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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.

Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties