Count of genus zero J -holomorphic curves in dimensions four and six

An application of Gromov–Witten invariants is that they distinguish the deformation types of symplectic structures on a smooth manifold. In this manuscript, it is proven that the use of Gromov–Witten invariants in the class of embedded J -holomorphic spheres is restricted. This restriction is in the sense that they cannot distinguish the deformation types of symplectic structures on X1 × S 2 and X2 × S 2 for two minimal, simply connected, symplectic 4-manifolds X1 and X2 with b + 2 (X1) > 1 and b + 2 (X2) > 1. The result employs the adjunction inequality for symplectic 4-manifolds which is derived from Seiberg–Witten theory

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