Research on canonical divisors of higher dimensional algebraic varieties

研究了代数簇的极小模型与其导出范畴之间的关系。自Kontsevich提出同调镜像对称猜想以来，代数簇的导出范畴吸引了众多数学分支的研究者。但是，从极小模型论的观点来看，派生范畴也是一个自然的研究课题。我们发现，显然复杂的导出范畴具有非常简单而优美的结构，并且与最小模型程序并行。在“D-等价与K-等价”一文中，我们考虑了代数簇的两种不同等价。两个代数簇被称为D-等价的，如果它们有等价的凝聚层的有界导出范畴。在它们是双有理等价的附加条件下，如果它们的标准因子在被拉回到 关于我们 共同的决议。K-等价的概念在极小模型理论中自然出现，但它与D-等价的定义完全无关。我们证明了这些等价是一致的，并在特殊情况下证明了这个猜想。例如，我们证明了由标准触发器或向井触发器连接的簇具有等价的导出范畴。我们证明了任何只具有Q-阶乘终端奇点且K-等价的三维簇都是D-等价的，这篇文章经常被其他人引用，并在2004/2005年的Bourbaki研讨会上将本文的结果与后来的结果结合起来进行了回顾。另一方面，在极小模型理论中，我们必须处理奇异簇。因此，我们考虑推广到奇异品种，并发现我们有时可以克服困难，考虑与给定品种相关的Deligne-Mumford堆栈。我们证明了如果两个光滑Deligne-Mumford栈与准光滑有毒簇有相同水平的典型因子，那么它们的导出范畴是等价的。证明了Grassmannian簇G（2，4）余切空间上的翻转诱导导出范畴的等价性。我们还证明了衍生类别的有毒品种所产生的异常收集束。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