Birational geometry for higher-dimensional algebraic varieties

2020 Fiscal Year Final Research Report

• Project/Area Number: 16H03925
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Fujino Osamu 大阪大学, 理学研究科, 教授 (60324711)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Keywords: 極小モデル理論 / 混合ホッジ構造 / トーリック多様体 / 双有理幾何学 / 小平消滅定理

Outline of Final Research Achievements

I am mainly interested in higher-dimensional complex projective varieties. I have already established some powerful generalizations of the Kodaira vanishing theorem by using the theory of mixed Hodge structures on cohomology with compact support. I have also studied the theory of variations of mixed Hodge structure on cohomology with compact support. Now I am interested in applications of the theory of variations of mixed Hodge structure for higher-dimensional algebraic varieties. I obtained a generalization of Kodaira's canonical bundle formula and now tries to apply this new result for some geometric problems. In 2020, I published a book on the Iitaka conjecture.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

この5年間の主な研究成果は、混合ホッジ構造の変動の理論を高次元代数多様体の研究に組織的に持ち込んだことである。すでにコンパクト台コホモロジーに入る混合ホッジ構造が高次元代数多様体論で有益であることは知っていたが、混合ホッジ構造の変動も考えることで様々な応用が考えられることに気づいた。安定多様体のモジュライ空間の射影性の証明は一番最初の素朴な応用である。