Asymptotic Invariants of Higher Cohomology Groups and Applications to Complex Geometry

2016 Fiscal Year Final Research Report

• Project/Area Number: 25800051
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tohoku University (2015-2016); Kagoshima University (2013-2014)
• Principal Investigator: Matsumura Shin-ichi 東北大学, 理学研究科, 准教授 (90647041)
• Research Collaborator: GONGYO Yoshinori 東京大学, 大学院数理科学研究科, 准教授 (70634210) ; FUJINO Osamu 大阪大学, 大学院理学研究科, 教授 (60324711) ; CAO Junyan Universite Paris 6, Institut de Mathematiques de Jussieu, Analyse complexe et geometrie ; DEMAILLY Jean-Pierre Laboratoire de Mathematiques, Institut Fourier
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: ケーラー幾何 / コホモロジー消滅定理 / 正則切断 / 特異計量 / 乗数イデアル層 / 調和積分論 / dbar, L2-理論 / 正値性

Outline of Final Research Achievements

From the viewpoint of complex analytic geometry, I studied vanishing theorems of higher cohomology groups that naturally appear in algebraic geometry, and generalized them to (holomorphic) line bundles with singular hermitian metrics. Specifically, I established a Nadel type vanishing theorem for singular hermitian metrics with minimal singularities on big line bundles, and I obtained injectivity theorems to pseudo-effective line bundles by using multiplier ideal sheaves. Furthermore, I generalized them to deformations of Kaehler varieties by using higher direct images. In the processes, I developed techniques of singular hermitian metrics and multiplier ideal sheaves with transcendental singularities and techniques of L2 methods for dbar-equations and the theory of harmonic integrals.

Free Research Field

複素幾何学