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Topological Methods in Stein Geometry

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Stein Manifolds and Holomorphic Mappings

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Abstract

In this chapter we introduce more advanced topological methods to the study of geometric problems on Stein manifolds and to Oka theory. We begin by considering complex points of smooth real surfaces in complex surfaces. After proving the Lai formulae and the Eliashberg-Harlamov cancellation theorem, we explore connections between the generalized adjunction inequality and the existence of embedded or immersed surfaces with a Stein neighborhood basis in a given complex surface. We then show how the Seiberg-Witten theory bears upon these questions by arguments similar to those leading to the proof of the generalized Thom conjecture. In the second part we present the Eliashberg-Gompf construction of integrable Stein structures on almost complex manifolds with a suitable handlebody decomposition, and we prove the soft Oka principle to the effect that every continuous map from a Stein manifold \(X\) to an arbitrary complex manifold \(Y\) is homotopic to a holomorphic map provided that we allow homotopic deformations of the Stein structure on \(X\) and, in real dimension four, also a change of the underlying smooth structure on \(X\).

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References

  1. Alexander, H.: Linking and holomorphic hulls. J. Differ. Geom. 38(1), 151–160 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 4. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  3. Bedford, E., Klingenberg, W.: On the envelope of holomorphy of a 2-sphere in \(\mathbf{C}^{2}\). J. Am. Math. Soc. 4(3), 623–646 (1991)

    MathSciNet  MATH  Google Scholar 

  4. Bharali, G.: Surfaces with degenerate CR singularities that are locally polynomially convex. Mich. Math. J. 53(2), 429–445 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bharali, G.: Polynomial approximation, local polynomial convexity, and degenerate CR singularities. J. Funct. Anal. 236(1), 351–368 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bishop, E.: Differentiable manifolds in complex Euclidean space. Duke Math. J. 32, 1–21 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chern, S.S., Spanier, E.H.: A theorem on orientable surfaces in four-dimensional space. Comment. Math. Helv. 25, 205–209 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back. Symplectic Geometry of Affine Complex Manifolds. Am. Math. Soc. Colloquium Publications, vol. 59. Am. Math. Soc., Providence (2012)

    MATH  Google Scholar 

  9. Demailly, J.-P., Lempert, L., Shiffman, B.: Algebraic approximations of holomorphic maps from Stein domains to projective manifolds. Duke Math. J. 76(2), 333–363 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Derdzinski, A., Januszkiewicz, T.: Totally real immersions of surfaces. Trans. Am. Math. Soc. 362(1), 53–115 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Elgindi, A.M.: On the topological structure of complex tangencies to embeddings of \(S^{3}\) into \(\mathbb {C}^{3}\). N.Y. J. Math. 18, 295–313 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Elgindi, A.M.: A topological obstruction to the removal of a degenerate complex tangent and some related homotopy and homology groups. Int. J. Math. 26(5), 16 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Elgindi, A.M.: Totally real perturbations and nondegenerate embeddings of \(S^{3}\). N.Y. J. Math. 21, 1283–1293 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Eliashberg, Y.: Classification of overtwisted contact structures on 3-manifolds. Invent. Math. 98(3), 623–637 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eliashberg, Y.: Filling by holomorphic discs and its applications. In: Geometry of Low-Dimensional Manifolds, 2, Durham, 1989. London Math. Soc. Lecture Note Ser., vol. 151, pp. 45–67. Cambridge Univ. Press, Cambridge (1990)

    Google Scholar 

  16. Eliashberg, Y.: Topological characterization of Stein manifolds of dimension \(>2\). Int. J. Math. 1(1), 29–46 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eliashberg, Y.: Legendrian and transversal knots in tight contact 3-manifolds. In: Topological Methods in Modern Mathematics, Stony Brook, NY, 1991, pp. 171–193. Publish or Perish, Houston (1993)

    Google Scholar 

  18. Eliashberg, Y., Kharlamov, V.: Some remarks on the number of complex points of a real surface in a complex one. In: Proc. Leningrad Int. Topology Conf., 1982. Nauka Leningrad Otdel., Leningrad (1983)

    Google Scholar 

  19. Fintushel, R., Stern, R.J.: Immersed spheres in 4-manifolds and the immersed Thom conjecture. Turk. J. Math. 19(2), 145–157 (1995)

    MathSciNet  MATH  Google Scholar 

  20. Forstnerič, F.: Complex tangents of real surfaces in complex surfaces. Duke Math. J. 67(2), 353–376 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Forstnerič, F.: Complements of Runge domains and holomorphic hulls. Mich. Math. J. 41(2), 297–308 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Forstnerič, F.: Stein domains in complex surfaces. J. Geom. Anal. 13(1), 77–94 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Forstnerič, F.: A complex surface admitting a strongly plurisubharmonic function but no holomorphic functions. J. Geom. Anal. 25(1), 329–335 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Forstnerič, F., Slapar, M.: Deformations of Stein structures and extensions of holomorphic mappings. Math. Res. Lett. 14(2), 343–357 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Forstnerič, F., Slapar, M.: Stein structures and holomorphic mappings. Math. Z. 256(3), 615–646 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Forstnerič, F., Stout, E.L.: A new class of polynomially convex sets. Ark. Mat. 29(1), 51–62 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. Freedman, M.H., Quinn, F.: Topology of 4-Manifolds. Princeton Mathematical Series, vol. 39. Princeton University Press, Princeton (1990)

    MATH  Google Scholar 

  28. Friedman, R., Morgan, J.W.: Smooth Four-Manifolds and Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 27. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  29. Fulton, W.: Intersection Theory, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 2. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  30. Givental’, A.: Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. Funct. Anal. Appl. 20, 197–203 (1986)

    Article  MATH  Google Scholar 

  31. Gompf, R.E.: Minimal genera of open 4-manifolds. Geom. Topol. 21(1), 107–155 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gompf, R.E.: Handlebody construction of Stein surfaces. Ann. Math. (2) 148(2), 619–693 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gompf, R.E.: Stein surfaces as open subsets of \({\mathbb {C}}^{2}\). J. Symplectic Geom. 3(4), 565–587 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gompf, R.E., Stipsicz, A.I.: 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, vol. 20. Am. Math. Soc., Providence (1999)

    MATH  Google Scholar 

  35. 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 

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

    Book  MATH  Google Scholar 

  37. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

    MATH  Google Scholar 

  38. Hatcher, A.E.: Notes on basic 3-manifold topology. http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html

  39. Hill, C.D., Taiani, G.: Families of analytic discs in \({\mathbb {C}}^{n}\) with boundaries on a prescribed CR submanifold. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 5(2), 327–380 (1978)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Hirzebruch, F., Hopf, H.: Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten. Math. Ann. 136, 156–172 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jöricke, B.: Local polynomial hulls of discs near isolated parabolic points. Indiana Univ. Math. J. 46(3), 789–826 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kallin, E.: Polynomial convexity: the three spheres problem. In: Proc. Conf. Complex Analysis, Minneapolis, 1964, pp. 301–304. Springer, Berlin (1965)

    Chapter  Google Scholar 

  44. Kasuya, N., Takase, M.: Knots and links of complex tangents. ArXiv e-prints (2016). arXiv:1606.03704

  45. Kenig, C.E., Webster, S.M.: The local hull of holomorphy of a surface in the space of two complex variables. Invent. Math. 67(1), 1–21 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kronheimer, P.B., Mrowka, T.S.: The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1(6), 797–808 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kronheimer, P.B., Mrowka, T.S.: Embedded surfaces and the structure of Donaldson’s polynomial invariants. J. Differ. Geom. 41(3), 573–734 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kruzhilin, N.G.: Two-dimensional spheres on the boundaries of pseudoconvex domains in \({\mathbb {C}}^{2}\). Izv. Akad. Nauk SSSR, Ser. Mat. 55(6), 1194–1237 (1991)

    MathSciNet  MATH  Google Scholar 

  49. Lai, H.F.: Characteristic classes of real manifolds immersed in complex manifolds. Trans. Am. Math. Soc. 172, 1–33 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  50. Lempert, L.: Algebraic approximations in analytic geometry. Invent. Math. 121(2), 335–353 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Lisca, P., Matić, G.: Tight contact structures and Seiberg-Witten invariants. Invent. Math. 129(3), 509–525 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  52. Massey, W.S.: Proof of a conjecture of Whitney. Pac. J. Math. 31, 143–156 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  53. Morgan, J.W.: The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. Mathematical Notes, vol. 44. Princeton University Press, Princeton (1996)

    MATH  Google Scholar 

  54. Morgan, J.W., Szabó, Z., Taubes, C.H.: A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture. J. Differ. Geom. 44(4), 706–788 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  55. Moser, J.K., Webster, S.M.: Normal forms for real surfaces in \({\mathbb {C}}^{2}\) near complex tangents and hyperbolic surface transformations. Acta Math. 150(3–4), 255–296 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  56. Nemirovski, S.: Adjunction inequality and coverings of Stein surfaces. Turk. J. Math. 27(1), 161–172 (2003)

    MathSciNet  MATH  Google Scholar 

  57. Nemirovski, S., Siegel, K.: Rationally convex domains and singular Lagrangian surfaces in \({\mathbb {C}}^{2}\). Invent. Math. 203(1), 333–358 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  58. Nemirovskiĭ, S.Y.: Complex analysis and differential topology on complex surfaces. Usp. Mat. Nauk 54(4(328)), 47–74 (1999)

    Article  MathSciNet  Google Scholar 

  59. Ozsváth, P., Szabó, Z.: The symplectic Thom conjecture. Ann. Math. (2) 151(1), 93–124 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  60. Prezelj, J., Slapar, M.: The generalized Oka-Grauert principle for 1-convex manifolds. Mich. Math. J. 60(3), 495–506 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  61. Rudin, W.: Totally real Klein bottles in \({\mathbb {C}}^{2}\). Proc. Am. Math. Soc. 82(4), 653–654 (1981)

    MathSciNet  MATH  Google Scholar 

  62. Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in \(N=2\) supersymmetric Yang-Mills theory. Nucl. Phys. B 426(1), 19–52 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  63. Shevchishin, V.V.: Lagrangian embeddings of the Klein bottle and the combinatorial properties of mapping class groups. Izv. Ross. Akad. Nauk Ser. Mat. 73(4), 153–224 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  64. Slapar, M.: On Stein neighborhood basis of real surfaces. Math. Z. 247(4), 863–879 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  65. Slapar, M.: Real surfaces in elliptic surfaces. Int. J. Math. 16(4), 357–363 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  66. Slapar, M.: Cancelling complex points in codimension two. Bull. Aust. Math. Soc. 88(1), 64–69 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  67. Slapar, M.: Modeling complex points up to isotopy. J. Geom. Anal. 23(4), 1932–1943 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  68. Slapar, M.: CR regular embeddings and immersions of compact orientable 4-manifolds into \({\mathbb {C}}^{3}\). Int. J. Math. 26(5), 5 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  69. Slapar, M.: On complex points of codimension 2 submanifolds. J. Geom. Anal. 26(1), 206–219 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  70. Stein, K.: Überlagerungen holomorph-vollständiger komplexer Räume. Arch. Math. (Basel) 7, 354–361 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  71. Stout, E.L.: Algebraic domains in Stein manifolds. In: Proceedings of the Conference on Banach Algebras and Several Complex Variables, New Haven, Conn., 1983. Contemp. Math., vol. 32, pp. 259–266. Am. Math. Soc., Providence (1984)

    Chapter  Google Scholar 

  72. Stout, E.L.: Polynomial Convexity. Progress in Mathematics, vol. 261. Birkhäuser Boston, Boston (2007)

    MATH  Google Scholar 

  73. Sukhov, A., Tumanov, A.: Pseudoholomorphic discs near an elliptic point. Tr. Mat. Inst. Steklova 253, 296–303 (2006). Kompleks. Anal. i Prilozh.

    MathSciNet  MATH  Google Scholar 

  74. Taubes, C.H.: The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett. 1(6), 809–822 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  75. Taubes, C.H.: \(\mathrm{SW}\Rightarrow\mathrm{Gr}\): from the Seiberg-Witten equations to pseudo-holomorphic curves. J. Am. Math. Soc. 9(3), 845–918 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  76. Webster, S.M.: Minimal surfaces in a Kähler surface. J. Differ. Geom. 20(2), 463–470 (1984)

    Article  MATH  Google Scholar 

  77. Weinstein, A.: Lectures on Symplectic Manifolds. Regional Conference Series in Mathematics, vol. 29. Am. Math. Soc., Providence (1977). Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976

    Book  MATH  Google Scholar 

  78. Weinstein, A.: Contact surgery and symplectic handlebodies. Hokkaido Math. J. 20(2), 241–251 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  79. Whitney, H.: The self-intersections of a smooth \(n\)-manifold in \(2n\)-space. Ann. Math. (2) 45, 220–246 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  80. Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1(6), 769–796 (1994)

    Article  MathSciNet  MATH  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). Topological Methods in Stein Geometry. 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_10

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