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Invariance of Quantum Rings under Ordinary Flops I: Quantum corrections and reduction to local models

Yuan-Pin Lee Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112 Hui-Wen Lin Taida Institute of Mathematical Sciences (TIMS), National Taiwan University, Taipei 106 Chin-Lung Wang Center for Advanced Study in Theoretical Sciences, National Taiwan University, Taipei 106

Algebraic Geometry mathscidoc:2204.45013

2011.9
This is the first of a sequence of papers proving the quantum invariance under ordinary flops over an arbitrary smooth base. In this first part, we determine the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. We then perform various reductions to reduce the problem to the local models. In Part II, we develop a quantum Leray--Hirsch theorem and use it to show that the big quantum cohomology ring is invariant under analytic continuations in the Kähler moduli space for ordinary flops of splitting type. In Part III, together with F. Qu, we remove the splitting condition by developing a quantum splitting principle, and hence solve the problem completely.
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@inproceedings{yuan-pin2011invariance,
  title={Invariance of Quantum Rings under Ordinary Flops I: Quantum corrections and reduction to local models},
  author={Yuan-Pin Lee, Hui-Wen Lin, and Chin-Lung Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220415112113843816031},
  year={2011},
}
Yuan-Pin Lee, Hui-Wen Lin, and Chin-Lung Wang. Invariance of Quantum Rings under Ordinary Flops I: Quantum corrections and reduction to local models. 2011. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220415112113843816031.
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