# On the local meromorphic extension of CR meromorphic mappings

Annales Polonici Mathematici (1998)

- Volume: 70, Issue: 1, page 163-193
- ISSN: 0066-2216

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topJoël Merker, and Egmont Porten. "On the local meromorphic extension of CR meromorphic mappings." Annales Polonici Mathematici 70.1 (1998): 163-193. <http://eudml.org/doc/262638>.

@article{JoëlMerker1998,

abstract = {Let M be a generic CR submanifold in $ℂ^\{m+n\}$, m = CR dim M ≥ 1, n = codim M ≥ 1, d = dim M = 2m + n. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,_f,[Γ_f])$, where: 1) $f: _f → Y$ is a ¹-smooth mapping defined over a dense open subset $_f$ of M with values in a projective manifold Y; 2) the closure $Γ_f$ of its graph in $ℂ^\{m+n\} × Y$ defines an oriented scarred ¹-smooth CR manifold of CR dimension m (i.e. CR outside a closed thin set) and 3) $d[Γ_f] = 0$ in the sense of currents. We prove that $(f,_f,[Γ_f])$ extends meromorphically to a wedge attached to M if M is everywhere minimal and $^ω$ (real-analytic) or if M is a $^\{2,α\}$ globally minimal hypersurface.},

author = {Joël Merker, Egmont Porten},

journal = {Annales Polonici Mathematici},

keywords = {CR generic currents; scarred CR manifolds; removable singularities for CR functions; deformations of analytic discs; CR meromorphic mappings},

language = {eng},

number = {1},

pages = {163-193},

title = {On the local meromorphic extension of CR meromorphic mappings},

url = {http://eudml.org/doc/262638},

volume = {70},

year = {1998},

}

TY - JOUR

AU - Joël Merker

AU - Egmont Porten

TI - On the local meromorphic extension of CR meromorphic mappings

JO - Annales Polonici Mathematici

PY - 1998

VL - 70

IS - 1

SP - 163

EP - 193

AB - Let M be a generic CR submanifold in $ℂ^{m+n}$, m = CR dim M ≥ 1, n = codim M ≥ 1, d = dim M = 2m + n. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,_f,[Γ_f])$, where: 1) $f: _f → Y$ is a ¹-smooth mapping defined over a dense open subset $_f$ of M with values in a projective manifold Y; 2) the closure $Γ_f$ of its graph in $ℂ^{m+n} × Y$ defines an oriented scarred ¹-smooth CR manifold of CR dimension m (i.e. CR outside a closed thin set) and 3) $d[Γ_f] = 0$ in the sense of currents. We prove that $(f,_f,[Γ_f])$ extends meromorphically to a wedge attached to M if M is everywhere minimal and $^ω$ (real-analytic) or if M is a $^{2,α}$ globally minimal hypersurface.

LA - eng

KW - CR generic currents; scarred CR manifolds; removable singularities for CR functions; deformations of analytic discs; CR meromorphic mappings

UR - http://eudml.org/doc/262638

ER -

## References

top- [1] E. Chirka, Complex Analytic Sets, Kluwer, Dordrecht, 1989. Zbl0683.32002
- [2] E. M. Chirka and E. L. Stout, Removable singularities in the boundary, in: Contributions to Complex Analysis and Analytic Geometry, Aspects of Math. E26, Vieweg, 1994, 43-104. Zbl0820.32008
- [3] T.-C. Dinh and F. Sarkis, Wedge removability of metrically thin sets and application to the CR meromorphic extension, preprint, 1997.
- [4] P. Dolbeault et G. M. Henkin, Chaînes holomorphes de bord donné dans $\u2102{\mathbb{P}}^{n}$, Bull. Soc. Math. France 125 (1997), 383-446.
- [5] F. R. Harvey and H. B. Lawson, On boundaries of complex analytic varieties, Ann. of Math., I: 102 (1975), 233-290; II: 106 (1977), 213-238. Zbl0317.32017
- [6] S. M. Ivashkovich, The Hartogs-type extension theorem for meromorphic maps into compact Kähler manifolds, Invent. Math. 109 (1992), 47-54. Zbl0738.32008
- [7] B. Jöricke, Removable singularities for CR-functions, Ark. Mat. 26 (1988), 117-143. Zbl0653.32013
- [8] B. Jöricke, Envelopes of holomorphy and CR-invariant subsets of CR-manifolds, C. R. Acad. Sci. Paris Sér. I 315 (1992), 407-411. Zbl0767.32010
- [9] B. Jöricke, Deformation of CR- manifolds, minimal points and CR-manifolds with the microlocal analytic extension property, J. Geom. Anal. 6 (1996), 555-611. Zbl0917.32007
- [10] B. Jöricke, Some remarks concerning holomorphically convex hulls and envelope of holomorphy, Math. Z. 218 (1995), 143-157. Zbl0816.32011
- [11] B. Jöricke, Boundaries of singularity sets, removable singularities, and CR-invariant subsets of CR-manifolds, preprint, 1996. Zbl0964.32031
- [12] G. Lupacciolu, A theorem on holomorphic extension of CR-functions, Pacific J. Math. 124 (1986), 177-191. Zbl0597.32014
- [13] C. Laurent-Thiébaut, Sur l'extension de fonctions CR dans une variété de Stein, Ann. Mat. Pura Appl. (4) 150 (1988), 141-151. Zbl0646.32010
- [14] J. Merker, Global minimality of generic manifolds and holomorphic extendibility of CR functions, Internat. Math. Res. Notices 8 (1994), 329-342. Zbl0815.32007
- [15] J. Merker, On removable singularities for CR functions in higher codimension, ibid. 1 (1997), 21-56. Zbl0880.32009
- [16] J. Merker and E. Porten, On removable singularities for integrable CR functions, preprint, 1997; available at: http://www.dmi.ens.fr/EDITION/preprints. Zbl0935.32010
- [17] E. Porten, thesis, Berlin, 1996.
- [18] E. Porten, A Hartogs-Bochner type theorem for continuous CR mappings, manuscript, 1997.
- [19] F. Sarkis, CR meromorphic extension and the non embedding of the Andreotti-Rossi CR structure in the projective space, Internat. J. Math., to appear. Zbl1110.32308
- [20] B. Shiffman, Separately meromorphic mappings into Kähler manifolds, in: Contributions to Complex Analysis and Analytic Geometry, Aspects of Math. E26, Vieweg, 1994, 243-250. Zbl0873.32024
- [21] J.-M. Trépreau, Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. France 118 (1990), 403-450.
- [22] A. E. Tumanov, Connections and propagation of analyticity for CR functions, Duke Math. J. 73 (1994), 1-24. Zbl0801.32005

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