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.

中文翻译：

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复曲面品种的量子同调和闭弦镜对称

通过直接显示Fukaya–Oh–Ohta–Ono的Kodaira–Spencer映射将同构定义到合适的Jacobian上，我们对任意光滑（半投射）复曲面变体| $ X $ |⁠的量子同调进行了简短的新计算。环。与以前的这种结果相反，| $ X $ | 不必紧凑。该证明基于纯粹的代数事实，即与| $ X $ |相关的一类广义Jacobian环 作为Novikov环上的模块免费提供。当| $ X $ | 是单调的，我们获得的表示是完全明确的，仅使用具有标准复杂结构的众所周知的计算即可。