Differential geometry of holomorphic vector bundles with Rizza structures and it applications

2020 Fiscal Year Final Research Report

• Project/Area Number: 16K05135
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kagoshima University
• Principal Investigator: Aikou Tadashi 鹿児島大学, 理工学域理学系, 教授 (00192831)
• Co-Investigator(Kenkyū-buntansha): 小櫃 邦夫 鹿児島大学, 理工学域理学系, 准教授 (00325763) ; 田中 恵理子 鹿児島大学, 理工学域理学系, 助教 (70376979)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Keywords: Rizza構造 / 複素Finsler構造 / Rizza-negativity / Griffith-negativity

Outline of Final Research Achievements

In this study, we mainly studied the negativity of holomorphic vector bundles over compact complex manifolds from the point of view of Finsler geometry. Given a Rizza structure in a holomorphic vector bundle, the notion of Rizza-negativity is naturally defined in terms of its curvature. On the other hand, there are two different types of negativity on holomorphic vector bundles, that is, the Griffith-negativity in the sense of Hermite geometry and the negativity in the sense of algebraic geometry. In this study, we have studied the relationship between these three negativities on holomorphic vector bundles. In particular, we have studied whether the Griffith-negativity leads the Rizza-negativity under the assumption that the given Rizza structure is a complex Berwald structure.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

この研究課題では，Hermite構造を一般化した複素Finsler構造の類の計量構造であるRizza構造を微分幾何学の手法を用いて研究し，代数幾何学的な概念であるコンパクト複素多様体上の正則ベクトル則のnegativity（負性）または，その双対的な概念であるampleness（豊富性）を議論したものである。特にRizza-negativityの概念を導入し，代数幾何学の意味でのnegativityとHermite幾何学の意味でのnegativityとの関係を構築できた。得られた結果は新たな研究課題を生み出し，今後の研究の方向性を示唆する結果となった。