Quotients of derived categories of smooth projective varieties by actions of finite groups of autoequivalences

本课题的研究目标是从同调代数中提取几何信息。更确切地说，目标是研究有限群的自等价作用对导出范畴的同分性的影响。代数几何中一类重要的对象是所谓的光滑射影簇。这些物体的几何形状可以用许多方法来研究。近年来，一个特殊的同调对象，即所谓的在簇X上的相干层的导出范畴，已经成为许多研究的焦点，其中一个动机是猜想某些物理现象在数学方面转化为关于导出范畴的陈述。人们相信，这个相当抽象的定义对象使得簇的某些对称性可见，这是用经典几何方法所不能看到的。研究导出范畴的一个重要可能性是计算它的自等价群。这在某些情况下是可以实现的，但一般来说，这个组很难计算。该项目的目标是一方面试图定义一个商的衍生类别X的任何有限组的自等价，从而推广已知的情况下，自等价实际上是自同态的品种X。另一方面，有趣的几何例子应进行研究，这也应该是由特设的方法访问。

The aim of the research project is the extraction of geometric information from homological algebra. To be more precise, the goal is the investigation of quotients of derived categories by actions of finite groups of autoequivalences.An important class of objects in algebraic geometry are so called smooth projective varieties. The geometry of these objects is investigated by many methods. In recent years one particular homological object, the so called derived category of coherent sheaves on a variety X as above, has been the focus of a lot of research, one of the motivations being the conjecture that certain physical phenomena translate on the mathematical side to statements about derived categories. It is believed that this fairly abstractly defined object makes certain symmetries of the variety visible, which cannot be seen with classical geometric methods.An important possibility to study the derived category is the computation of its group of autoequivalences. This was accomplished in some cases, but in general this group is hard to compute. The goal of the project is on the one hand the attempt to define a quotient of the derived category of X by any finite group of autoequivalences, thus generalising the known case where the autoequivalences are in fact endomorphisms of the variety X. On the other hand interesting geometric examples shall be studied, which should also be accessible by adhoc methods.