Braid Group Actions on Derived Categories of Coherent Sheaves
نویسنده
چکیده
This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dimX ≥ 2, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.
منابع مشابه
Braiding via Geometric Lie Algebra Actions
We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov-Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s...
متن کامل2 3 M ay 2 00 7 Braid cobordisms , triangulated categories , and flag varieties Mikhail Khovanov and Richard
We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.
متن کاملSe p 20 06 Braid cobordisms , triangulated categories , and flag varieties Mikhail Khovanov and Richard
We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct an action of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.
متن کامل0 Mirror symmetry and actions of braid groups on derived categories
After outlining the conjectural relationship between the conjec-tural mirror symmetry programmes of Kontsevich and Strominger-Yau-Zaslow, I will describe some natural consequences of this which are proved from scratch in joint work with Mikhail Khovanov and Paul Seidel. Namely, actions of braid groups are found on derived categories of coherent sheaves, dual to Seidel's braid group of symplecti...
متن کاملBraid group actions on categories of coherent sheaves
In this talk we will construct, following the recent paper [BR] by Bezrukavnikov and Riche, actions of certain groups on certain categories. Both the groups and the categories are relevant in representation theory, but all currently known constructions of the action are given by explicitly constructing action by generators and checking relations between them — this is what we will do. There are...
متن کامل