New Recursions for Genus-zero Gromov-witten Invariants

New relations among the genus-zero Gromov-Witten invariants of a complex projective manifold X are exhibited. When the cohomology of X is generated by divisor classes and classes “with vanishing one-point invariants,” the relations determine many-point invariants in terms of one-point invariants. 0. Introduction The localization theorem for equivariant cohomology has recently been used with great success to compute the genus-zero Gromov-Witten invariants relevant to the mirror conjecture [12, 19, 5]. Zero-point invariants count expected numbers of rational curves on a projective manifold X , while the more general m-point invariants count expected numbers of rational curves meeting m given submanifolds (or cohomology classes). For the the mirror conjecture, only the zero and one-point invariants are computed, though for the construction of the quantum product (even the small version), one needs more general invariants. In this paper we will apply the localization theorem to study genus-zero GromovWitten invariants involving any number of marked points. A straightforward generalization of Givental’s (one variable) J-function yields homology-valued J-functions in any number of variables t1, . . . ,tm which encode all the (generalized) m-point genus-zero invariants. Our main theorem is a collection of relations among these J-functions expressing a part of the J-function for a fixed curve class and number of variables in terms of the J-functions involving fewer variables and/or “smaller” curve classes. When the cohomology of X is generated by divisor classes, or, more generally, when every class orthogonal to the subring generated by divisor classes annihilates (via cap product) all one-variable J-functions, then these new relations completely determine all m-point genus-zero Gromov-Witten invariants (of classes generated by divisor classes) in terms of one-point invariants. That is, in this setting, the one-variable J-function determines all the others. A complete intersection in P has this orthogonality property, and in that case we exhibit a formula expressing “mixed” two-point invariants in terms of one-point invariants. We apply this new formula to compute previously unknown quantum products of cohomology classes on Fano complete intersections, where the one-variable J-function is known. Since our recursions do not require any positivity of X , they would apply just as well to general-type complete intersections. Unfortunately, in those cases, the one-variable J-function is not known. 1991 Mathematics Subject Classification. Primary: 14N35. Secondary: 14A20, 14D20, 14H10, 53D45.