2018 Fiscal Year Final Research Report
Study of the gamma structure in Gromov-Witten theory
| Project/Area Number |
16K05127
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | グロモフ・ウィッテン理論 / ガンマ整構造 / ミラー対称性 / 量子コホモロジー / トロピカル幾何 / 保型性 / 軌道体 / 導来圏 |
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| Outline of Final Research Achievements |
Gromov-Witten theory concerns counting curves in a given space, and the Gamma-integral structure is a mysterious integral structure appearing in Gromov-Witten theory. We studied the problem that how the Gamma-integral structures are related under birational transformations. We also searched for the origin of the Gamma-integral structure. In joint work with Coates, Corti and Tseng, we established Hodge-theoretic mirror symmetry for toric orbifolds. In joint work with Abouzaid, Ganatra and Sheridan, we explained the Gamma class from a viewpoint of tropical geometry. Also, in joint work with Coates, we showed the quasi-modularity of the Gromov-Witten potential of the local projective plane.
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| Free Research Field |
幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
ガンマ整構造がグロモフ・ウィッテン理論の間の関係（関手性）とどのようにかかわっているかについての理解が深まった．特に連接層の導来圏の半直交分解と，量子コホモロジーの分解とが対応することがトーリック軌道体などの例を通じて理解できたことは重要である．また，ガンマ整構造はガンマ類と呼ばれる超越的な特性類により定まるものであり，その起源は明らかになっていなかったが，トロピカル幾何とStrominger-Yau-Zaslow描像を通じてガンマ類がどのように現れるかについて大きな理解の進展があった．
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