Skip to main content
SpringerLink
Log in

Preliminaries

  • Chapter
  • First Online:
Stein Manifolds and Holomorphic Mappings

Abstract

In this preliminary chapter we review the basic notions concerning complex manifolds, complex spaces with singularities, holomorphic mappings, holomorphic fibre bundles and especially vector bundles, the tangent and cotangent bundle of a complex manifold, differential forms, de Rham and Dolbeault cohomology, plurisubharmonic functions, the Levi form, vector fields and their flows. We also recall a basic form of Gromov’s homotopy principle for first order partial differential relations that are ample in the coordinate directions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Applied Mathematical Sciences, vol. 75. Springer, New York (1988)

    Book  MATH  Google Scholar 

  2. Adams, J.F.: Lectures on Lie Groups. W. A. Benjamin, New York (1969)

    MATH  Google Scholar 

  3. Ahern, P., Flores, M., Rosay, J.-P.: On \({\mathbb {R}}^{+}\) and ℂ complete holomorphic vector fields. Proc. Am. Math. Soc. 128(10), 3107–3113 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematical Series, vol. 47. Princeton University Press, Princeton (1999)

    Book  MATH  Google Scholar 

  5. Behnke, H., Stein, K.: Modifikation komplexer Mannigfaltigkeiten und Riemannscher Gebiete. Math. Ann. 124, 1–16 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups. Graduate Texts in Mathematics, vol. 98. Springer, New York (1995). Corrected reprint of the 1985 translation

    MATH  Google Scholar 

  7. Brody, R.: Compact manifolds and hyperbolicity. Trans. Am. Math. Soc. 235, 213–219 (1978)

    MathSciNet  MATH  Google Scholar 

  8. Cartan, H.: Sur les fonctions de plusieurs variables complexes: les espaces analytiques. In: Proc. Internat. Congress Math., vol. 1958, pp. 33–52. Cambridge Univ. Press, New York (1960)

    Google Scholar 

  9. Chirka, E.: Complex Analytic Sets. Kluwer Academic Publishers, Dordrecht (1989). Translated from the Russian by R.A.M. Hoksbergen

    Book  MATH  Google Scholar 

  10. Demailly, J.-P.: Complex analytic and differential geometry. https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

  11. Donaldson, S.: Riemann Surfaces. Oxford Graduate Texts in Mathematics, vol. 22. Oxford University Press, Oxford (2011)

    Book  MATH  Google Scholar 

  12. Eisenman, D.A.: Intrinsic Measures on Complex Manifolds and Holomorphic Mappings. Memoirs of the Am. Math. Soc., vol. 96. Am. Math. Soc., Providence (1970)

    MATH  Google Scholar 

  13. Eliashberg, Y., Mishachev, N.: Introduction to the \(h\)-Principle. Graduate Studies in Mathematics, vol. 48. Am. Math. Soc., Providence (2002)

    MATH  Google Scholar 

  14. Fischer, G.: Complex Analytic Geometry. Lecture Notes in Math., vol. 538. Springer, Berlin (1976)

    MATH  Google Scholar 

  15. Fischer, W., Grauert, H.: Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten. Nachr. Akad. Wiss. Gött. Math.-Phys. Kl., 2B 1965, 89–94 (1965)

    MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Forstnerič, F.: Actions of \((\mathbb {R},+)\) and \((\mathbb {C},+)\) on complex manifolds. Math. Z. 223(1), 123–153 (1996)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Grauert, H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. (2) 68, 460–472 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grauert, H., Remmert, R.: Komplexe Räume. Math. Ann. 136, 245–318 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grauert, H., Remmert, R.: Theory of Stein Spaces. Grundlehren Math. Wiss., vol. 236. Springer, Berlin (1979). Translated from the German by Alan Huckleberry

    MATH  Google Scholar 

  23. Grauert, H., Remmert, R.: Coherent Analytic Sheaves. Grundlehren Math. Wiss., vol. 265. Springer, Berlin (1984)

    MATH  Google Scholar 

  24. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. John Wiley & Sons, New York (1994). Reprint of the 1978 original

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  26. Gromov, M.L.: Stable mappings of foliations into manifolds. Izv. Akad. Nauk SSSR, Ser. Mat. 33, 707–734 (1969)

    MathSciNet  Google Scholar 

  27. Gromov, M.L.: Convex integration of differential relations. I. Izv. Akad. Nauk SSSR, Ser. Mat. 37, 329–343 (1973)

    MathSciNet  MATH  Google Scholar 

  28. Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. AMS Chelsea Publishing, Providence (2009). Reprint of the 1965 original

    Book  MATH  Google Scholar 

  29. Hirsch, M.W.: Immersions of manifolds. Trans. Am. Math. Soc. 93, 242–276 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland Mathematical Library, vol. 7. North-Holland Publishing Co., Amsterdam (1990)

    MATH  Google Scholar 

  31. Hörmander, L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhäuser Boston, Boston (1994)

    MATH  Google Scholar 

  32. Josefson, B.: On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on \({\mathbb {C}}^{n}\). Ark. Mat. 16(1), 109–115 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  33. Jouanolou, J.P.: Une suite exacte de Mayer-Vietoris en \(K\)-théorie algébrique. In: Algebraic \(K\)-Theory, I: Higher \(K\)-Theories, Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972. Lecture Notes in Math., vol. 341, pp. 293–316. Springer, Berlin (1973)

    Google Scholar 

  34. Kaliman, S.: Some facts about Eisenman intrinsic measures. Complex Var. Theory Appl. 27(2), 163–173 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kelley, J.L.: General Topology. Graduate Texts in Mathematics, vol. 27. Springer, New York (1975). Reprint of the 1955 edition, Van Nostrand, Toronto, Ont.

    MATH  Google Scholar 

  36. Klimek, M.: Pluripotential Theory. London Mathematical Society Monographs. New Series, vol. 6. The Clarendon Press/Oxford University Press, New York (1991)

    MATH  Google Scholar 

  37. Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings. Pure and Applied Mathematics, vol. 2. Marcel Dekker, New York (1970)

    MATH  Google Scholar 

  38. Kobayashi, S.: Intrinsic distances, measures and geometric function theory. Bull. Am. Math. Soc. 82(3), 357–416 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  39. 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 

  40. Koebe, P.: Über die Uniformisierung beliebiger analytischer Kurven. Zweite Mitteilung. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1907, 633–669 (1907)

    MATH  Google Scholar 

  41. Lelong, P.: Définition des fonctions plurisousharmoniques. C. R. Acad. Sci. Paris 215, 398–400 (1942)

    MathSciNet  MATH  Google Scholar 

  42. Lelong, P.: Fonctions plurisousharmoniques et formes différentielles positives. Gordon & Breach, Paris (1968)

    MATH  Google Scholar 

  43. McDuff, D., Salamon, D.: \(J\)-Holomorphic Curves and Symplectic Topology, 2nd edn. Amer. Math. Soc. Colloquium Publications, vol. 52. Am. Math. Soc., Providence (2012)

    MATH  Google Scholar 

  44. Narasimhan, R.: Analysis on Real and Complex Manifolds. North-Holland Mathematical Library, vol. 35. North-Holland Publishing Co., Amsterdam (1985). Reprint of the 1973 edition

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Oka, K.: Collected Papers. Springer, Berlin (1984). Translated from the French by R. Narasimhan. With commentaries by H. Cartan. Edited by R. Remmert

    Book  MATH  Google Scholar 

  47. Phillips, A.: Submersions of open manifolds. Topology 6, 171–206 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  48. Poincaré, H.: Sur l’uniformisation des fonctions analytiques. Acta Math. 31, 1–64 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  49. Remmert, R.: Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes. C. R. Acad. Sci. Paris 243, 118–121 (1956)

    MathSciNet  MATH  Google Scholar 

  50. Remmert, R.: Holomorphe und meromorphe Abbildungen komplexer Räume. Math. Ann. 133, 328–370 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  51. Rudin, W.: Function Theory in the Unit Ball of \(\mathbb {C}^{n}\). Classics in Mathematics. Springer, Berlin (2008). Reprint of the 1980 edition

    MATH  Google Scholar 

  52. Serre, J-P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier 6, 1–42 (1955/1956)

    Google Scholar 

  53. Smale, S.: The classification of immersions of spheres in Euclidean spaces. Ann. Math. (2) 69, 327–344 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  54. Spring, D.: Convex Integration Theory. Solutions to the \(h\)-Principle in Geometry and Topology. Monographs in Mathematics, vol. 92. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  55. 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 

  56. Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer, New York (1983). Corrected reprint of the 1971 edition

    MATH  Google Scholar 

  57. Wells, R.O. Jr.: Differential Analysis on Complex Manifolds, 3rd edn. Graduate Texts in Mathematics, vol. 65. Springer, New York (2008)

    Book  MATH  Google Scholar 

  58. 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 

  59. Whitney, H., Bruhat, F.: Quelques propriétés fondamentales des ensembles analytiques-réels. Comment. Math. Helv. 33, 132–160 (1959)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Franc Forstnerič

Authors
  1. Franc Forstnerič
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Forstnerič, F. (2017). Preliminaries. 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_1

Download citation

Publish with us

Policies and ethics

Search

Navigation