Real and complex analytic sets. The relevance of Segre varieties

Diederich, Klas; Mazzilli, Emmanuel

Diederich, Klas; Mazzilli, Emmanuel

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 3, p. 447-454

Abstract

Let $X \subset {ℂ^{n}}^{n}$ be a closed real-analytic subset and put $𝒜 : = {z \in X ∣ \exists A \subset X, germ of a complex-analytic set, z \in A, {dim}_{z} A > 0}$ 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.

MR 2466436 | Zbl 1178.32006

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/

References

[1] E. Bishop, Condition for the analyticity of certain sets, Michigan Math. J. 11 (1964), 289-304. | MR 168801 | Zbl 0143.30302

[2] E. M. Chirka, “Complex Analytic Sets”, Kluwer, Dordrecht, 1990. | MR 1111477 | Zbl 0683.32002

[3] J. D'Angelo Real hypersurfaces, orders of contact, and applications, Ann. of Math. 115 (1982), 615-637. | MR 657241 | Zbl 0488.32008

[4] J. P. D'Angelo, “Several Complex Variables and the Geometry of Real Hypersurfaces”, CRC Press, Boca Raton, FL, 1993. | MR 1224231 | Zbl 0854.32001

[5] K. Diederich and J. E. Fornæss, Pseudoconvex domains with real analytic boundary, Ann. of Math. 107 (1978), 371-384. | MR 477153 | Zbl 0378.32014

[6] K. Diederich and S. Pinchuk, Uniform volume estimates for holomorphic families of analytic sets, Proc. Steklov Inst. Math., 235 (2001), 52-56. | MR 1886572 | Zbl 1005.32012

[7] J. Kohn, Subellipticity of the $\bar{\partial}$ -Neumann problem on pseudoconvex domains: Sufficient conditions, Acta. Math. 142 (1979), 79-122. | MR 512213 | Zbl 0395.35069

[8] E. Mazzilli, Germes d'ensembles analytiques dans une hypersurface algébrique, Ark. Mat. 44 (2006), 327-333. | MR 2292725 | Zbl 1158.32301