Extremal Transition and Quantum Cohomology: Examples of Toric Degeneration
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.
Keywords
Cite
@article{arxiv.1504.05013,
title = {Extremal Transition and Quantum Cohomology: Examples of Toric Degeneration},
author = {Hiroshi Iritani and Jifu Xiao},
journal= {arXiv preprint arXiv:1504.05013},
year = {2016}
}
Comments
29 pp, in v2 and v3, license statement changed. in v3 thanks to C.Z. Li added. in v4, the comparison of the cohomology of Gr(2,5) and the cohomology of the small resolution of toric degeneration of Gr(2,5) is added. a conjecture on general cases is added