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Algebraic and symplectic Gromov-Witten invariants coincide. (English) Zbl 0970.14030
The purpose of this paper is to prove the following
Theorem: Let \(M\) be a complex projective manifold, \(R\in H_2(M,\mathbb Z)\), \(g,k\geq 0\). Let \({\mathcal C}_{R,g,k}(M)\) be the moduli space of stable maps of genus \(g\) with \(k\) marked points and representing class \(R\). Then the homology class associated to the algebraic virtual fundamental class (a Chow class on \({\mathcal C}_{R,g,k}(M))\) as in the paper by K. Behrend [Invent. Math. 127, 601-617 (1997; Zbl 0909.14007)] and the symplectic virtual fundamental class as in the paper by B. Siebert [“Gromov-Witten invariants for general symplectic manifolds”, preprint http://arXiv.org/abs/dg-ga/?9608005] coincide.

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14F35 Homotopy theory and fundamental groups in algebraic geometry
14D22 Fine and coarse moduli spaces
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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