nLab
pseudoholomorphic curve

Contents

Contents

Idea

Given a smooth Riemann surface Σ\Sigma with complex structure jj, and an almost complex manifold MM with almost complex structure JJ, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map f:ΣMf:\Sigma\to M whose differential commutes with the almost complex structure in the sense that the equation

Jdf=dfj J\circ d f = d f\circ j

holds. One can also consider a symplectic manifold (N,ω)(N,\omega) instead of MM, in which case one chooses an almost complex structure compatible with the symplectic form ω\omega.

A more general notion is of a pseudoholomorphic map.

Counting pseudoholomorphic curves with constraints to obtain topological invariants has been pioneered by Mikhail Gromov and later Andreas Floer.

References

Related nnLab entries include symplectic category, Lagrangian correspondence, Fukaya category, Gromov-Witten invariant, quantum cohomology.

  • S. K. Donaldson, What is…a pseudoholomorphic curve?, Notices AMS 52 (9): pp.1026–1027, pdf
  • wikipedia pseudoholomorphic curve
  • Mikhail Gromov, Pseudo-holomorphic curves in symplectic manifolds, Inventiones Math. 82 (1985) 307-347 pdf
  • Andreas Floer, Holomorphic curves and a Morse theory for fixed points of exact symplectomorphisms, in: Aspects dynamiques et topologiques des groupes infinis de transformation de la

    mécanique (Lyon, 1986), Travaux en Cours, vol. 25, pp. 49–60. Hermann, Paris, 1987

  • Kenji Fukaya, Kaoru Ono, Floer homology and Gromov-Witten invariant over integer of general symplectic manifolds—summary, in: Taniguchi Conference on Mathematics Nara ‘98,

    Adv. Stud. Pure Math. 31, 75–91. Math. Soc. Japan, Tokyo, 2001.

  • Brett Parker, Integral counts of pseudo-holomorphic curves, arxiv/1309.0585

Last revised on August 22, 2018 at 09:53:27. See the history of this page for a list of all contributions to it.