QUANTUM COHOMOLOGY AND CLOSED-STRING MIRROR SYMMETRY FOR TORIC VARIETIES

We give a short new computation of the quantum cohomology of an arbitrary smooth (semiprojective) toric variety X, by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. In contrast to previous results of this kind, X need not be compact. The proof is based on the purely algebraic fact that a class of generalized Jacobian rings associated to X are free as modules over the Novikov ring. When X is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.