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 equations. 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.
Definition Let X {\displaystyle X} be an almost complex manifold with almost complex structure J {\displaystyle J} . Let C {\displaystyle C} be a smooth Riemann surface (also called a complex curve) with complex structure j {\displaystyle j} . A pseudoholomorphic curve in X {\displaystyle X} is a map f : C → X {\displaystyle f:C\to X} that satisfies the Cauchy–Riemann equation
∂ ¯ j , J f := 1 2 ( d f + J ∘ d f ∘ j ) = 0. {\displaystyle {\bar {\partial }}_{j,J}f:={\frac {1}{2}}(df+J\circ df\circ j)=0.}
Since J 2 = − 1 {\displaystyle J^{2}=-1} , this condition is equivalent to
J ∘ d f = d f ∘ j , {\displaystyle J\circ df=df\circ j,}
which simply means that the differential d f {\displaystyle df} is complex-linear, that is, J {\displaystyle J} maps each tangent space
T x f ( C ) ⊆ T x X {\displaystyle T_{x}f(C)\subseteq T_{x}X}
to itself. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term ν {\displaystyle \nu } and to study maps satisfying the perturbed Cauchy–Riemann equation
∂ ¯ j , J f = ν . {\displaystyle {\bar {\partial }}_{j,J}f=\nu .}
A pseudoholomorphic curve satisfying this equation can be called, more specifically, a ( j , J , ν ) {\displaystyle (j,J,\nu )} -holomorphic curve. The perturbation ν {\displaystyle \nu } 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 X {\displaystyle X} , 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 C {\displaystyle C} of fixed genus g {\displaystyle g} and we introduce n {\displaystyle n} marked points (or punctures) on C {\displaystyle C} . As soon as the punctured Euler characteristic 2 − 2 g − n {\displaystyle 2-2g-n} is negative, there are only finitely many holomorphic reparametrizations of C {\displaystyle C} that preserve the marked points. The domain curve C {\displaystyle C} is an element of the Deligne–Mumford moduli space of curves.
Analogy with the classical Cauchy–Riemann equations The classical case occurs when X {\displaystyle X} and C {\displaystyle C} are both simply the complex number plane. In real coordinates
j = J = [ 0 − 1 1 0 ] , {\displaystyle j=J={\begin{bmatrix}0&-1\\1&0\end{bmatrix}},}
and
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