Nakajima--Grojnowski-Operatoren und derivierte Autoäquivalenzen von Hibertschema von Punkten auf Flächen
Zusammenfassung der Projektergebnisse
In the area of mathematics known as algebraic geometry, one usually studies geometric objects given by polynomial equations, called varieties, by means of their invariants, that means numbers or algebraic structures naturally attached to them. A classical example of an invariant of a variety is its dimension. The derived category is a much finer invariant, meaning it contains a lot more information about the variety. One of the reasons for the recent interest in derived categories is a conjectured relationship to physical string theory. One can say that there are four main goals in the study of derived categories of varieties: (i) Determine under which circumstances the derived categories of two different varieties agree (‘derived equivalence’, ‘Fourier–Mukai partners’). (ii) Describe all symmetries of the derived category of a given variety (‘groups of autoequivalences’). (iii) If possible, decompose the derived category into smaller pieces (‘semi–orthogonal decompositions’, ‘exceptional sequences’). (iv) Lift interesting structures known to exist at the level of coarser invariants to the higher level of the derived categories (‘categorification’). The last problem, known as ‘categorification’, is the least precise one. On the other hand, it often inspires progress towards goals (i), (ii), and (iii). This was exactly the intended route of the project: Given a surface, one can associate in a natural way a series of higher-dimensional varieties, the Hilbert schemes of points on that surface. There is a very important structure on the level of the cohomology of these Hilbert schemes, namely the Nakajima–Grojnowski operators. The goal of this project was to categorify these operators in a meaningful way and, in consequence, make progress towards goal (ii) by describing the symmetries of the derived categories of the Hilbert schemes. These goals ought to be realised using the McKay correspondence, which is a striking principle relating geometry and the representation theory of finite groups. During the course of the project, I found two different ways to lift the Nakajima–Grojnowski constructions to the level of the derived categories. One of them gives a categorification of the algebraic structure induced by the Nakajima–Grojnowski operators; that of a module over the Heisenberg algebra. The other one lifts the Nakajima–Grojnowski correspondences themselves which gives a series of P-functors between derived categories of the Hilbert schemes. These, in turn, induce interesting new symmetries of the derived categories. Some steps in the direction of a verification of a conjecture describing the full groups of autoequivalences where taken, but, so far, I could not accomplish a complete proof. In joint work with Pawel Sosna, we were able to make contributions towards problems (i) and (iii) in the context of the McKay correspondence, that were not expected prior to the start of the project. Concerning (i), we gave constructions of semi-orthogonal decompositions of the Hilbert schemes induced by semi-orthogonal decompositions of the underlying surface. In the case of an Enriques surface, these decompositions again induce symmetries of the derived category. Furthermore, we systematically studied functors between equivariant derived categories, i.e. the kind of categories which show up in the derived McKay correspondence, also known as Bridgeland– King–Reid equivalence. One main application is a result which can be seen as a partial converse of the Bridgeland–King–Reid equivalence; namely a necessary condition for a quotient stack to be derived equivalent to a variety.
Projektbezogene Publikationen (Auswahl)
- P-functor versions of the Nakajima operators. 41 pages
Andreas Krug
- Equivalences of equivariant derived categories. J. Lond. Math. Soc. (2) 92 (2015), no. 1, 19–40
Andreas Krug (with P. Sosna)
(Siehe online unter https://doi.org/10.1112/jlms/jdv014) - On the derived category of the Hilbert scheme of points on an Enriques surface. Selecta Math. (N.S.) 21 (2015), no. 4, 1339–1360.1
Andreas Krug (with P. Sosna)
(Siehe online unter https://doi.org/10.1007/s00029-015-0178-x) - Symmetric quotient stacks and Heisenberg actions. 11 pages
Andreas Krug
