According to Professor Igor Dolgachev, it has been a long standing problem if there is a smooth projective surface 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 reviewing some known results (for finite generation), I would like to show the following answers:

There is a smooth complex projective surface, birational to some K3 surface, such that the automorphism group is (discrete and) non-finitely generated and with infinitely many non-isomorphic real forms (joint work with Professor Dinh). We also would like to discuss positive characteristic case.