Tameness of complex dimension in a real analytic set (Q2841815)

Let \(X\) be a closed real analytic subset of an open set in \(\mathbb C^n\). Let \(\mathcal A^d\) be the set of points \(p \in X\) such that the germ of \(X\) at \(p\) contains a complex analytic germ of dimension \(d\). It is proved that \(\mathcal A^d\) is a closed semianalytic subset of \(X\), for every integer \(d \geq 1\). If \(X\) is real algebraic, then \(\mathcal A^d\) is semialgebraic in \(X\). The set \(\mathcal A^d\) is not real analytic in general. An important tool in the proof are the properties of Segre varieties. If \(\rho(z, \overline z)\) is a real analytic function on some open polydisc \(V\) in \(\mathbb C^n\) defining \(X\) in \(V\), then the Segre variety of \(w \in V\) is the set \(S_w = \{z \in V: \rho(z, \overline w)=0\}\).NEWLINENEWLINEIn the appendix, the authors show that the set of points of infinite D'Angelo type coincides with \(\mathcal A^1\). A proof of this fact had been given in [\textit{J. P. D'Angelo}, Several complex variables and the geometry of real hypersurfaces. Boca Raton, FL: CRC Press (1993; Zbl 0854.32001)], but its validity was questioned in [\textit{K. Diederich} and \textit{E. Mazzilli}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 7, No. 3, 447--454 (2008; Zbl 1178.32006)].