Can a~Good Manifold Come to a~Bad End?

C. L. Epstein ; G. M. Henkin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and geometric issues of complex analysis, Tome 235 (2001), pp. 71-93.

Résumé

Two notions of cobordism are defined for compact CR-manifolds. The weaker notion, complex cobordism realizes two CR-manifolds as the boundary of a complex manifold; in the stronger notion, strict complex cobordism there is a strictly plurisubharmonic function defined on the total space of the cobordism with the boundary components as level sets of this function. We show that the embeddability for a 3-dimensional, strictly pseudoconvex CR-manifold is a strict cobordism invariant. De Oliveira has recently shown that this is false for complex cobordisms. His construction is described in the appendix.

Bibliographie

[1] Airapetyan R. A., Khenkin G. M., “Integralnye predstavleniya differentsialnykh form na mnogoobraziyakh Koshi–Rimana i teoriya CR-funktsii”, UMN, 39:3 (1984), 41–118 | MR | Zbl

[2] Andreotti A., Grauert H., “Théoremes de finitude pour la cohomologie des espaces complexes”, Bull. Soc. Math. France, 90 (1962), 193–259 | MR

[3] Andreotti A., Siu Y.-T., “Projective embedding of pseudoconcave spaces”, Ann. Scuola Norm. Super. Pisa, 24 (1970), 231–278 | MR | Zbl

[4] Andreotti A., Vesentini E., “Carleman estimates for the Laplace–Beltrami equation on complex manifolds”, Publ. Math. IHES, 1965, no. 25, 81–150 | MR

[5] Boutet de Monvel L., “Intégration des équations de Cauchy–Riemann induites”, Sem. Goulaouic–Schwartz, no. 9, 1974/75

[6] Burns D. M. , Jr., “Global behavior of some tangential Cauchy–Riemann equations”, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977), Lecture Notes in Pure and Appl. Math., 48, Dekker, New York, 51–56 | MR

[7] Cartan H., “Quotient d'un espace analytique par un groupe d'automorphisms”, Algebraic geometry and topology, Symp. in honor of S. Lefschetz, Princeton Univ. Press, Princeton, 1957, 90–102 | MR | Zbl

[8] Chirka E. M., “Analiticheskoe predstavlenie CR-funktsii”, Mat. sb., 98 (1975), 591–623 | Zbl

[9] Demailly J.-P., “Estimations $L^2$ pour l'opérateur $\overline\partial$ d'un fibré vectoriel holomorphe semi-positif au dessus d'une variété Kählérienne compléte”, Ann. Sci. Ecole Norm. Super, 15 (1982), 457–511 | MR | Zbl

[10] De Oliveira B., Complex cobordism and embeddability of CR-manifolds, Preprint Univ. Pennsylvania, 2001 | MR

[11] Epstein C. L., “A relative index on the space of embeddable CR-structures, I, II”, Ann. Math., 147 (1998), 1–59, 61–91 | DOI | MR | Zbl

[12] Epstein C. L., Henkin G. M., “Two lemmas in local analytic geometry”, Analysis, geometry, number theory: The mathematics of Leon Ehrenpreis (Proc. Conf. Philadelphia, 1998), Contemp. Math., 251, Amer. Math. Soc., Providence, RI, 2000, 189–195 | MR | Zbl

[13] Epstein C. L., Henkin G. M., “Embeddings for 3-dimensional CR-manifolds”, Complex analysis and geometry, Proc. Intern. Conf. in honor of Pierre Lelong on the occasion of his 85th birthday (Paris, Sept. 22–26, 1997), Progr. Math., 188, eds. P. Dolbeault et al., Birkhäuser, Basel, 2000, 223–236 | MR | Zbl

[14] Epstein C. L., Henkin G. M., “Stability of embeddings for pseudoconcave surfaces and their boundaries”, Acta Math., 185:2 (2000), 161–237 | DOI | MR | Zbl

[15] Grauert H., “Über Modifikationen und exzeptionelle analytische Mengen”, Math. Ann., 146 (1962), 331–368 | DOI | MR

[16] Grauert H., “On {L}evi's problem and the imbedding of real analytic manifolds”, Ann. Math., 68 (1958), 460–472 | DOI | MR | Zbl

[17] Harvey R., Lawson H. B., “On the boundaries of complex analytic varieties, I”, Ann. Math., 102 (1975), 223–290 | DOI | MR | Zbl

[18] Khenkin G. M., “Uravnenie Levi i analiz na psevdovypuklykh mnogoobraziyakh”, UMN, 32:3 (1977), 57–118 | MR | Zbl

[19] Henkin G. M., Leiterer J., Theory of functions on complex manifolds, Monogr. Math., 79, Birkhäuser, Basel, 1984, 226 pp. | MR

[20] Henkin G. M., Leiterer J., Andreotti–Grauert theory by integral formulas, Progr. Math., 74, Birkhäuser, Boston, 1988, 270 pp. | MR | Zbl

[21] Kodaira K., “On compact complex analytic surfaces, I”, Ann. Math., 71 (1960), 111–152 | DOI | MR | Zbl

[22] Kohn J. J., “Boundaries of complex manifolds”, Proc. Conf. on Complex Analysis (Minneapolis, 1964), Springer-Verl., Berlin, 1965, 81–94 | MR

[23] Kohn J. J., Rossi H., “On the extension of holomorphic functions from the boundary of a complex manifold”, Ann. Math., 81 (1965), 451–472 | DOI | MR | Zbl

[24] Lempert L., “On algebraic approximations in analytic geometry”, Invent. Math., 121 (1995), 335–354 | DOI | MR | Zbl

[25] McShane E. J., “Extension of the range of functions”, Bull. Amer. Math. Soc., 40 (1934), 837–842 | DOI | MR | Zbl

[26] Morrow J., Rossi H., “Some general results on equivalence of embeddings”, Recent developments in several complex variables (Proc. Conf. Princeton, 1979), Ann. Math. Stud., 100, Princeton Univ. Press, Princeton, 1981, 299–325 | MR | Zbl

[27] Narasimhan R., “The Levi problem for complex spaces, II”, Math. Ann., 146 (1962), 195–216 | DOI | MR | Zbl

[28] Pardon W., Stern M., “$L^2$-$\overline\partial$-cohomology for complex projective varieties”, J. Amer. Math. Soc., 4 (1991), 603–621 | DOI | MR | Zbl

[29] Prill D., “Local classification of quotients of complex manifolds by discontinuous groups”, Duke Math. J., 34 (1967), 375–386 | DOI | MR | Zbl

[30] Rossi H., “Attaching analytic spaces to an analytic space along a pseudoconcave boundary”, Proc. Conf. on Complex Analysis (Minneapolis, 1964), Springer-Verl., Berlin, 1965, 242–256 | MR

[31] Tougeron J. C., Ideaux de fonctions differentielles, Springer-Verl., Berlin, 1972 | MR | Zbl

[32] Whitney H., “Analytic extension of differentiable functions defined in closed sets”, Trans. Amer. Math. Soc., 36 (1934), 63–89 | DOI | MR | Zbl