nLab pseudoholomorphic curve

Idea

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

$$J\circ d f = d f\circ j$$

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

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 $n$Lab 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