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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.

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.

Reviewer: Qingtao Chen (Trieste)

### MSC:

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