Gromov Compactness Theorem

Theorem7.1Gromov Compactness

Let $(M, \omega)$ be a compact symplectic manifold and $J_n$ a sequence of $\omega$ -compatible almost complex structures converging to $J_\infty$ . If $u_n: (\Sigma_n, j_n) \to (M, J_n)$ is a sequence of $J_n$ -holomorphic curves with $[u_n] = A$ and $E(u_n) = \omega(A)$ uniformly bounded, then a subsequence converges (in the Gromov topology) to a stable map: a $J_\infty$ -holomorphic map from a nodal Riemann surface $\Sigma_\infty$ representing the class $A$ .

The convergence is smooth away from finitely many points where bubbling occurs. At each bubble point, energy concentrates and a sphere component is created.

ExampleBubble Tree

A sequence of degree-2 holomorphic maps $u_n: \mathbb{CP}^1 \to \mathbb{CP}^2$ may degenerate to a "bubble tree": two degree-1 maps (lines) joined at a node, when energy $\pi$ concentrates at a point. The total energy $2\pi$ is preserved.

RemarkStable Maps and Compactification

The Kontsevich moduli space $\overline{\mathcal{M}}_{g,k}(M, A)$ of stable maps provides a compactification of the moduli space of smooth holomorphic curves. A stable map is a holomorphic map from a nodal curve such that each component has finitely many automorphisms (stability). This compactification is essential for defining Gromov-Witten invariants as intersection numbers.