Universal Models For Real Submanifolds

V. K. Beloshapka

Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 507-522

Résumé

In previous papers by the present author, a machinery for calculating automorphisms, constructing invariants, and classifying real submanifolds of a complex manifold was developed. The main step in this machinery is the construction of a “nice” model surface. The nice model surface can be treated as an analog of the osculating paraboloid in classical differential geometry. Model surfaces suggested earlier possess a complete list of the desired properties only if some upper estimate for the codimension of the submanifold is satisfied. If this estimate fails, then the surfaces lose the universality property (that is, the ability to touch any germ in an appropriate way), which restricts their applicability. In the present paper, we get rid of this restriction: for an arbitrary type $(n,K)$ (where $n$ is the dimension of the complex tangent plane, and $K$ is the real codimension), we construct a nice model surface. In particular, we solve the problem of constructing a nondegenerate germ of a real analytic submanifold of a complex manifold of arbitrary given type $(n,K)$ with the richest possible group of holomorphic automorphisms in the given class.