Compact complex homogeneous manifolds with large automorphism groups

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 K?hler then dimAut (X)≤n(n+2) with equality only when X is complex projective space. In this article examples of non-K?hler compact complex homogeneous manifolds X are given that demonstrate dimAut(X) can depend exponentially on n. Let X be a connected compact complex manifold of dimension n. The group of holomorphic automorphisms of X, Aut(X), is a complex Lie group [3]. For a fixed n>1, the dimension of Aut(X) can be arbitrarily large compared to n. Simple examples are provided by the Hirzebruch surfaces F _m , m∈N, for which dimAut(F _m )=m+5, see, e.g. [2, Example 2.4.2]. If X is homogeneous, that is, any point of X can be mapped to any other point of X under a holomorphic automorphism, then the dimension of the automorphism group of X is bounded by an integer that depends only on n, see [1, 2, 6]. The estimate given in [2, Theorem 3.8.2] is roughly dimAut(X)≤(n+2) ⁿ . For many classes of manifolds, however, the dimension of the automorphism group never exceeds n(n+2). For example, it follows directly from the classification given by Borel and Remmert [4], that if X is a compact homogeneous K?hler manifold, then dimAut(X)≤n(n+2) with equality only when X is complex projective space P ⁿ . It is an old question raised by Remmert, see [2, p. 99], [6], whether this same bound applies to all compact complex homogeneous manifolds. In this note we show that this is not the case by constructing non-K?hler compact complex homogeneous manifolds whose automorphism group has a dimension that depends exponentially on n. The simplest case among these examples has n=3m+1 and dimAut(X)=3m+3 ^m , so the above conjectured bound is exceeded when n≥19. These manifolds have the structure of non-trivial fiber bundles over products of flag manifolds with parallelizable fibers given as the quotient of a solvable group by a discrete subgroup. They are constructed using the original ideas of Otte [6, 7] and are surprisingly similar to examples found there. Generally, a product of manifolds does not result in an automorphism group with a large dimension relative to n. Nevertheless, products are used in an essential way in the construction given here, and it is perhaps this feature that caused such examples to be previously overlooked. Oblatum 13-X-97 & 24-X-1997