Compact complex homogeneous manifolds with large automorphism groups (Q1127868)

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 \(\dim\Aut(X)\leq n(n+2)\) with equality only when \(X\) is complex projective space. It is an old question raised by Remmert whether this is also true in the non-Kähler case. In this article we answer this question by providing examples of non-Kähler compact complex homogeneous manifolds \(X\) for which \(\dim\Aut(X)\) depends exponentially on \(n\).