### Regularization of birational group operations in sense of Weil.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

We survey results on unique determination of local $CR$-automorphisms of smooth $CR$-manifolds and of local biholomorphisms of real-analytic $CR$-submanifolds of complex spaces by their jets of finite order at a given point. Examples generalizing [28] are given showing that the required jet order may be arbitrarily high.

We present a large class of homogeneous 2-nondegenerate CR-manifolds $M$, both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains $U$, $V$ in $M$ extends to a global real-analytic CR-automorphism of $M$. We show that this class contains $G$-orbits in Hermitian symmetric spaces $Z$ of compact type, where $G$ is a real form of the complex Lie group $Aut{\left(Z\right)}^{0}$ and $G$ has an open orbit that is a bounded symmetric domain of tube type.

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in ${C}^{n}$ whose differentials have one-dimensional family of resonances in the first $m$ eigenvalues, $m\le n$ (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.

We show the uniqueness of local and global decompositions of abstract CR-manifolds into direct products of irreducible factors, and a splitting property for their CR-diffeomorphisms into direct products with respect to these decompositions. The assumptions on the manifolds are finite non-degeneracy and finite-type on a dense subset. In the real-analytic case, these are the standard assumptions that appear in many other questions. In the smooth case, the assumptions cannot be weakened by replacing...

**Page 1**