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

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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.