Skip to main content
SpringerLink
Log in

On genericity for holomorphic curves in four-dimensional almost-complex manifolds

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We consider spaces of immersed (pseudo-)holomorphic curves in an almost complex manifold of dimension four. We assume that they are either closed or compact with boundary in a fixed totally real surface, so that the equation for these curves is elliptic and has a Fredholm index. We prove that this equation is regular if the Chern class is ≥ 1 (in the case with boundary, if the ambient Maslov number is ≥ 1). Then the spaces of holomorphic curves considered will be manifolds of dimension equal to the index.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Gromov, M. Pseudo-holomorphic curves in symplectic manifolds.Invent. Math. 82, 307–347 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  2. Griffiths, P., and Harris, J.Principles of Algebraic Geometry. Wiley, New York, 1978.

    MATH  Google Scholar 

  3. Hofer, H. Pseudo-holomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three.Invent. Math. 114, 515–563 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  4. Ivashkovich, S., and Shevchishin, V. Pseudoholomorphic curves and envelopes of meromorphy of two-spheres in CP2, preprint, Essen/Bochum/Dusseldorf SFB237, January 1995.

  5. McDuff, D. Rational and ruled symplectic manifolds.J. Am. Math. Soc. 3, 679–711(1990).

    Article  MathSciNet  MATH  Google Scholar 

  6. McDuff, D. The local behaviour of holomorphic curves in almost-complex 4-manifolds.J. Diff. Geom. 34, 143–164 (1991).

    MathSciNet  MATH  Google Scholar 

  7. McDuff, D., and Salamon, D. Pseudo-holomorphic curves. Manuscript.

  8. McDuff, D., and Salamon, D.J-holomorphic curves and quantum cohomology.Amer. Math. Soc. Lect. Notes 6 (1994).

  9. Sikorav, J. C. Some properties of holomorphic curves in almost complex manifolds. Chapter V ofHolomorphic Curves in Symplectic Geometry (ed. M. Audin and J. Lafontaine). Birkhäuser, Basel, 1994.

    Google Scholar 

  10. Vekua, N. I.Generalized Analytic Functions. Pergamon, London, 1962.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute, New-York University, 251 Mercer Street, 10012, New-York, NY, USA

    Helmut Hofer

  2. U.F.R. Sciences, Mathématiques, I.U.F.M. de Toulouse, 118, route de Narbonne, F-31078, Toulouse cedex 4

    Véronique Lizan

  3. Laboratoire Emile Picard, UMR CNRS 5578, Université Paul Sabatier, 118, route de Narbonne, F-31062, Toulouse

    Jean-Claude Sikorav

Authors
  1. Helmut Hofer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Véronique Lizan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-Claude Sikorav
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helmut Hofer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hofer, H., Lizan, V. & Sikorav, JC. On genericity for holomorphic curves in four-dimensional almost-complex manifolds. J Geom Anal 7, 149–159 (1997). https://doi.org/10.1007/BF02921708

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02921708

Math Subject Classification

Key Words and Phrases

Search

Navigation