Semi-positive holomorphic line bundles and complex dynamics

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14313
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Osaka Metropolitan University (2022-2023); Osaka City University (2020-2021)
• Principal Investigator: Koike Takayuki 大阪公立大学, 大学院理学研究科, 准教授 (30784706)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 半正正則直線束 / 正則葉層構造

Outline of Final Research Achievements

We have succeeded in determining the geometric structure of complex manifolds associated with semi-positive line bundles and obtaining its applications by using a method based on a technique completely different from conventional dynamical approaches. We mainly deal with dynamical properties (especially linearization problems) on a neighborhood of submanifolds based on the semi-positivity of line bundles. First we obtained some results by applying techniques from the theory of several complex functions. By additionally applying differential geometrical techniques on holomorphic foliations from the viewpoint of algebraic geometry, we succeeded in obtaining an affirmative answer to the conjecture we have posed as a goal of this program, and also in obtaining its applications especially on complex surfaces. Concurrently, in collaboration with Takato Uehara at Okayama University, we have solved the realizability problem of projective K3 surfaces by our gluing construction.

Free Research Field

複素解析幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究では, 複素多様体の複素解析幾何学的構造の解明を行った. 複素多様体は局所的に複素数によってパラメータ付けられる対象であり, 多項式たちの共通ゼロ点集合の様な非常に基礎的かつ重要な幾何学的対象である.私の研究成果では, 複素多様体研究に於いて金字塔ともいえる小平の埋め込み定理の深化にあたる結果を得ている. この成果は複素多様体の解析的・幾何学的構造の詳細を明らかにするものであり, 複素多様体が登場する数学, 延いては関連する数理科学全般に於ける学術的意義は大きい. また本研究成果により新たな射影的K3曲面の構成方法が判明したことに対しては, 数理物理学的応用が大いに期待される.