A Reconstruction Theorem for Genus Zero Gromov-witten Invariants of Stacks

We generalize the First Reconstruction Theorem of Kontsevich and Manin in two respects. First, we allow the target space to be a DeligneMumford stack. Second, under some convergence assumptions, we show it suffices to check the hypothesis of H-generation not on the cohomology ring, but on an any quantum ring in the family given by small quantum cohomology. Introduction The main goal of this article is to prove an appropriate extension of the First Reconstruction Theorem of Kontsevich and Manin [10, Theorem 3.1] in genus zero Gromov-Witten theory to the case where the target is a Deligne-Mumford stack. More precisely, consider the following version of [ibid]: Theorem 0.1. Let X be a smooth projective variety. Suppose that H(X) is generated by H(X), then all genus zero Gromov-Witten invariants can be uniquely reconstructed starting from 3-point invariants. The proof relies mainly on the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation in genus zero Gromov-Witten theory and uses the hypothesis in two ways. First, note that the divisor axiom applies to classes in H(X). Second, note that the degree zero 3-point invariants on X form the structure constants of the cup product on H(X), so the hypothesis can be viewed as a condition on degree zero 3-point invariants. Now, let X be a smooth Deligne-Mumford stack with projective coarse moduli space X . Gromov-Witten invariants on X pull back classes from H(Īμ(X )), where Īμ(X ) is a stack naturally associated to X called the rigidified inertia stack. Īμ(X ) contains X as an open and closed substack, and the subspace H(X ) ⊆ H(Īμ(X )) is called the untwisted sector. To see how Theorem 0.1 might be generalized to this context consider the following results. In [1] and [2], the WDVV equation is extended to the case to the case of Deligne-Mumford stacks. There is a divisor axiom which applies to classes in H(X ) ⊂ H(Īμ(X )), and degree zero 3-point invariants determine a new product on the vector space H(Īμ(X )). The new ring is called the orbifold cohomology ring of X and is denoted H orb(X ) [7]. The following proposition then follows via the same technique as the proof of Theorem 0.1. Proposition 0.2. Let X be a smooth Deligne-Mumford stack with projective coarse moduli space. Suppose that H orb(X ) is generated by H (X ), then all genus zero Gromov-Witten invariants can be reconstructed starting from 3-point invariants. However, this is only a theoretical generalization: unless X is a scheme H orb(X ) is never generated by H(X ). Thus we search for a more useful generalization. 1