2004 Fiscal Year Final Research Report Summary
Research on canonical divisors of higher dimensional algebraic varieties
| Project/Area Number |
14340003
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | The University of Tokyo |
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Principal Investigator |
KAWAMUTA Yujiro The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (90126037)
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| Co-Investigator(Kenkyū-buntansha) |
KATSURA Toshiyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40108444)
MIYAOKA Yoichi The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50101077)
ODA Takayuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (10109415)
SAITO Takesi The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70201506)
OGUISO Keiji The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (40224133)
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| Project Period (FY) |
2002 – 2004
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| Keywords | minimal model / derived category / algebraic variety / birational / toric variety / stack / flip / flop |
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| Research Abstract |
We investigated the relationship between the minimal models of algebraic varieties and the derived categories. The derived categories of algebraic varieties attracted many researchers of different branches of mathematics after Kontsevich proposed his homological mirror symmetry conjecture. But the derived category is also a natural subject for investigation from the view point of the theory of minimal models. We found that the apparently complicated derived categories have very simple and beautiful structures and behave in a parallel way with the minimal model program.
We considered two different kinds of equivalences of algebraic varieties in the article "D-equivalence and K-equivalence". Two algebraic varieties are said to be D-equivalent if they have equivalent bounded derived categories of coherent sheaves. Under the additional condition that they are birationally equivalent, they are said to be K-equivalent if their canonical divisors become linearly equivalent when pulled back to
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a common resolution. The concept of K-equivalence arises naturally in the theory of minimal models, but it is totally unrelated to the D-equivalence by definition. We conjectured that these equivalences coincide, and proved this conjecture in special cases. For example, we proved that the varieties connected by a standard flop or a Mukai flop have equivalent derived categories. We proved that any 3-dimensional varieties with only Q-factorial terminal singularities which are K-equivalent are D-equivalent.
This article has been frequently cited by others, and the results obtained in this paper combined with the subsequent ones were reviewed in Bourbaki seminar in the year 2004/2005.
The theory of derived categories works well for smooth varieties, but not for singular ones. On the other hand, we have to deal with singular varieties in the theory of minimal models. So we considered the generalization to the singular varieties, and found that we can sometimes overcome the difficulty by considering the Deligne-Mumford stack associated to the given variety. We proved that if two smooth Deligne-Mumford stacks associated to quasi-smooth toxic varieties have the same level of the canonical divisors, then their derived categories are equivalent. We proved that the flop over the cotangent space of the Grassmannian variety G(2,4) induces an equivalence of derived categories. We also proved that the derived categories of toxic varieties are generated by exceptional collections of sheaves. Less
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