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

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