Tameness of complex dimension in a real analytic set

TL;DR

This paper proves that for a real analytic set in a complex manifold, the set of points where a complex analytic germ of a fixed dimension exists is a closed semianalytic subset, revealing structural properties of such sets.

Contribution

It establishes that the set of points with a complex analytic germ of fixed dimension in a real analytic set is a closed semianalytic subset, advancing understanding of their geometric structure.

Findings

A(d) is a closed semianalytic subset of X.

The structure of points with complex analytic germs is well-characterized.

Provides a new insight into the geometry of real analytic sets in complex manifolds.

Abstract

Given a real analytic set X in a complex manifold and a positive integer d, denote by A(d) the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that A(d) is a closed semianalytic subset of X.