compact manifold, a symplectic form on it induces a nonzero cohomology class But this second cohomology group is zero unless . So the symplectic sum is possible only along a submanifold of codimension two. Identity element Given with codimension-two symplectic submanifold , one may projectively complete the normal bundle of in to the -bundle This contains two canonical copies of : the zero-section , which has normal bundle equal to that of in , and the infinity-section , which has opposite normal bundle. Therefore, one may symplectically sum with ; the result is again , with now playing the role of : So for any particular pair there exists an identity element for the symplectic sum. Such identity elements have been used both in establishing theory and in computations; see below. Symplectic sum and cut as deformation It is sometimes profitable to view the symplectic sum as a family of manifolds. In this framework, the given data , , , , , determine a unique smooth -dimensional symplectic manifold and a fibration in which the central fiber is the singular space obtained by joining the summands along , and the generic fiber is a symplectic sum of the . (That is, the generic fibers are all members of the unique isotopy class of the symplectic sum.) Loosely speaking, one constructs this family as follows. Choose a nonvanishing holomorphic section of the trivial complex line bundle Then, in the direct sum with representing a normal vector to in , consider the locus of the quadratic equation for a chosen small . One can glue both (the summands with deleted) onto this locus; the result is the symplectic sum . As varies, the sums naturally form the family described above. The central fiber is the symplectic cut of the generic fiber. So the symplectic sum and cut can be viewed together as a quadratic deformation of symplectic manifolds. An important example occurs when one of the summands is an identity element . For then the generic fiber is a symplectic manifold and the central fiber is with the normal bundle of "pinched off at infinity" to form the -bundle . This is analogous to deformation to the normal cone along a smooth divisor in algebraic geometry. In fact, symplectic treatments of Gromov–Witten theory often use the symplectic sum/cut for "rescaling the target" arguments, while algebro-geometric treatments use deformation to the normal cone