×

Stability of embeddings for pseudoconcave surfaces and their boundaries. (English) Zbl 0983.32035

Let \(M\) be a compact, strictly pseudoconvex, 3-dimensional CR-manifold. The CR-structure on \(M\) can be described as a subbundle \(T^{0,1} M\) of the complexified tangent bundle, with fiber dimension 1, such that \(T_p^{ 0,1} M\cap T_p^{1,0}M= \{0\}\) for \(p\in M\), where \(T_p^{1,0}M= \overline {T_p^{ 0,1}M}\). Such CR-structure is induced on a strictly pseudoconvex, real hypersurface in a complex surface or as the boundary of a 2-dimensional Stein space. In the latter case we say that the CR-manifold is embeddable.
Assuming that \((M,T^{0,1}M)\) is an embeddable CR-manifold, the authors consider the set of all deformations of this CR-structure and investigate, among other things, the question whether the set of small embeddable deformations is closed in the \(C^\infty\)-topology. For that purpose, the authors introduce the notion of weak embeddability and almost embeddability for the pair \((X_-,Z)\), where \(X_-\) is a pseudoconcave surface and \(Z\) is a positively embedded, smooth, compact curve in \(X_-\). In fact, if the boundary of \(X_-\) is embeddable, then \((X_-,Z)\) is almost embeddable.
The authors prove, as one of the main results, that the converse is also true: If \((X_-,Z)\) is almost embeddable, then \(bX_-\) is embeddable.
Furthermore, the authors prove that almost embeddability is closed under convergence in the \(C^\infty\)-topology. From this, the authors show that the set of small, embeddable perturbations of the CR-structure on \(bX_-\) is closed in the \(C^\infty\)-topology for many new classes of CR-manifolds.

MSC:

32V30 Embeddings of CR manifolds
32V15 CR manifolds as boundaries of domains
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andreotti, A., Théorèmes de dépendence algébrique sur les espaces complexes pseudoconcave.Bull. Soc. Math. France, 91 (1963), 1–38.
[2] Andreotti, A. &Siu, Y.-T., Projective embedding of pseudoconcave spaces.Ann. Scuola Norm. Sup. Pisa (3), 24 (1970), 231–278. · Zbl 0195.36901
[3] Andreotti, A. &Tomassini, G., Some remarks on pseudoconcave manifolds, inEssays on Topology and Related Topics (mémoires dédiés à Georges de Rham), pp. 85–104. Springer-Verlag, New York, 1970.
[4] Bishop, E., Conditions for the analyticity of certain sets.Michigan Math. J., 11 (1964), 289–304. · Zbl 0143.30302 · doi:10.1307/mmj/1028999180
[5] Bland, J., Contact geometry and CR structures onS 3.Acta Math., 172 (1994), 1–49. · Zbl 0814.32002 · doi:10.1007/BF02392789
[6] Bland, J. &Duchamp, T., Deformation theory for the hyperplane line bundle on P1, inCR-Geometry and Overdetermined Systems (Osaka, 1994), pp. 41–59. Adv. Stud. Pure Math., 25. Math. Soc. Japan, Tokyo, 1997. · Zbl 0892.32016
[7] Bland, J. &Epstein, C. L., Embeddable CR-structures and deformations of pseudoconvex surfaces, I. Formal deformations.J. Algebraic Geom., 5 (1996), 277–368. · Zbl 0851.32012
[8] Bogomolov, F., On fillability of contact structures on 3-dimensional manifolds. Preprint, Göttingen, 1993.
[9] Bott, R., Homogeneous vector bundles.Ann. of Math., 66 (1957), 203–248. · Zbl 0094.35701 · doi:10.2307/1969996
[10] Barth, W., Peters, C. &Van de Ven, A.,Compact Complex Surfaces. Ergeb. Math. Grenzgeb. (3), 4. Springer-Verlag, Berlin-New York, 1984.
[11] Burns, D. M. &Epstein, C. L., Embeddability for three-dimensional CR-manifolds.J. Amer. Math. Soc., 3 (1990), 809–841. · Zbl 0736.32017 · doi:10.1090/S0894-0347-1990-1071115-4
[12] Chow, W.-L. &Kodaira, K., On analytic surfaces with two independent meromorphic functions.Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 319–325. · Zbl 0046.30903 · doi:10.1073/pnas.38.4.319
[13] Catlin, D. &Lempert, L., A note on the instabilityof embeddings of Cauchy-Riemann manifolds.J. Geom. Anal., 2 (1992), 99–104. · Zbl 0748.32012
[14] Docquier, F. &Grauert, H., Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten.Math. Ann., 140 (1960), 94–123. · Zbl 0095.28004 · doi:10.1007/BF01360084
[15] Epstein, C. L. &Henkin, G. M., Extension of CR-structures for 3-dimensional pseudoconcave manifolds, inMultidimensional Complex Analysis and Partial Differential Equations (São Carlos, 1995), pp. 51–67. Contemp. Math., 205. Amer. Math. Soc., Providence, RI, 1997.
[16] –, Two lemmas in local analytic geometry, inAnalysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998), pp. 189–195. Contemp. Math., 251. Amer. Math. Soc., Providence, RI, 2000.
[17] –, Embeddings for 3-dimensional CR-manifolds, inComplex Analysis and Geometry (Paris, 1997), pp. 223–236. Progr. Math., 188. Birkhäuser, Basel, 2000.
[18] Eliashberg, Y., Filling by holomorphic discs and its applications, inGeometry of Low-Dimensional Manifolds, 2 (Durham, 1989), pp. 45–67. London Math. Soc. Lecture Note Ser., 151. Cambridge Univ. Press, Cambridge, 1990.
[19] Epstein, C. L., CR-structures on three-dimensional circle bundles.Invent. Math., 109 (1992), 351–403. · Zbl 0786.32013 · doi:10.1007/BF01232031
[20] –, A relative index for embeddable CR-structures, I; II.Ann. of Math., 147 (1998), 1–59; 61–91. · Zbl 0942.32025 · doi:10.2307/120982
[21] Fabre, B., Sur l’intersection d’une surface de Riemann avec des hypersurfaces algébriques.C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 371–376. · Zbl 0866.32011
[22] Griffiths, P. &Harris, J.,Principles of Algebraic Geometry. Wiley-Interscience, New York, 1978. · Zbl 0408.14001
[23] Griffiths, P., The extension problem in complex analysis, II. Embeddings with positive normal bundle.Amer. J. Math., 88 (1966), 366–446. · Zbl 0147.07502 · doi:10.2307/2373200
[24] Grauert, H. &Riemenschneider, O., Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen.Invent. Math., 11 (1970), 263–292. · Zbl 0202.07602 · doi:10.1007/BF01403182
[25] Gunning, R. C.,Introduction to Holomorphic Functions of Several Variables, Vol. II. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1990. · Zbl 0699.32001
[26] Hartshorne, R.,Algebraic Geometry. Graduate Texts in Math., 52. Springer-Verlag, New York-Heidelberg, 1977. · Zbl 0367.14001
[27] Harvey, R., Holomorphic chains and their boundaries, inSeveral Complex Variables (Williamstown, MA, 1975), pp. 309–382. Proc. Sympos. Pure Math., 30. Amer. Math. Soc., Providence, RI, 1977.
[28] Harvey, F. R. &Lawson, H. B., Jr., On boundaries of complex analytic varieties, I.Ann. of Math., 102 (1975), 223–290. · Zbl 0317.32017 · doi:10.2307/1971032
[29] Kato, T.,Perturbation Theory of Linear Operators. Grundlehren Math. Wiss., 132. Springer-Verlag, New York, 1966. · Zbl 0148.12601
[30] Kodaira, K., On compact complex analytic surfaces, I; II; III.Ann. of Math., 71; 77; 78 (1960; 1963; 1963), 111–152; 563–626; 1–40. · Zbl 0098.13004 · doi:10.2307/1969881
[31] –, On stability of compact submanifolds of complex manifolds.Amer. J. Math., 85 (1963), 79–94. · Zbl 0173.33101 · doi:10.2307/2373187
[32] Kiremidjian, G., A direct extension method for CR-structures.Math. Ann., 242 (1979), 1–19. · Zbl 0399.32015 · doi:10.1007/BF01420478
[33] Kronheimer, P. B. &Mrowka, T. S., Monopoles and contact structures.Invent. Math., 130 (1997), 209–255. · Zbl 0892.53015 · doi:10.1007/s002220050183
[34] Kohn, J. J., The range of the tangential Cauchy-Riemann operator.Duke Math. J., 53 (1986), 525–545. · Zbl 0609.32015 · doi:10.1215/S0012-7094-86-05330-5
[35] Kohn, J. J. &Rossi, H., On the extension of holomorphic functions from the boundary of a complex manifold.Ann. of Math., 81 (1965), 451–472. · Zbl 0166.33802 · doi:10.2307/1970624
[36] Kuranishi, M.,Deformations of Compact Complex Manifolds. Séminaire de Mathématiques Supérieures, 39. Les Presses de l’Université de Montréal, Montreal, 1971.
[37] Lempert, L., On three-dimensional Cauchy-Riemann manifolds.J. Amer. Math. Soc., 5 (1992), 923–969. · Zbl 0781.32014 · doi:10.1090/S0894-0347-1992-1157290-3
[38] –, Embeddings of three-dimensional Cauchy-Riemann manifolds.Math. Ann., 300 (1994), 1–15. · Zbl 0817.32009 · doi:10.1007/BF01450472
[39] –, Algebraic approximations in analytic geometry.Invent. Math., 121 (1995), 335–353. · Zbl 0837.32008 · doi:10.1007/BF01884302
[40] Li, H.-L., The stability of embeddings of Cauchy-Riemann manifolds. Thesis, Purdue University, 1995.
[41] Morrow, J. &Rossi, H., Some general results on equivalence of embeddings, inRecent Developments in Several Complex Variables (Princeton, NJ, 1979), pp. 299–325. Ann. of Math. Stud., 100. Princeton Univ. Press, Princeton, NJ, 1981.
[42] –, Some theorems of algebraicity for complex spaces.J. Math. Soc. Japan, 27 (1975), 167–183. · Zbl 0306.32003 · doi:10.2969/jmsj/02720167
[43] Ouyang, Y., Ph.D. Thesis, University of Pennsylvania, 1999.
[44] Pinkham, H. C., Deformations of cones with negative grading.J. Algebra, 30 (1974), 92–102. · Zbl 0284.14009 · doi:10.1016/0021-8693(74)90194-X
[45] Pardon, W. &Stern, M. A.,L 2- \(\bar \partial \) -cohomology of complex projective varieties.J. Amer. Math. Soc., 4 (1991), 603–621. · Zbl 0751.14011
[46] Rossi, H., Attaching analytic spaces to an analytic space along a pseudoconcave boundary, inProceedings of the Conference on Complex Analysis (Minneapolis, MN, 1964), pp. 242–256. Springer-Verlag, Berlin, 1965.
[47] Siu, Y.-T., Every Stein subvariety admits a Stein neighborhood.Invent. Math., 38 (1976), 89–100. · Zbl 0343.32014 · doi:10.1007/BF01390170
[48] Shiffman, B. &Sommese, A. J.,Vanishing Theorems on Complex Manifolds. Progr. Math., 56. Birkhäuser, Boston, Boston, MA, 1985. · Zbl 0578.32055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.