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.

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.
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