Pseudo-Holomorphic Curves

Pseudo-holomorphic (or $J$ -holomorphic) curves, introduced by Gromov in 1985, are the primary analytical tool in modern symplectic topology. They are maps from Riemann surfaces to symplectic manifolds satisfying a Cauchy-Riemann-type equation determined by a compatible almost complex structure.

Basic Definitions

Definition7.1Almost Complex Structure

An almost complex structure on a manifold $M$ is a bundle endomorphism $J: TM \to TM$ with $J^2 = -\mathrm{id}$ . On a symplectic manifold $(M, \omega)$ , $J$ is $\omega$ -compatible if $g(v,w) = \omega(v, Jw)$ defines a Riemannian metric ( $g$ is symmetric and positive definite). The space $\mathcal{J}(M, \omega)$ of compatible almost complex structures is contractible.

Definition7.2$J$-Holomorphic Curve

A $J$ -holomorphic curve is a smooth map $u: (\Sigma, j) \to (M, J)$ from a Riemann surface $(\Sigma, j)$ satisfying the Cauchy-Riemann equation: $\bar{\partial}_J u = \frac{1}{2}(du + J \circ du \circ j) = 0$ . Equivalently, $du$ is complex-linear: $du \circ j = J \circ du$ .

ExampleHolomorphic Spheres in $\mathbb{CP}^n$

For the standard complex structure $J_0$ on $\mathbb{CP}^n$ , $J_0$ -holomorphic curves are ordinary holomorphic maps. A degree- $d$ rational curve $u: \mathbb{CP}^1 \to \mathbb{CP}^n$ satisfies $\int u^*\omega_{FS} = d \cdot \pi$ and $\mu(u) = 2(n+1)d$ (Maslov index in the appropriate sense).

Energy and Compactness

Definition7.3Energy of a $J$-Holomorphic Curve

The energy of a $J$ -holomorphic curve $u: \Sigma \to M$ is $E(u) = \int_\Sigma u^*\omega = \frac{1}{2}\int_\Sigma |du|_g^2 \, \mathrm{dvol}_\Sigma$ . The equality of topological and analytic energy is a consequence of the Cauchy-Riemann equation and is the key to compactness.

RemarkGromov Compactness

The Gromov compactness theorem states that a sequence of $J$ -holomorphic curves with bounded energy has a subsequence converging to a stable map (a nodal curve with holomorphic components). Energy can concentrate at points, creating "bubbles" — sphere components attached at nodes. This compactness is essential for defining Gromov-Witten invariants.