Extremal Transition and Quantum Cohomology: Examples of Toric Degeneration
Abstract
When a singular projective variety Xsing admits a projective crepant resolution Xres and a smoothing Xsm, we say that Xres and Xsm are related by extremal transition. In this paper, we study a relationship between the quantum cohomology of Xres and Xsm in some examples. For three dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing Xsm is isomorphic to a certain subquotient of the quantum cohomology of a resolution Xres with the quantum variables of exceptional curves specialized to one. We observe that similar phenomena happen for toric degenerations of Fl(1,2,3), Gr(2,4) and Gr(2,5) by explicit computations.
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