A modification of Poincare's construction and its application to the CR geometry of hypersurfaces in ${\bf C}^4$

TL;DR

This paper extends Poincare's construction to estimate the automorphism Lie algebra dimension of real analytic hypersurfaces in ${f C}^4$, providing bounds that distinguish between degenerate and nondegenerate cases.

Contribution

It generalizes the homological Poincare's operator to derive new bounds on the automorphism algebra dimension for hypersurfaces in ${f C}^4$, including special cases.

Findings

Dimension is either infinite or at most 24, with 24 only for specific hyperquadrics.

For 2-nondegenerate hypersurfaces, the bound is 17.

For 3-nondegenerate hypersurfaces, the bound is 20.

Abstract

A generalization of the homological Poincare's operator was used to estimate the dimension of the Lie algebra of infinitesimal holomorphic automorphisms of an arbitrary germ of a real analytic hypersurface in ${\bf C}^4$. The following alternative is proved: either this dimension is infinite, or it does not exceed 24. Value 24 takes place only for one of two nondegenerate hyperquadrics. If the hypersurface is 2-nondegenerate at a generic point, then the dimension does not exceed 17, and if the hypersurface is 3-nondegenerate at a generic point, then the estimate is 20.