Annals of Global Analysis and Geometry

, Volume 11, Issue 2, pp 125–133 | Cite as

Local invariants of singular surfaces in an almost complex four-manifold

  • Goo Ishikawa
  • Toru Ohmoto
Article

Abstract

In this paper we define two local invariants, the local self-intersection index and the Maslov index, for singular surfaces in an almost complex four-manifold and prove formulae involving these invariants, which generalize formulae of Lai and Givental.

MSC 1991

58C27 58F05 

Key words

Singular surface local self-intersection index Maslov index open Whitney umbrella 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Arnol'd, V.I.:Singularities of Caustics and Wave Fronts. Kluwer Academic Publishers, 1990.Google Scholar
  2. [2]
    Banchoff, T.; Farris, F.: Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space.Pacific J. Math. To appear.Google Scholar
  3. [3]
    Bennequin, B.: Entrelacements et équations de Pfaff.Astérisque 107–108 (1983), 87–161.Google Scholar
  4. [4]
    Bennequin, B.: Topologie symplectique, convexité holomorphe et structure de contact.Astérisque 189–190 (1990), 285–323.Google Scholar
  5. [5]
    Bishop, E.: Differentiable manifolds in complex Euclidean space.Duke Math. J. 32 (1965), 1–21.Google Scholar
  6. [6]
    Chern, S.S.;Spanier, E.: A theorem on orientable surfaces in four-dimensional space.Comment. Math. Helv. 25 (1951), 205–209.Google Scholar
  7. [7]
    Eisenbud, D.;Levine, H.I.: An algebraic formula for the degree of aC map germ.Ann. of Math. 106 (1977), 19–44.Google Scholar
  8. [8]
    Fiedler, T.: Exceptional points on surfaces in four-spaces.Ann. Global Anal. Geom. 3 (1985), 219–231.Google Scholar
  9. [9]
    Fiedler, T.: A characteristic class for totally real surfaces in the Grassmannian of two-planes in four-space.Ann. Global Anal. Geom. 4 (1986), 121–132.Google Scholar
  10. [10]
    Fiedler, T.: Complex plane curves in the ball.Invent. Math. 95 (1989), 479–506.Google Scholar
  11. [11]
    Friedrich, Th.: On surfaces in four-spaces.Ann. Global Anal. Geom. 2 (1984), 257–287.Google Scholar
  12. [12]
    Fukuda, T.: Local topological properties of differentiable mappings, I.Invent. Math. 65 (1981/82), 227–250.Google Scholar
  13. [13]
    Givental, A.B.: Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. Funktsional. Anal. i Prilozhen.20 (1986), 35–41.Google Scholar
  14. [14]
    Givental, A.B.: Singular Lagrangian varieties and their Lagrangian mappings. In:„Itogi Nauki i Tekhniki., Ser. Sovrem. Probl. Mat. (Contemporary Problems of Mathematics) 33“, VINITI, 1988, pp. 55–112.Google Scholar
  15. [15]
    Hirzebruch, H.:Topological Methods in Algebraic Geometry. Springer-Verlag, 1956.Google Scholar
  16. [16]
    Ishikawa, G.: The local model of an isotropic map-germ arising from one dimensional symplectic reduction.Math. Proc. Camb. Phil. Soc. 111 (1992), 103–112.Google Scholar
  17. [17]
    Lai, H.-F.: Characteristic classes of real manifolds immersed in complex manifolds.Trans. Amer. Math. Soc. 172 (1972), 1–33.Google Scholar
  18. [18]
    McDuff, D.: The local behavior of holomorphic curves in almost complex 4-manifolds.J. Diff. Geom. 34 (1991), 143–164.Google Scholar
  19. [19]
    Moser, J.;Webster, S.: Normal forms for real surfaces in\(\mathbb{C}^2 \) near complex tangents and hyperbolic surface transformations.Acta Math. 150 (1983), 255–296.Google Scholar
  20. [20]
    Ohmoto, T.:Thom polynomials for isotropic mappings. Preprint.Google Scholar
  21. [21]
    Viro, O.Y.:Some integral calculus based on Euler characteristics. In: „Lecture Notes in Math. vol. 1346“, Springer, pp. 127–138.Google Scholar
  22. [22]
    Weinstein, A.:Lectures on symplectic manifolds. Regional Conference Series in Math. 29, Amer. Math. Soc., 1977.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Goo Ishikawa
    • 1
  • Toru Ohmoto
    • 2
    • 3
  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan
  2. 2.Department of MathematicsTokyo Institute of TechnologyOhokayamaJapan
  3. 3.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations