Kyoto Journal of Mathematics

Extremal transition and quantum cohomology: Examples of toric degeneration

Hiroshi Iritani and Jifu Xiao

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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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1478509222

Digital Object Identifier
doi:10.1215/21562261-3664959

Mathematical Reviews number (MathSciNet)
MR3568645

Zentralblatt MATH identifier
1360.14132

Subjects
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)

Keywords
quantum cohomology extremal transition conifold transition toric degeneration partial flag variety

Citation

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. https://projecteuclid.org/euclid.kjm/1478509222


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