Blow-ups and infinitesimal automorphisms of CR-manifolds

TL;DR

This paper classifies the symmetry algebras of real-analytic CR-hypersurfaces with Levi-nondegeneracy, establishing bounds on their dimensions and constructing explicit examples, including models with large symmetry via blow-up techniques.

Contribution

It provides a classification of symmetry algebras for CR-hypersurfaces, including explicit examples and the use of blow-up methods to realize large symmetry groups.

Findings

Either the symmetry algebra dimension reaches the maximal value for spherical hypersurfaces.

Bound on the symmetry algebra dimension for non-spherical hypersurfaces.

Construction of explicit models with large symmetry groups using blow-up techniques.

Abstract

Let $M$ be a real-analytic connected CR-hypersurface of CR-dimension $n>0$ having a point of Levi-nondegeneracy. The following alternative is demonstrated for both the symmetry algebra $s$ and the automorphism group $G$ of $M$. Denote by $d$ the dimension of $s$ or $G$. Then (i) either $d=n^2+4n+3$ and $M$ is spherical everywhere; (ii) or $d\le n^2+2n+2+\delta_{2,n}$ and in the case of equality $M$ is spherical of fixed Levi signature in the open dense subset of Levi-nondegenerate points. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.