Smooth projective surfaces with discrete and non-finitely generated automorphism group

According to Professor Igor Dolgachev, it has been a long standing problem if there is a smooth projective variety, especially surfaces, such that the automorphism group is discrete and non-finitely generated and/or it has infinitely many non-isomorphic real forms.

In this talk, after recalling previously known results, I would like to explain current status of this problem with an idea of construction and sensitive aspect of the field of definitions. My talk is based on my joint works with T.C. Dinh, J.H. Keum and by myself.