Applications of the Monge Ampere equation on Kahler manifolds to entire curves

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K14200
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Ochanomizu University
• Principal Investigator: Tiba Yusaku お茶の水女子大学, 基幹研究院, 准教授 (90635616)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: 多重劣調和函数 / ケーラー多様体 / 正則ベクトル束

Outline of Final Research Achievements

We show that the set where the Levi form of a plurisubharmonic function on Kahler manifolds does not vanish (non-pluriharmonic locus) has a relation to the topology and analytic properties of the Kahler manifolds. Especially, we show that there exist isomorphisms and an injective map from a cohomology of a holomorphic vector bundle of Kahler manifolds to a cohomology of the restriction of holomorphic vector bundle on non-pluriharmonic locus. This result implies some special case of the Hartogs extension theorem. We also show that the restriction map from the de Rham cohomology of Kahler manifolds and that of non-pluriharmonic locus are isomorphic if the degree of cohomology is smaller that the dimension of the Kahler manifold. This result is an analytic analogy of the Lefschetz hyperplane theorem.

Free Research Field

多変数関数論

Academic Significance and Societal Importance of the Research Achievements

多重劣調和函数の非調和領域が空間全体のコホモロジーと関わるという結果は、多変数関数論や複素幾何学に新たな視点をもたらした。非調和領域は複雑であり、何か統一的な性質を持つということはこれまでの研究では触れられていなかった。しかし本研究により、非多重調和領域はケーラー多様体全体の性質と深い関わりが明らかになり、この結果は今後さまざまな数学の研究分野に応用できると期待できる。例えば、複素力学系や正則曲線の定めるアールフォルスカレントの研究、さらにトロピカル幾何学に応用を持つと言える。