Research on canonical divisors of higher dimensional algebraic varieties

我们研究了代数簇的极小模型与派生范畴之间的关系。在Kontsevich提出他的同调镜像对称猜想之后，代数簇的派生范畴吸引了许多不同数学分支的研究者。但从极小模型理论的角度来看，派生范畴也是一个自然的研究对象。在D-等价和K-等价一文中，我们考虑了代数簇的两种不同的等价关系。称两个代数簇是D-等价的，如果它们具有等价的有界相干层派生范畴。在出生等价的附加条件下，如果它们的标准因子拉回到…时成为线性等价的，则称它们是K等价的更常见的解决办法。K-等价的概念是在极小模型理论中自然产生的，但从定义上讲它与D-等价是完全无关的。我们猜想这些等价关系是一致的，并在特殊情况下证明了这一猜想。例如，我们证明了由标准Flop或Mukai Flop连接的簇具有等价的派生范畴。证明了仅有q-阶乘末端奇点且K-等价的三维簇是D-等价的，这篇文章经常被人引用，并在2004/2005年的Bourbaki研讨会上回顾了本文所得到的结果。另一方面，我们必须处理极小模型理论中的奇异变元。所以我们考虑了对奇异变种的推广，发现有时我们可以通过考虑与给定变种相关的Deligne-Mumford堆栈来克服这个困难。证明了如果两个与拟光滑有毒簇相关的光滑Deligne-Mumford堆栈具有相同水平的正则因子，则它们的派生范畴是等价的。证明了Grassmanian簇G(2，4)在余切空间上的Flop诱导出了一个等价的派生范畴。我们还证明了有毒变种的派生范畴是由例外集合生成的。较少

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