Effective Actions of the Unitary Group on Complex Manifolds
نویسنده
چکیده
We are interested in classifying all connected complex manifolds M of dimension n ≥ 2 admitting effective actions of the unitary group Un by biholomorphic transformations. One motivation for our study was the following question that we learned from S. Krantz: assume that the group Aut(M) of all biholomorphic automorphisms of M and the group Aut(C) of all biholomorphic automorphisms of C are isomorphic as topological groups equipped with the compact-open topology; does it imply that M is biholomorphically equivalent to C? The group Aut(C) is very large (see, e.g., [AL]), and it is not that clear from the start what automorphisms of C one can use to approach the problem. The isomorphism between Aut(M) and Aut(C) induces a continuous effective action on M of any subgroup G ⊂ Aut(C). If G is a Lie group, then this action is in fact real-analytic. We consider G = Un which, as it turns out, results in a very short list of manifolds that can occur. In Section 1 we find all possible dimensions of orbits of a Un-action on M . It turns out (see Proposition 1.1) that an orbit is either a point (hence Un has a fixed point in M), or a real hypersurface in M , or a complex hypersurface in M , or the whole of M (in which case M is homogeneous). Manifolds admitting an action with fixed point were found in [K] (see Remark 1.2). In Section 2 we classify manifolds with a Un-action such that all orbits are real hypersurfaces. We show that such a manifold is either a spherical
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