Research on canonical divisors of higher dimensional algebraic varieties

研究了代数变量的极小模型与派生范畴之间的关系。自Kontsevich提出同调镜像对称猜想后，代数变体的派生范畴吸引了众多数学分支的研究者。但从最小模型理论的角度来看，衍生范畴也是一个自然的研究对象。我们发现，表面上复杂的衍生范畴具有非常简单和美观的结构，并与最小模型程序并行运行。在“d -等价和k -等价”一文中，我们考虑了代数变量的两种不同的等价。如果两个代数变体具有相干束的等价有界派生范畴，则称它们是d等价的。在它们是双等效的附加条件下，如果它们的正则因子在拉回时变为线性等效，则它们被称为k等效。k等价的概念是在最小模型理论中自然产生的，但从定义上看，它与d等价是完全无关的。我们推测这些等价重合，并在特殊情况下证明了这一推测。例如，我们证明了由标准翻牌或Mukai翻牌所连接的变量具有等价的派生范畴。我们证明了任何只有q !端点奇点且是k等价的三维变异都是d等价的。这篇文章经常被别人引用，并且在2004/2005年的Bourbaki研讨会上回顾了本文所获得的结果和随后的结果。派生范畴理论适用于光滑变量，但不适用于奇异变量。另一方面，我们必须处理极小模型理论中的奇异变量。因此，我们考虑了对奇异品种的泛化，并发现我们有时可以通过考虑与给定品种相关的delign - mumford堆栈来克服这一困难。证明了如果与拟光滑有毒品种相关的两个光滑Deligne-Mumford堆具有相同的正则因子水平，则它们的派生范畴是等价的。证明了G（2,4）的协切空间上的翻牌可以推导出派生范畴的等价性。我们还证明了有毒品种的衍生类别是由特殊的捆集产生的。少

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