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Applications of Oka Theory and Its Methods

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Stein Manifolds and Holomorphic Mappings

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Abstract

In this chapter we apply the methods and results of Oka theory to a variety of problems in Stein geometry. In particular, we discuss the structure of holomorphic vector bundles and their homomorphisms over Stein spaces, find the minimal number of generators of coherent analytic sheaves over Stein spaces, consider the problem of complete intersections and, more generally, of elimination of intersections of holomorphic maps with complex subvarieties, present the solution of the holomorphic Vaserstein problem, discuss transversality theorems for holomorphic and algebraic maps from Stein manifolds, and obtain estimates on the dimension of singularity (degeneration) sets of generic holomorphic maps from Stein manifolds to Oka manifolds. In the last part of the chapter we further develop the method of local holomorphic sprays of maps from strongly pseudoconvex Stein domains. We apply this technique to prove an up-to-the-boundary version of the Oka principle on such domains, and we establish a Banach manifold structure theorem for certain spaces of holomorphic maps from strongly pseudoconvex domains to arbitrary complex manifolds.

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References

  1. Abraham, R.: Transversality in manifolds of mappings. Bull. Am. Math. Soc. 69, 470–474 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bănică, C., Forster, O.: Complete intersections in Stein manifolds. Manuscr. Math. 37(3), 343–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bochnak, J., Kucharz, W.: Complete intersections in differential topology and analytic geometry. Boll. Unione Mat. Ital., B (7) 10(4), 1019–1041 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Cartan, H.: Sur les matrices holomorphes de \(n\) variables complexes. J. Math. Pures Appl. 19, 1–26 (1940)

    MathSciNet  MATH  Google Scholar 

  5. Cartan, H.: Espaces fibrés analytiques. In: Symposium internacional de topología algebraica (International Symposium on Algebraic Topology), pp. 97–121. Universidad Nacional Autónoma de México and UNESCO, Mexico City (1958)

    Google Scholar 

  6. Cohn, P.M.: On the structure of the \(\mathrm{GL}_{2}\) of a ring. Publ. Math. IHÉS 30, 5–53 (1966)

    Article  MathSciNet  Google Scholar 

  7. Cornalba, M., Griffiths, P.: Analytic cycles and vector bundles on non-compact algebraic varieties. Invent. Math. 28, 1–106 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Demailly, J.-P.: Différents exemples de fibrés holomorphes non de Stein. In: Séminaire Pierre Lelong-Henri Skoda (Analyse), Année 1976/77. Lecture Notes in Math., vol. 694, pp. 15–41. Springer, Berlin (1978)

    Chapter  Google Scholar 

  9. Demailly, J.-P.: Un exemple de fibré holomorphe non de Stein à fibre \({\mathbb {C}}^{2}\) ayant pour base le disque ou le plan. Invent. Math. 48(3), 293–302 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Drinovec Drnovšek, B., Forstnerič, F.: Holomorphic curves in complex spaces. Duke Math. J. 139(2), 203–253 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Drinovec-Drnovšek, B., Forstnerič, F.: Approximation of holomorphic mappings on strongly pseudoconvex domains. Forum Math. 20(5), 817–840 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Eisenbud, D., Evans, E.G. Jr.: Every algebraic set in \(n\)-space is the intersection of \(n\) hypersurfaces. Invent. Math. 19, 107–112 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Forster, O.: Some remarks on parallelizable Stein manifolds. Bull. Am. Math. Soc. 73, 712–716 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  14. Forster, O.: Zur Theorie der Steinschen Algebren und Moduln. Math. Z. 97, 376–405 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  15. Forster, O.: Plongements des variétés de Stein. Comment. Math. Helv. 45, 170–184 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  16. Forster, O.: Complete intersections in affine algebraic varieties and Stein spaces. In: Complete Intersections, Acireale, 1983. Lecture Notes in Math., vol. 1092, pp. 1–28. Springer, Berlin (1984)

    Chapter  Google Scholar 

  17. Forster, O.: Lectures on Riemann Surfaces. Graduate Texts in Mathematics, vol. 81. Springer, New York (1991). Translated from the 1977 German original by Bruce Gilligan. Reprint of the 1981 English translation

    Google Scholar 

  18. Forster, O., Ramspott, K.J.: Singularitätenfreie analytische Raumkurven als vollständige Durchschnitte. Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B. 1965(II), 1–10 (1966)

    MATH  Google Scholar 

  19. Forster, O., Ramspott, K.J.: Analytische Modulgarben und Endromisbündel. Invent. Math. 2, 145–170 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  20. Forster, O., Ramspott, K.J.: Okasche Paare von Garben nicht-abelscher Gruppen. Invent. Math. 1, 260–286 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  21. Forster, O., Ramspott, K.J.: Homotopieklassen von Idealbasen in Steinschen Algebren. Invent. Math. 5, 255–276 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  22. Forster, O., Ramspott, K.J.: Über die Anzahl der Erzeugenden von projektiven Steinschen Moduln. Arch. Math. (Basel) 19, 417–422 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  23. Forstnerič, F.: On complete intersections. Ann. Inst. Fourier (Grenoble) 51(2), 497–512 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Forstnerič, F.: Holomorphic flexibility properties of complex manifolds. Am. J. Math. 128(1), 239–270 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Forstnerič, F.: Manifolds of holomorphic mappings from strongly pseudoconvex domains. Asian J. Math. 11(1), 113–126 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)

    MATH  Google Scholar 

  27. Goresky, M., MacPherson, R.: Stratified Morse Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 14. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  28. Grauert, H.: Charakterisierung der holomorph vollständigen komplexen Räume. Math. Ann. 129, 233–259 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  29. Grauert, H.: Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen. Math. Ann. 133, 450–472 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  30. Grauert, H.: Analytische Faserungen über holomorph-vollständigen Räumen. Math. Ann. 135, 263–273 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gromov, M.: Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 9. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  32. Gromov, M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2(4), 851–897 (1989)

    MathSciNet  MATH  Google Scholar 

  33. Grunewald, F., Mennicke, J., Vaserstein, L.: On the groups \(\mathrm{SL}_{2}(\mathbf{Z}[x])\) and \(\mathrm{SL}_{2}(k[x,y])\). Isr. J. Math. 86(1–3), 157–193 (1994)

    Article  MATH  Google Scholar 

  34. Henkin, G., Leiterer, J.: The Oka-Grauert principle without induction over the base dimension. Math. Ann. 311(1), 71–93 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Henkin, G.M., Leiterer, J.: Theory of functions on complex manifolds. Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 60. Akademie-Verlag, Berlin (1984)

    Google Scholar 

  36. Heunemann, D.: An approximation theorem and Oka’s principle for holomorphic vector bundles which are continuous on the boundary of strictly pseudoconvex domains. Math. Nachr. 127, 275–280 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  37. Heunemann, D.: Extension of the complex structure from Stein manifolds with strictly pseudoconvex boundary. Math. Nachr. 128, 57–64 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  38. Heunemann, D.: Theorem B for Stein manifolds with strictly pseudoconvex boundary. Math. Nachr. 128, 87–101 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Husemoller, D.: Fibre Bundles, 3rd edn. Graduate Texts in Mathematics, vol. 20. Springer, New York (1994)

    Book  MATH  Google Scholar 

  40. Ivarsson, B., Kutzschebauch, F.: Holomorphic factorization of mappings into \(\mathrm {SL}_{n}(\mathbb {C})\). Ann. Math. (2) 175(1), 45–69 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ivarsson, B., Kutzschebauch, F.: On the number of factors in the unipotent factorization of holomorphic mappings into \(\mathrm{SL}_{2}(\mathbb {C})\). Proc. Am. Math. Soc. 140(3), 823–838 (2012)

    Article  MATH  Google Scholar 

  42. Kaliman, S., Zaĭdenberg, M.: A transversality theorem for holomorphic mappings and stability of Eisenman-Kobayashi measures. Trans. Am. Math. Soc. 348(2), 661–672 (1996)

    Article  MATH  Google Scholar 

  43. Klein, M., Ramspott, K.J.: Ein Transformationssatz für Idealbasen holomorpher Funktionen. Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber., 93–100 (1988), 1987

    Google Scholar 

  44. Kobayashi, S., Ochiai, T.: Meromorphic mappings onto compact complex spaces of general type. Invent. Math. 31(1), 7–16 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kronecker, L.: Grundzüge einer arithmetischen Theorie der algebraische Grössen. J. Reine Angew. Math. 92, 1–122 (1882)

    MathSciNet  MATH  Google Scholar 

  46. Laufer, H.B.: On the infinite dimensionality of the Dolbeault cohomology groups. Proc. Am. Math. Soc. 52, 293–296 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  47. Leiterer, J.: Theorem B für analytische Funktionen mit stetigen Randwerten. Beitr. Anal. 8, 95–102 (1976)

    MathSciNet  MATH  Google Scholar 

  48. Leiterer, J.: Holomorphic vector bundles and the Oka-Grauert principle. In: Several Complex Variables. IV. Algebraic Aspects of Complex Analysis. Encyclopaedia Math. Sci., vol. 10, 63–103 (1990); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 10, 75–121 (1986)

    Google Scholar 

  49. Lempert, L.: Holomorphic functions on (generalized) loop spaces. Math. Proc. R. Ir. Acad. 104A(1), 35–46 (electronic) (2004)

    Article  MathSciNet  MATH  Google Scholar 

  50. Murolo, C., Trotman, D.J.A., du Plessis, A.A.: Stratified transversality by isotopy. Trans. Am. Math. Soc. 355(12), 4881–4900 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  51. Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. (2) 65, 391–404 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  52. Ohsawa, T.: Holomorphic embedding of compact s.p.c. manifolds into complex manifolds as real hypersurfaces. In: Differential Geometry of Submanifolds, Kyoto, 1984. Lecture Notes in Math., vol. 1090, pp. 64–76. Springer, Berlin (1984)

    Chapter  Google Scholar 

  53. Palais, R.S.: Foundations of Global Non-linear Analysis. W. A. Benjamin, New York (1968)

    MATH  Google Scholar 

  54. Palais, R.S.: Banach manifolds of fiber bundle sections. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 243–249. Gauthier-Villars, Paris (1971)

    Google Scholar 

  55. Poletsky, E.A.: Stein neighborhoods of graphs of holomorphic mappings. J. Reine Angew. Math. 684, 187–198 (2013)

    MathSciNet  MATH  Google Scholar 

  56. Prezelj, J.: Positivity of metrics on conic neighborhoods of 1-convex submanifolds. Int. J. Math. 27(5), 24 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  57. Putinar, M.: A reciprocal to the Oka-Grauert principle. Math. Nachr. 159, 149–152 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  58. Rosay, J.-P.: Extension of holomorphic bundles to the disc (and Serre’s problem on Stein bundles). Ann. Inst. Fourier (Grenoble) 57(2), 517–523 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  59. Rosenberg, J.: Algebraic \(K\)-Theory and Its Applications. Graduate Texts in Mathematics, vol. 147. Springer, New York (1994)

    Book  MATH  Google Scholar 

  60. Schneider, M.: Vollständige, fast-vollständige und mengen-theoretisch-vollständige Durchschnitte in Steinschen Mannigfaltigkeiten. Math. Ann. 260(2), 151–174 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  61. Schneider, M.: On the number of equations needed to describe a variety. In: Complex Analysis of Several Variables, Madison, Wis., 1982. Proc. Sympos. Pure Math., vol. 41, pp. 163–180. Am. Math. Soc., Providence (1984)

    Chapter  Google Scholar 

  62. Skoda, H.: Fibrés holomorphes à base et à fibre de Stein. C. R. Acad. Sci. Paris Sér. A-B 284(19), A1199–A1202 (1977)

    MATH  Google Scholar 

  63. Skoda, H.: Fibrés holomorphes à base et à fibre de Stein. Invent. Math. 43(2), 97–107 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  64. Stein, K.: Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem. Math. Ann. 123, 201–222 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  65. Stopar, K.: Approximation of holomorphic mappings on 1-convex domains. Int. J. Math. 24(14), 33 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  66. Sukhov, A., Tumanov, A.: Deformations and transversality of pseudo-holomorphic discs. J. Anal. Math. 116, 1–16 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  67. Suslin, A.A.: The structure of the special linear group over rings of polynomials. Izv. Akad. Nauk SSSR, Ser. Mat. 41(2), 235–252 (1977). 477

    MathSciNet  MATH  Google Scholar 

  68. Thom, R.: Les singularités des applications différentiables. Ann. Inst. Fourier 6, 43–87 (1955/1956)

    Google Scholar 

  69. Thom, R.: Un lemme sur les applications différentiables. Bol. Soc. Mat. Mexicana (2) 1, 59–71 (1956)

    MathSciNet  MATH  Google Scholar 

  70. Thurston, W., Vaserstein, L.: On \(K_{1}\)-theory of the Euclidean space. Topol. Appl. 23(2), 145–148 (1986)

    Article  MATH  Google Scholar 

  71. Trotman, D.J.A.: Stability of transversality to a stratification implies Whitney \((a)\)-regularity. Invent. Math. 50(3), 273–277 (1978/1979)

    Google Scholar 

  72. Vaserstein, L.N.: Reduction of a matrix depending on parameters to a diagonal form by addition operations. Proc. Am. Math. Soc. 103(3), 741–746 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  73. Whitney, H.: Local properties of analytic varieties. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 205–244. Princeton University Press, Princeton (1965)

    Google Scholar 

  74. Wright, D.: The amalgamated free product structure of \(\mathrm {GL}_{2}(k[X_{1},\cdots,X_{n}])\) and the weak Jacobian theorem for two variables. J. Pure Appl. Algebra 12(3), 235–251 (1978)

    Article  MathSciNet  Google Scholar 

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  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Franc Forstnerič

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Forstnerič, F. (2017). Applications of Oka Theory and Its Methods. In: Stein Manifolds and Holomorphic Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-319-61058-0_8

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