Boundaries of analytic varieties

TL;DR

This paper proves that smooth CR hypersurface manifolds in complex space can be extended to complex varieties with boundary, with properties depending on pseudoconvexity and orientation, generalizing classical boundary theorems.

Contribution

It establishes the existence of complex varieties bounded by smooth CR hypersurfaces, extending previous results to include pseudoconvex and embedded cases, with detailed singularity analysis.

Findings

Every smooth CR hypersurface has a complex strip-manifold extension.

Pseudoconvex-oriented hypersurfaces form the boundary of an embedded variety.

Singularities of the variety are isolated when the hypersurface is pseudoconvex oriented.

Abstract

We prove that every smooth CR manifold $M\subset\subset \C^n$, of hypersurface type, has a complex strip-manifold extension in $\C^n$. If $M$ is, in addition, pseudoconvex-oriented, it is the "exterior" boundary of the strip. In turn, the strip extends to a variety with boundary $M$ (Rothstein-Sperling Theorem); in case $M$ is contained in a pseudoconvex boundary with no complex tangencies, the variety is embedded in $\C^n$. Altogether we get: $M$ is the boundary of a variety (Harvey-Lawson Theorem); if $M$ is pseudoconvex oriented the singularities of the variety are isolated in the interior; if $M$ lies in a pseudoconvex boundary, the variety is embedded in $\C^n$ (and is still smooth at $M$)