Study of canonical divisors on higher dimensional algebraic variety

2015 Fiscal Year Final Research Report

• Project/Area Number: 22244002
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: Kawamata Yujiro 東京大学, 数理(科)学研究科(研究院), 教授 (90126037)
• Project Period (FY): 2010-04-01 – 2016-03-31
• Keywords: 代数多様体 / 標準因子 / 導来圏 / 極小モデル / トーリック多様体 / DK予想 / アバンダンス予想

Outline of Final Research Achievements

I described the change of the derived categories under the minimal model program in the case of toric varieties, and proved the DK conjecture in this case. I gave similar description for the divisorial extractions, and proved the derived McKay correspondence for finite abelian groups. I also proved the derived McKay correspondence for any finite subgroup of general linear group in the case of dimension 2, and described the semi-orthogonal complements.
I proved the abundance conjecture in the case of numerical Kodaira dimension 0. I proved that a Mori dream space is of Calabi-Yau type if and only if the Cox ring has only log canonical singularities. I extended the semi-positivity theorem of the Hodge bundle in the case of reducible fibers. I described the decomposition of the space of boundary divisors according to the minimal models and the canonical models.

Free Research Field

代数幾何学