective varieties with non-finitely generated discrete automorphism group and infinitely many real forms

We show, among other things, that for each integer $n\ge 3$, there is a smooth complex projective rational variety of dimension $n$, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.