Symplectic submanifold

A submanifold $S\subset (M,\omega )$ is symplectic when $\omega {|}_{TS}$ is nondegenerate.

Two symplectic surfaces meeting positively and transversely can be replaced near an intersection by a smooth symplectic surface representing the sum of their homology classes. In compatible complex coordinates the local union is $\{xy=0\}$ and its smoothing is $\{xy=ϵ\}$; a cutoff joins this local model to the original surfaces.

For a connected embedded symplectic surface $C$ in a symplectic four-manifold,where $K_{\omega }=-c_1(TM,J)$ is the symplectic canonical class for any compatible almost complex structure.

Distinct simple $J$-holomorphic curves in an almost complex four-manifold have positive local intersection multiplicities. Consequently their homological intersection number is nonnegative whenever they intersect in finitely many points.

The symplectic normal bundle of $S$ is the symplectic orthogonal complement $(TS{)}^{\omega }\subset TM{|}_S$. It is a symplectic vector bundle and is naturally isomorphic to the ordinary normal bundle.

If a symplectomorphism between compact symplectic submanifolds lifts to an isomorphism of their symplectic normal bundles, then it extends to a symplectomorphism between neighborhoods. Thus the germ of a symplectic neighborhood is determined by the restricted form and the symplectic normal bundle.

A symplectic sum removes tubular neighborhoods of symplectic submanifolds whose normal bundles have opposite Euler classes and glues their boundaries by a fiber-reversing identification.

A symplectic sphere of self-intersection $-4$ can be rationally blown down by taking the symplectic sum with ${\mathbb{CP}}^2$ along the sphere and a smooth conic of self-intersection $+4$. Equivalently, its disk-bundle neighborhood is replaced by the rational ball ${\mathbb{CP}}^2\setminus \nu (Q)$ whose boundary is the lens space $L(4,1)$.