Abstract
The paper is devoted to the study of necessary and sufficient topological conditions for an embedded real surface to lie in a strictly pseudoconvex domain on a complex surface. These results are used to construct Stein domains on algebraic manifolds and to describe envelopes of holomorphy of real surfaces in ℂP 2 and in some other complex surfaces.
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References
S. Yu. Nemirovskii, “Stein domains on algebraic manifolds,”Mat. Zametki [Math. Notes],60, No. 2, 295–298 (1996).
Y.-T. Siu, “Every Stein subvariety admits a Stein neighborhood,”Invent.Math.,38, No. 1, 89–100 (1976).
L. Lempert, “Algebraic approximation in analytic geometry,”Invent. Math.,121, No. 2, 335–354 (1995).
D. Bennequin, “Topologie symplectique, convexité holomorphe et structure de contact (d'après Y.Eliashberg, D.McDuff et al.),”Astérisque,189–190, 285–323 (1990).
R. Fujita, “Domaines sans point critique intérieur sur l'espace projectif complexe,”J. Math. Soc. Japon,15, No. 4, 443–473 (1963).
R. Fujita, “Domaines sans point critique intérieur sur l'espace produit,”J. Math. Kyoto Univ.,4, No. 3, 493–514 (1965).
S. Ivashkovich and V. Shevchishin, “Pseudoholomorphic curves and envelopes of meromorphy of two-spheres in ℂP 2,” Preprint (1995).
E. Chirka, “The generalized Hartogs lemma and the nonlinear\(\bar \partial\)-equation,” in:Complex Analysis in Contemporary Mathematics (dedicated to the memory of B. Shabat), Moscow, ‘Fasis’ (1998), pp. 19–30.
E. Hirai, “Domaine d'holomorphie sur un espace projectif complexe,”J. Math. Kyoto Univ.,10, No. 1, 83–102 (1970).
B. Fabre, “Intersections of an algebraic curve with algebraic hypersurfaces,”Comptes Rendus de l'Ac. des Sciences, Serie 1,322, 371–376 (1996).
P. Kronheimer and T. Mrowka, “The genus of embedded surfaces in the projective plane,”Math. Research Letters,1, 797–808 (1994).
J. W. Morgan,The Seiberg-Witten liquations and Applications to the Topology of Smooth Four-Manifolds, Mathematical Notes, Vol. 44, Princeton Univ. Press, Princeton, N.J. (1996).
E. Witten, “Monopoles and four-manifolds,”Math. Research Letters,1, 769–796 (1994).
J. W. Morgan, Z. Szabó, and C. H. Taubes, “A product formula for Seiberg-Witten invariants and the generalized Thom conjecture,”J. Differential Geom.,44, No. 4, 706–788 (1996).
Y. Matsugu, “The Levi problem for a product manifold,”Pacific J. Math.,46, No. 1, 231–233 (1973).
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Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 599–606, April, 1998.
The author is grateful to A. Vitushkin for the statement of the problem and for constant encouragement.
This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01002 and by the International Science Foundation under grant No. a97-1850.
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Nemirovskii, S.Y. Holomorphic functions and embedded real surfaces. Math Notes 63, 527–532 (1998). https://doi.org/10.1007/BF02311256
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DOI: https://doi.org/10.1007/BF02311256