Gromov-Witten invariants of a class of toric varieties.

Abstract

1.1. Background. Toric varieties admit a combinatorial description, which allows many invariants to be expressed in terms of combinatorial data. Batyrev [Ba2] and Morrison and Plesser [MP] describe the quantum cohomology rings of certain toric varieties, in terms of generators (divisors and formal q variables) and relations (linear relations and q-deformed monomial relations). The relations are easily obtained from the combinatorial data. Unfortunately, the relations alone do not tell us how to multiply cohomology classes in the quantum cohomology ring QH∗(X), or even how to express ordinary cohomology classes in H∗(X,Q) in terms of the given generators. In this paper, we give a formula that expresses any class inH∗(X,Q)—as a polynomial in divisor classes and formal q variables—for anyX belonging to a certain class of toric varieties. These expressions, along with the presentation of QH∗(X) via generators and relations, permit computation of any product of cohomology classes in QH∗(X). Let X be a complete toric variety of dimension n over the complex numbers (all varieties in this paper are over the complex numbers). This means X is a normal variety with an action by the algebraic torus (C∗)n and a dense equivariant embedding (C∗)n → X. By the theory of toric varieties (cf. [F]), such X are characterized by a fan ∆ of strongly convex polyhedral cones in N ⊗Z R, where N is the lattice Z. The cones are rational, that is, generated by lattice points. In particular, to every ray (1-dimensional cone) σ there is a unique generator ρ ∈ N such that σ ∩N = Z>0 · ρ. There is a one-to-one correspondence between such ray generators ρ and toric (i.e., torus-invariant) divisors of X. Given toric divisors D1, . . . , Dk, with corresponding ray generators ρ1, . . . , ρk, we have D1 ∩ · · · ∩Dk 6= ∅ if and only if ρ1, . . ., ρk span a cone in ∆. Hypotheses on X translate as follows into conditions on ∆: (i) X is nonsingular if and only if every cone is generated by a part of a Z-basis of N ; (ii) given that X is nonsingular: X is Fano (i.e., X has ample anticanonical class) if and only if the set of ray generators is strictly convex.