Extremal transition and quantum cohomology: Examples of toric degeneration

Abstract

When a singular projective variety $X_{\mathrm{sing}}$ admits a projective crepant resolution $X_{\mathrm{res}}$ and a smoothing $X_{\mathrm{sm}}$, we say that $X_{\mathrm{res}}$ and $X_{\mathrm{sm}}$ are related by extremal transition. In this article, we study a relationship between the quantum cohomology of $X_{\mathrm{res}}$ and $X_{\mathrm{sm}}$ in some examples. For $3$-dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing $X_{\mathrm{sm}}$ is isomorphic to a certain subquotient of the quantum cohomology of a resolution $X_{\mathrm{res}}$ 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.