2023 Fiscal Year Final Research Report
A deeper understanding of moduli theory integrated by special Riemannian metrics and convex polytopes
| Project/Area Number |
18K13406
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Kagawa University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | 球形多様体 / 複素構造変形理論 / トーリック多様体 / 凸多面体の三角形分割 / ケーラー計量 / Bott多様体 / 凸多面体上のBernstein測度 / GKZ-理論 |
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| Outline of Final Research Achievements |
I clarified the diffeomorphism classes of the Calabi-Yau 3-folds we constructed in 2014. Furthermore, we proved that if a simple normal crossing complex surface with the trivial canonical bundle satisfies a certain condition, there exists a family of global smoothings in a differential geometrical sense. Through this collaborative work, we also constructed some concrete examples.
Meanwhile, I proved that asymptotic Chow semistability implies that Ding polystability for any Gorensitein toric fano varieties. Also, I proved that the additivity of the Mabuchi constant for the product of toric manifolds in terms of the associated moment polytopes. Building upon these research achievements and my experience in these calculations, we classified (a) all strong Calabi dream Bott manifolds , and (b) 3 or 4-dimensional toric Fano manifolds in terms of relative K/Ding stabilities.
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| Free Research Field |
ケーラー幾何学,複素幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
一般に与えられた代数多様体がGIT-安定か否かを判定する事は,トーリックの場合でも非常に難しい.これは対応する凸多面体上の積分値を具体的に計算する必要性が生じるためであり,この困難を部分的ではあるが克服した点や,複素構造変形理論の重要性を再吟味しつつ,正規交叉カラビ-ヤウ多様体の代数幾何的スムージング理論を特殊な状況下で微分幾何的に再現・具体例を発見した点は意義深い.特に「凸多面体の幾何学的不変量を如何に効率的に測るか?」という問題は純粋数学のみならず,工学や確率論とも密接な関係がある.ゆえに,幾何学の枠組みに捉われず分野を超えた交流による相互のインプットが大きな社会的意義を与える筈である.
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