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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.

##### MSC:
 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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