2021 Fiscal Year Annual Research Report
K-theoretic enumerative invariants and q-difference equations
| Project/Area Number |
19F19802
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MILANOV Todor 東京大学, カブリ数物連携宇宙研究機構, 教授 (80596841)
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| Co-Investigator(Kenkyū-buntansha) |
ROQUEFEUIL ALEXIS 東京大学, カブリ数物連携宇宙研究機構, 外国人特別研究員
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| Project Period (FY) |
2019-11-08 – 2022-03-31
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| Keywords | quantum K-theory / q-difference equations |
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| Outline of Annual Research Achievements |
We got two interesting results. The first one is related to the problem of confluence in the theory of q-difference equations. Namely, we proved that the small K-theoretic J-functions of a smooth projectve variety with non-negative first Chern class has a limit as q->1 and this limit coincides with the small cohomological J-function. Here, non-negative first Chern class means that the natural pairing of the 1st Chern class of the tangent bundle and the homology class of an irreducible curve is a non-negative number. The limit is taken after rescaling each Novikov variable in the K-theoretic J-function by an appropriate power of q-1.Moreover, we expect that our argument can be generalized so one can prove the confluence of the big J-function and the confluence of the quantum q-difference equations. It is also expected that the positivity condition of the 1st Chern class is redundant but removing this condition seems to be a challenging problem. Our second result is in the settings of toric geometry. We were able to identify explicitly the small J-function of a Fano toric manifold of Picard rank 2 with a certain q-oscillatory integral. The latter was introduced by Givental in order to provide a solution of the quantum q-difference equations and it can be viewed as a first step towards constructing or fomrulating mirror symmetry in quantum K-theory.
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| Research Progress Status |
令和3年度が最終年度であるため、記入しない。
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| Strategy for Future Research Activity |
令和3年度が最終年度であるため、記入しない。
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