  We show that there is a smooth complex projective variety, of any dimension
greater than or equal to two, whose automorphism group is discrete and not
finitely generated. Moreover, this variety admits infinitely many real forms
which are mutually non-isomorphic over the real number field. Our result is
inspired by the work of Lesieutre and answers questions by Dolgachev, Esnault
and Lesieutre.
