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

Abstract

The first part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $({\mathbb{C}}^n,0)$. 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 $({\mathbb{C}}^n,0)$ 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 $\mathcal{L}$. 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).

34 pages

Cited by in corpus (1)

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