Construction of complex analytic theory for LC singularities and studies of positively curved varieties based on the extension problem

2023 Fiscal Year Final Research Report

• Project/Area Number: 19KK0342
• Research Category: Fund for the Promotion of Joint International Research (Fostering Joint International Research (A))
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Matsumura Shin-ichi 東北大学, 理学研究科, 教授 (90647041)
• Project Period (FY): 2020 – 2023
• Keywords: 双有理幾何 / LC特異点 / 極小モデル理論 / アバンダンス予想 / 非消滅予想 / 正曲率性 / 調和積分論 / 葉層構造

Outline of Final Research Achievements

Developing the theory of harmonic integrals for LC strata, I generalized the injectivity theorem (an extension of Kodaira's vanishing theorem) to the complex geometric setting for varieties with LC singularities, thus solving the Fujino conjecture. I also solved the abundance conjecture for minimal projective manifolds with vanishing second Chern class. Furthermore, I studied the non-vanishing conjecture in the framework of the generalized minimal model program and solved it for the nef anti-canonical bundles of three-dimensional varieties. Additionally, I clarified a relation between the geometry of algebraic fiber spaces and the positivity of relative (pluri-)anti-canonical bundles, aiming to establish structural theorems for varieties whose anti-canonical divisor or tangent vector bundle satisfies certain positivity.

Free Research Field

複素幾何学, 複素解析学, 代数幾何学

Academic Significance and Societal Importance of the Research Achievements

代数幾何学(特に双有理幾何学)では, Birkar-Cascini-Hacon-McKernanの大結果以降, 半正値性と特異点の重要性が増している. 本研究は, 特異点と半正値性に対する超越的な手法を発展させた点で意義があり, 将来的にはさらなる進展が期待できる. 例えば, LC特異点を扱う複素解析理論を構築し, 消滅定理を一般化した. これにより, LC特異点の解析的側面が明らかになり, ホッジ構造の複素解析的側面の探求が期待できる. また, アバンダンス予想や非消滅予想といった当該分野の大問題に対しても, 部分的ではあるが新たな成果を上げた点で価値があると思われる.