When a singular projective variety admits a projective crepant resolution and a smoothing , we say that and are related by extremal transition. In this article, we study a relationship between the quantum cohomology of and in some examples. For -dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing is isomorphic to a certain subquotient of the quantum cohomology of a resolution with the quantum variables of exceptional curves specialized to one. We observe that similar phenomena happen for toric degenerations of , , and by explicit computations.
"Extremal transition and quantum cohomology: Examples of toric degeneration." Kyoto J. Math. 56 (4) 873 - 905, December 2016. https://doi.org/10.1215/21562261-3664959