Let X be a compact complex homogeneous manifold and let Aut(X) be the complex Lie group of holomorphic automorphisms of X. It is well-known that the dimension of Aut(X) is bounded by an integer that depends only on n = dim X. Moreover, if X is Kähler then dim Aut(X)<= n(n + 2) with equality only when X is complex projective space. It has been an old question raised by Remmert whether this is true for non Kähler manifolds as well. In this article we give a negative answer to this question: We construct sequences of non-Kähler compact complex homogeneous manifolds X for which dim Aut(X) depends exponentially on dim X.
Inventiones mathematicae134, 139 - 144 (1998)