A mirror theorem for toric complete intersections

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions. 0. Introduction. Let X denote a non-singular compact Kähler toric variety with the Picard number k. The variety X can be obtained by the symplectic reduction of the standard Hermitian space C by the action of a subtorus T k in the torus T of diagonal unitary matrices on a generic level of the momentum map (see Section 3). The coordinate hyperplanes in C are T -invariant, and their T -reductions on the same level of the momentum map define N compact toric hypersurfaces in X. We denote u1, ..., uN the classes in H (X) Poincare-dual to the fundamental cycles of these hypersurfaces. It is known that H(X) is a free abelian group of rank k spanned by u1, ..., uN , that it multiplicatively generates the ring H (X), and that the 1-st Chern class c(TX) of the tangent bundle to X is equal to u1 + ...+ uN . Let us consider the sum V of l ≥ 0 non-negative line bundles over X with the 1-st Chern classes v1, ..., vl and denote Y the non-singular complete intersection in X of dimension N − k − l defined by global holomorphic sections of these line bundles. The inclusion Y ⊂ X induces the homomorphism ∗Research is partially supported by Alfred P. Sloan Foundation and by NSF grant DMS-9321915