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2018 Fiscal Year Final Research Report

The study of the existence problem in the Donaldson-Tian-Yau conjecture

Research Project

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Project/Area Number 25287010
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypePartial Multi-year Fund
Section一般
Research Field Geometry
Research InstitutionOsaka University

Principal Investigator

MABUCHI Toshiki  大阪大学, その他部局等, 名誉教授 (80116102)

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)
Project Period (FY) 2013-04-01 – 2019-03-31
KeywordsDonaldson-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
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.

Free Research Field

複素微分幾何学

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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Published: 2020-03-30  

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