Extremal Transition and Quantum Cohomology: Examples of Toric Degeneration

TL;DR

This paper explores the relationship between quantum cohomology of resolutions and smoothings in extremal transitions, providing explicit examples involving toric degenerations of certain Grassmannians and flag varieties.

Contribution

It demonstrates that similar quantum cohomology phenomena observed in conifold transitions also occur in toric degenerations of specific algebraic varieties through explicit computations.

Findings

Quantum cohomology of smoothings relates to that of resolutions via specialization.

Explicit examples show similar phenomena in toric degenerations of Grassmannians and flag varieties.

Results extend understanding of extremal transitions in algebraic geometry.

Abstract

When a singular projective variety X_sing admits a projective crepant resolution X_res and a smoothing X_sm, we say that X_res and X_sm are related by extremal transition. In this paper, we study a relationship between the quantum cohomology of X_res and X_sm in some examples. For three dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing X_sm is isomorphic to a certain subquotient of the quantum cohomology of a resolution X_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.