Tameness of holomorphic closure dimension in a semialgebraic set

TL;DR

This paper proves that the holomorphic closure of a semialgebraic set in complex space is a Nash germ and introduces a semialgebraic stratification of such sets into CR manifolds with a strong frontier condition.

Contribution

It establishes the Nash germ nature of holomorphic closures for semialgebraic sets and constructs a semialgebraic stratification into CR manifolds.

Findings

Holomorphic closure of semialgebraic sets is a Nash germ.

Semialgebraic sets admit a stratification into CR manifolds.

Holomorphic closure dimension varies semialgebraically.

Abstract

Given a semianalytic set S in a complex space and a point p in S, there is a unique smallest complex-analytic germ at p which contains the germ of S, called the holomorphic closure of S at p. We show that if S is semialgebraic then its holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic filtration by the holomorphic closure dimension. As a consequence, every semialgebraic subset of a complex vector space admits a semialgebraic stratification into CR manifolds satisfying a strong version of the condition of the frontier.