2023 Fiscal Year Final Research Report
Variational approach to Kobayashi-Hitchin correspondence and Higgs bundles by means of Quot-schemes
| Project/Area Number |
19K14524
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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 | Osaka Metropolitan University (2022-2023)
Tokyo Institute of Technology (2019-2021) |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 小林-Hitchin対応 / 正則ベクトル束の安定性 / Kaehler多様体上の標準計量 |
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| Outline of Final Research Achievements |
This project concerned the Kobayashi-Hitchin correspondence for holomorphic vector bundles over a smooth complex projective variety. In place of the previous method using the theory of nonlinear partial differential equations, I analysed the asymptotic behaviour of an energy functional called the Donaldson functional, and employed the Quot-schemes in algebraic geometry, to clarify the relationship between differential and algebraic geometry from a variational point of view. I also worked on many related topics.
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| Free Research Field |
幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
小林-Hitchin対応は，Hermite-Einstein計量と呼ばれる非線形偏微分方程式の解の存在が代数幾何学的な安定性条件と同値であることを主張する非常に重要な定理である．本研究では，このような定理がなぜ成立するのかを，変分法の観点からより幾何学的直感に訴える形で理解するために重要な結果を示した．特に，Fubini-Study計量のQuotスキーム極限を定義することにより，極限Donaldson汎関数の漸近挙動を記述する代数的不変量を求めることに成功した．
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