Symplectic submanifold

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

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 $L(4,1)$.