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Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties
Quarterly Journal of Mathematics ( IF 0.704 ) Pub Date : 2020-01-11 , DOI: 10.1093/qmathj/haz056
Jack Smith

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.
更新日期:2020-01-11

 

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