The study of the existence problem in the Donaldson-Tian-Yau conjecture
| Project/Area Number |
25287010
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Partial Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Osaka University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
後藤 竜司 大阪大学, 理学研究科, 教授 (30252571)
中川 泰宏 佐賀大学, 工学(系)研究科(研究院), 教授 (90250662)
新田 泰文 東京工業大学, 理工学研究科, 助教 (90581596)
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| Project Period (FY) |
2013-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥12,870,000 (Direct Cost: ¥9,900,000、Indirect Cost: ¥2,970,000)
Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2016: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2015: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2014: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2013: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
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| Keywords | Donaldson-Tian-Yau予想 / K-安定性 / テスト配位 / Donaldson-二木不変量 / 偏極代数多様体 / 定スカラー曲率Kaehler計量 / extremal Kaehler計量 / extremal Kaehler 計量 / 定スカラー曲率ケーラー計量カラー曲率ケーラー計量 / 定スカラー曲率ケーラー計量 / 強K-安定性 |
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| Outline of Final Research Achievements |
The Donaldson-Tian-Yau conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, this conjecture is still open for general polarizations or more generally in extremal Kaehler cases. We mainly worked in this unsolved cases of the conjecture.
For a polarized algebraic manifold, by introducing the concept of strong relative K-stability, we showed that strong relative K-stability implies asymptotic relative Chow stability. In particular, a strong relative K-stable polarized algebraic manifold always admits a sequence of polybalanced metric that are expected to converge to an extremal Kaehler metric in the polarization class. This then shows that the sequence of polybalanced metrics admits a suitable a priori bound. Moreover, we obtained a couple of results on the related problems.
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| Academic Significance and Societal Importance of the Research Achievements |
特殊計量の存在に関する Donaldson-Tian-Yau 予想は，複素幾何における中心問題のひとつとして知られている．たとえば，この予想の Kaehler-Einstein 計量の場合は，最近 Chen-Donaldson-Sun や Tian によって肯定的に解決されたが，これはケーラー幾何において、フィールズ賞受賞者である Yau がカラビ予想を解決したとき以来の，初めての本格的な結果として知られている．さらに予想を，より一般の extremal Kaehler 計量の場合に解決することも非常に大きな学術的意義があり，その意味でも我々の研究には少なからぬ価値があると考えられる．
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Report
(7 results)
Research Products
(18 results)
