# Non compact boundaries of complex analytic varieties in Hilbert spaces

Samuele Mongodi; Alberto Saracco

Complex Manifolds (2014)

- Volume: 1, Issue: 1, page 34-44, electronic only
- ISSN: 2300-7443

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topSamuele Mongodi, and Alberto Saracco. "Non compact boundaries of complex analytic varieties in Hilbert spaces." Complex Manifolds 1.1 (2014): 34-44, electronic only. <http://eudml.org/doc/276957>.

@article{SamueleMongodi2014,

abstract = {We treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly convex, open subsets of a complex Hilbert space H.We deal with the problem by cutting with a family of complex hyperplanes and applying the already known result for the compact case.},

author = {Samuele Mongodi, Alberto Saracco},

journal = {Complex Manifolds},

keywords = {Boundary problem; convexity; maximally complex submanifold; complex Hilbert spaces; CRgeometry; boundary problem for complex manifolds; strongly convex subsets of Hilbert spaces},

language = {eng},

number = {1},

pages = {34-44, electronic only},

title = {Non compact boundaries of complex analytic varieties in Hilbert spaces},

url = {http://eudml.org/doc/276957},

volume = {1},

year = {2014},

}

TY - JOUR

AU - Samuele Mongodi

AU - Alberto Saracco

TI - Non compact boundaries of complex analytic varieties in Hilbert spaces

JO - Complex Manifolds

PY - 2014

VL - 1

IS - 1

SP - 34

EP - 44, electronic only

AB - We treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly convex, open subsets of a complex Hilbert space H.We deal with the problem by cutting with a family of complex hyperplanes and applying the already known result for the compact case.

LA - eng

KW - Boundary problem; convexity; maximally complex submanifold; complex Hilbert spaces; CRgeometry; boundary problem for complex manifolds; strongly convex subsets of Hilbert spaces

UR - http://eudml.org/doc/276957

ER -

## References

top- [1] L. Ambrosio, B. Kirchheim, Currents in metric spaces, Acta Math., 185 1 (2000), 1–80. Zbl0984.49025
- [2] L. Ambrosio, B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 3 (2000), 527–555. Zbl0966.28002
- [3] V. Aurich, Bounded analytic sets in Banach spaces, Ann. Inst. Fourier, 36 4 (1986), 229–243. [Crossref] Zbl0591.46005
- [4] G. Della Sala, Geometric properties of non-compact CR manifolds, Tesi 14, Edizioni della Normale, Pisa (2009), 103+xv. Zbl1200.32022
- [5] G. Della Sala, A. Saracco, Non-compact boundaries of complex analytic varieties, Int. J. Math. 18 2 (2007), 203–218. [Crossref][WoS] Zbl1140.32025
- [6] G. Della Sala, A. Saracco, Semi-global extension of maximally complex submanifolds, Bull. Aust. Math. Soc. 84 (2011), 458–474. [WoS] Zbl1239.32028
- [7] T.-C. Dinh, Conjecture de Globevnik-Stout et théorème de Morera pur une chaîne holomorphe, Ann. Fac. Sci. Toulouse Math. 8 (1999) 235–257.
- [8] P. Dolbeault, G. Henkin, Surfaces de Riemann de bord donné dans CPn, in Contributions to complex analysis and analytic geometry, Aspects Math. (Vieweg, Braunschweig, 1994), pp. 163–187. Zbl0821.32008
- [9] P. Dolbeault, G. Henkin, Chaînes holomorphes de bord donné dans CPn, Bull. Soc. Math. France 125 (1997) 383–445.
- [10] M. P. Gambaryan, Regularity condition for complex films, Uspekhi Mat. Nauk 40 (1985) 203–204. Zbl0576.32016
- [11] F. R. Harvey, H. B. Lawson Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975), 223–290. Zbl0317.32017
- [12] F. R. Harvey, H. B. Lawson, Jr., On boundaries of complex analytic varieties. II, Ann. of Math. 106 (1977) 213–238. Zbl0361.32010
- [13] F. R. Harvey, H. B. Lawson, Jr., Addendum to Theorem 10.4 in “Boundaries of analytic varieties”,
- [arXiv: math.CV/0002195] (2000).
- [14] H. Lewy, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math. 64 (1956) 514–522. Zbl0074.06204
- [15] S. Mongodi, Some applications of metric currents to complex analysis, Man. Math. 141 (2013) 363–390. Zbl1267.32038
- [16] S. Mongodi, Positive metric currents and holomorphic chains in Hilbert spaces, Zbl1332.32015
- [arXiv:1207.5244], to appear in Rev. Mat. Iberoam. 31 (2015).
- [17] G. Ruget, A propos des cycles analytiques de dimension infinie, Inv. Math. 8 (1969) 267–312. Zbl0188.25102
- [18] A. Saracco, Extension problems in complex and CR-geometry, Tesi 9, Edizioni della Normale, Pisa (2008), 153+xiv. Zbl1165.32001
- [19] G. Stolzenberg, Uniform approximation on smooth curves, Acta Math. 115 (1966) 185–198. Zbl0143.30005
- [20] J. Wermer, The hull of a curve in Cn, Ann. of Math. 68 (1958) 550–561. Zbl0084.33402
- [21] R. Williamson, L. Janos, Constructing metrics with the Heine-Borel property, Proc. A.M.S. 100 3 (1987), 567–573. Zbl0626.54035