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Research on canonical divisors of higher dimensional algebraic varieties

高维代数簇的正则因数研究

基本信息

批准号：
14340003
负责人：
KAWAMUTA Yujiro
金额：
$ 5.25万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

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 … More 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

我们研究了代数簇的最小模型和派生类别之间的关系。康采维奇提出同调镜像对称猜想后，代数簇的派生范畴吸引了许多数学不同分支的研究者。但从最小模型理论的角度来看，派生范畴也是一个自然的研究课题。我们发现表面上复杂的派生范畴具有非常简单和优美的结构，并且与最小模型程序以并行的方式运行。我们在“D-等价性和K-等价性”一文中考虑了代数簇的两种不同等价性。如果两个代数簇具有等价的相干滑轮有界派生类别，则称它们是 D 等价的。在它们双有理等价的附加条件下，如果它们的规范因数在拉回到共同分辨率时变得线性等价，则称它们是 K-等价的。 K-等价的概念自然出现在最小模型理论中，但从定义上来说，它与D-等价完全无关。我们猜想这些等价性是一致的，并在特殊情况下证明了这个猜想。例如，我们证明了由标准翻牌或 Mukai 翻牌连接的品种具有等效的派生类别。我们证明了任何只有 Q 阶乘末端奇点且 K 等价的 3 维簇都是 D 等价的。这篇文章被其他人频繁引用，本文的结果与后续的结果一起在 2004/2005 年的 Bourbaki 研讨会上进行了回顾。派生范畴理论适用于光滑簇，但不适用于奇异簇。另一方面，我们必须处理最小模型理论中的奇异变体。因此，我们考虑了对奇异品种的泛化，并发现有时可以通过考虑与给定品种相关的德利涅-芒福德堆栈来克服困难。我们证明，如果与准光滑有毒品种相关的两个光滑 Deligne-Mumford 堆栈具有相同水平的规范除数，则它们的派生类别是等效的。我们证明了格拉斯曼簇 G(2,4) 余切空间上的翻转会导致派生范畴的等价。我们还证明了有毒品种的派生类别是由特殊的滑轮集合产生的。较少的