Symplectic sum

Suppose symplectic manifolds $(M_1^{2n},{\omega }_1)$ and $(M_2^{2n},{\omega }_2)$ contain compact codimension-two symplectic submanifolds $V_1,V_2$. A symplectomorphism $V_1\to V_2$ identifies their restricted forms. To form their symplectic sum, their symplectic normal bundles, viewed as complex line bundles, must have opposite First Chern classes under this identification. Choosing a fiber-reversing normal-bundle identification then permits removal of tubular neighborhoods and gluing of their circle boundaries.^[ref]

The local smoothing resembles $xy=\epsilon$ in the two normal directions. After adjusting the forms near the gluing locus, the joined manifold carries a symplectic form. The identification is part of the construction, so the result is not determined solely by the two ambient manifolds.^[ref]

In real dimension four, the surfaces must have matching genus and symplectic area, and their self-intersection numbers must sum to zero. For square-zero surfaces the normal bundles are trivial, giving the symplectic fiber sum along a square-zero surface.

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 $L(4,1)$.