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A product formula for Gromov-Witten invariants. (English) Zbl 1273.53071

Y. Ruan and G. Tian [J. Differ. Geom. 42, No. 2, 259–367 (1995; Zbl 0860.58005)], M. Kontsevich and Yu. Manin [Invent. Math. 124, No. 1–3, 313–339 (1996; Zbl 0853.14021)] dealt with the special case when the Hamiltonian fibration is a product of two symplectic manifolds of more general situation considered than in this article. They proved that when both the base and the fiber are semi-simple, the GW-invariants of the total space are given by products of related GW-invariants in the base and in the fiber. Anyway, the general case is much more difficult to study.
The author of this article resolved the problem by giving a product formula for Gromov-Witten invariants by assuming that the reference fiber of the Hamiltonian fibration is semi-positive relative to the total space with connected fiber and over any connected symplectic base. Technically, he first establishes a correspondence between the moduli space of pseudo-holomorphic maps with marked points in the total space of the Hamiltonian fibration and the corresponding moduli space of pseudo-holomorphic maps with marked point in the base. Then he uses it to find the product formula by integration. Also, several interesting applications are discussed in this article.

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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