Study on the dimension of the global sections of adjoint bundles for polarized manifolds via their invariants

2020 Fiscal Year Final Research Report

• Project/Area Number: 16K05103
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kochi University
• Principal Investigator: Yoshiaki Fukuma 高知大学, 教育研究部自然科学系理工学部門, 教授 (20301319)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Keywords: 代数学 / 代数幾何学 / 偏極多様体 / 豊富な因子 / 随伴束 / nefかつbigな因子

Outline of Final Research Achievements

Let X be a smooth projective variety defined over the field of complex numbers and let L be an ample Cartier divisor on X. Then the pair (X,L) is called a polarized manifold. We conducted studies on the dimension of the global sections of adjoint bundles K+tL, where K denotes the canonical divisor of X and t is a positive integer. In particular, we studied the following conjecture and its related topics: Let (X,L) be an n-dimensional polarized manifold. If K+(n-1)L is nef, then the dimension of the global sections of K+(n-1)L is positive. Consequently we obtained that the above conjecture is true for some special cases. We also were able to achieve some research results about the dimension of the global sections of adjoint bundles and invariants of polarized manifolds.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

偏極多様体の随伴束は, 射影多様体の研究においていろいろな場面で使われており, 射影多様体の分類や高次元代数幾何学の研究においてとても重要な役割を果たしている. 偏極多様体の随伴束の持つ性質に関する研究については, 例えば基点自由性に関する研究があり, いわゆる藤田予想といわれる予想の解決に向けた研究成果が高次元代数多様体論の研究に大きな役割を果たしていることを考えると, 随伴束の大域切断のなす次元に関する研究が今後の代数幾何学、特に偏極多様体のさらなる研究に活かされていくことが大いに期待される.