On the holomorphic closure dimension of real analytic sets

TL;DR

This paper investigates the holomorphic closure dimension of real analytic sets in complex spaces, establishing its constancy outside a proper analytic subset and exploring its relation to CR dimension.

Contribution

It introduces the concept of holomorphic closure dimension for real analytic sets and proves its constancy on large parts of the set, linking it to CR dimension.

Findings

Holomorphic closure dimension is constant outside a proper analytic subset.

The paper establishes a relationship between holomorphic closure dimension and CR dimension.

Provides a framework for understanding complex analytic germs containing real analytic sets.

Abstract

Given a real analytic (or, more generally, semianalytic) set R in the n-dimensional complex space, there is, for every point p in the closure of R, a unique smallest complex analytic germ X_p that contains the germ R_p. We call the complex dimension of X_p the holomorphic closure dimension of R at p. We show that the holomorphic closure dimension of an irreducible R is constant on the complement of a closed proper analytic subset of R, and discuss the relationship between this dimension and the CR dimension of R.