Research on rationally connected varieties and vector bundles on them from projective-geometric and categorical points of view

2014 Fiscal Year Final Research Report

• Project/Area Number: 22540043
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Electro-Communications
• Principal Investigator: OHNO Masahiro 電気通信大学, 情報理工学(系)研究科, 准教授 (70277820)
• Co-Investigator(Kenkyū-buntansha): TERAKAWA Hiroyuki 都留文科大学, 文学部, 教授 (80277863) ; YAMAGUCHI Kohhei 電気通信大学, 大学院情報理工学研究科, 教授 (00175655) ; KIDA Masanari 電気通信大学, 大学院情報理工学研究科, 教授 (20272057)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: ベクトル束 / 例外列 / 局所自由分解 / スペクトル系列 / 導来圏 / 射影多様体 / ネフ

Outline of Final Research Achievements

A. Bondal shows that if the bounded derived category of coherent sheaves on a smooth projective variety admits a full strong exceptional sequence then it is exact equivalent to the bounded derived category of modules over some finite dimensional algebra. We deduced a spectral sequence from this theorem of Bondal. We also gave a resolution of a coherent sheaf in terms of a full strong exceptional sequence of vector bundles if the derived category admits such a sequence of vector bundles. As an application, we obtained a new proof of the classification due to Peternell-Surek-Wisniewski of nef vector bundles on projective spaces with the first Chern class less than three and on smooth hyperquadrics with the first Chern class less than two. Our proof is based only on this resolution and some cohomological study of nef vector bundles, whereas the original proof due to Peternell-Surek-Wisniewski is based on the theory of extremal rays and analysis of the corresponding contraction morphisms.

Free Research Field

代数幾何学