Kyoto Journal of Mathematics

Extremal transition and quantum cohomology: Examples of toric degeneration

Hiroshi Iritani and Jifu Xiao

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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 article, we study a relationship between the quantum cohomology of Xres and Xsm in some examples. For 3-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.

Article information

Kyoto J. Math., Volume 56, Number 4 (2016), 873-905.

Received: 23 April 2015
Revised: 20 October 2015
Accepted: 26 November 2015
First available in Project Euclid: 7 November 2016

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 14E30: Minimal model program (Mori theory, extremal rays)

quantum cohomology extremal transition conifold transition toric degeneration partial flag variety


Iritani, Hiroshi; Xiao, Jifu. Extremal transition and quantum cohomology: Examples of toric degeneration. Kyoto J. Math. 56 (2016), no. 4, 873--905. doi:10.1215/21562261-3664959.

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