Studies on Vanishing Theorems and Extension Problems of Holomorphic Sections based on Singular Hermitian Metrics

2020 Fiscal Year Final Research Report

• Project/Area Number: 17H04821
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Matsumura Shin-ichi 東北大学, 理学研究科, 准教授 (90647041)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 特異エルミート計量 / 正則切断の拡張問題 / 極小モデル理論 / 構造定理 / 葉層構造 / 接べクトル束 / 正則断面曲率 / 反標準束

Outline of Final Research Achievements

I have studied semi-positivity/singularities in algebraic geometry from the viewpoint of complex geometry/analysis. As results, I gave an extension theorem of holomorphic sections from subvarieties and structure theorems for projective manifolds with “non-negative curvature”. Specifically, I proved an extension theorem formulated for non-reduced and singular subvarieties, and also established structure theorems for maximal rationally connected fibrations associated with projective manifolds with pseudo-effective tangent bundle, non-negative holomorphic sectional curvature, or nef anti-canonical divisor. In the process, I had applied and developed the theory of singular metrics of vector bundles, positivity of direct image sheaves, holomorphic foliations, stability of coherent sheaves, and Hermite-Einstein metrics.

Free Research Field

複素幾何学, 多変数複素解析学

Academic Significance and Societal Importance of the Research Achievements

本研究では, 双有理幾何の半正値性と特異点を複素幾何/解析の立場から研究し, 拡張定理/消滅定理を与えた. 拡張定理/消滅定理は汎用性が高く半正値性は近年重要性が増しつつあり, 本研究の理論/技術はさらなる応用が見込める. “非負曲率”の構造定理は幾何学の究極の目標のひとつである分類理論に寄与する. 特に, 非負の正則断面曲率の構造定理は決定的な成果であり, 微分幾何的な曲率と代数幾何的な有理連結性を結びつける点で価値がある. 本研究のいくつかの成果は純粋な代数幾何の問題だが超越的な証明しか知られておらず, 本研究は代数的手法の不足部分を補い両分野の調和を促す点でも意義があると思われる.