Family of intersecting totally real manifolds of $(\Bbb C^n,0)$ and CR-singularities

Abstract

The first part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $(\Bbb C^n,0)$. We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which interesection with the family can be holomorphically staightened. The second part is devoted to the study real analytic $(n+r)$-submanifolds of $(\Bbb C^n,0)$ having a CR-singularity at the origin ($r$ is a nonnegative integer). We consider deformations of quadrics and we define generalized Bishop invariants. Such a quadric intersects the complex linear manifold ${z_{p+1}=...=z_n=0}$ along some real linear set ${\cal L}$. We study what happens to this intersection when the quadric is analytically perturbed. On the other hand, we show, under some assumptions, that if such a submanifold is formally equivalent to its associated quadric then it is holomorphically equivalent to it. All these results rely on a result stating the existence (and caracterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms (not tangent to the identity at the origin).