The symplectic sum formula for Gromov-Witten invariants. (English) Zbl 1075.53092

Ionel, Eleny-Nicoleta; Parker, Thomas H.

Ann. Math. (2) 159, No. 3, 935-1025 (2004).

For two symplectic \(2n\)-manifolds \(X\) and \(Y\) with a symplectic \((2n-2)\)-submanifold \(V\) and a symplectic identification between two copies of \(V\) and a complex anti-linear isomorphism between normal bundles \(N_XV\) and \(N_YV\) of \(V\) in \(X\) and \(Y\), the symplectic sum \(Z= X\# Y\) is constructed. The symplectic sum formula expresses the Gromov-Witten invariants of \(Z\) in terms of relative GW invariants of \(X\) and \(Y\). Applications: new proofs for recent results in enumerative geometry: see E.-N. Ionel and Th. H. Parker [Math. Res. Lett. 5, No. 5, 563–576 (1998; Zbl 0943.53046) and Math. Ann. 314, No. 1, 127–158 (1999; Zbl 0948.53047)] and L. Lafforgue [Invent. Math. 136, No. 1, 233–271 (1999; Zbl 0965.14024) and Erratum, Invent. Math. 145, No. 3, 619–620 (2001; Zbl 1041.14004)].

Reviewer: Maido Rahula (Tartu)

MSC:

53D45

Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Keywords:

symplectic sum; Gromov-Witten invariants

Citations:

Zbl 0943.53046; Zbl 0948.53047; Zbl 0965.14024; Zbl 1041.14004