Real and complex analytic sets. The relevance of Segre varieties
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 3, p. 447-454

Let $X\subset {{ℂ}^{n}}^{n}$ be a closed real-analytic subset and put $𝒜:=\left\{z\in X\mid \exists \phantom{\rule{4pt}{0ex}}A\subset X,\phantom{\rule{4pt}{0ex}}\text{germ}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{4pt}{0ex}}\text{a}\phantom{\rule{4pt}{0ex}}\text{complex-analytic}\phantom{\rule{4pt}{0ex}}\text{set,}\phantom{\rule{4pt}{0ex}}z\in A,\phantom{\rule{0.166667em}{0ex}}{dim}_{z}A>0\right\}$ This article deals with the question of the structure of $𝒜$. In the main result a natural proof is given for the fact, that $𝒜$ always is closed. As a main tool an interesting relation between complex analytic subsets of $X$ of positive dimension and the Segre varieties of $X$ is proved and exploited.

Classification:  32B10,  32C07
@article{ASNSP_2008_5_7_3_447_0,
author = {Diederich, Klas and Mazzilli, Emmanuel},
title = {Real and complex analytic sets. The relevance of Segre varieties},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 7},
number = {3},
year = {2008},
pages = {447-454},
zbl = {1178.32006},
mrnumber = {2466436},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2008_5_7_3_447_0}
}

Diederich, Klas; Mazzilli, Emmanuel. Real and complex analytic sets. The relevance of Segre varieties. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 3, pp. 447-454. http://www.numdam.org/item/ASNSP_2008_5_7_3_447_0/

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