Hyperbolic Manifolds of Dimension n with Automorphism Group of Dimension

We consider complex Kobayashi-hyperbolic manifolds of dimension n ≥ 2 for which the dimension of the group of holomorphic au-tomorphisms is equal to n 2 − 1. We give a complete classification of such manifolds for n ≥ 3 and discuss several examples for n = 2. 0 Introduction Let M be a connected complex manifold and Aut(M) the group of holomor-phic automorphisms of M. If M is Kobayashi-hyperbolic, Aut(M) is a Lie group in the compact-open topology [Ko], [Ka]. Let d(M) := dim Aut(M). It is well-known (see [Ko], [Ka]) that d(M) ≤ n 2 + 2n, and that d(M) = n 2 + 2n if and only if M is holomorphically equivalent to the unit ball B n ⊂ C n , where n := dim C M. In [IKra] we studied lower automorphism group dimensions and showed that, for n ≥ 2, there exist no hyperbolic manifolds with n 2 + 3 ≤ d(M) ≤ n 2 + 2n − 1, and that the only manifolds with n 2 < d(M) ≤ n 2 + 2 are, up to holomorphic equivalence, B n−1 × ∆ (where ∆ is the unit disc in C) and the 3-dimensional Siegel space (the symmetric bounded domain of type (III 2) in C 3). Further, in [I1] all manifolds with d(M) = n 2 were determined (for partial classifications in special cases see also [GIK] and [KV]). The classification in this situation is substantially richer than that for higher automorphism group dimensions. Observe that a further decrease in d(M) almost immediately leads to unclassifiable cases. For example, no good classification exists for n = 2 and d(M) = 2, since the automorphism group of a generic Reinhardt domain in C 2 is 2-dimensional (see also [I1] for a more specific statement). While it is possible that there is some classification for d(M) = n 2 − 2, n ≥ 3 as well as for particular pairs d(M), n with d(M) < n 2 − 2 (see [GIK] in this