Concept
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In particular, they lead to the Gromov–Witten invariants and Floer homology, and play a prominent role in string theory. Let be an almost complex manifold with almost complex structure . Let be a smooth Riemann surface (also called a complex curve) with complex structure . A pseudoholomorphic curve in is a map that satisfies the Cauchy–Riemann equation Since , this condition is equivalent to which simply means that the differential is complex-linear, that is, maps each tangent space to itself. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term and to study maps satisfying the perturbed Cauchy–Riemann equation A pseudoholomorphic curve satisfying this equation can be called, more specifically, a -holomorphic curve. The perturbation is sometimes assumed to be generated by a Hamiltonian (particularly in Floer theory), but in general it need not be. A pseudoholomorphic curve is, by its definition, always parametrized. In applications one is often truly interested in unparametrized curves, meaning embedded (or immersed) two-submanifolds of , so one mods out by reparametrizations of the domain that preserve the relevant structure. In the case of Gromov–Witten invariants, for example, we consider only closed domains of fixed genus and we introduce marked points (or punctures) on . As soon as the punctured Euler characteristic is negative, there are only finitely many holomorphic reparametrizations of that preserve the marked points. The domain curve is an element of the Deligne–Mumford moduli space of curves. The classical case occurs when and are both simply the complex number plane. In real coordinates and where .