Sergey Arkhipov

(Aarhus University)

(Aarhus University)

Seminar

Tuesday, 27 January, 2015, at 15:00, in Aud. D1 (1531-113)

Abstract:

A year ago we announced a generalization of the famous construction of a categorical braid group action due to Bezrukavnikov and Riche. Now we outline details of the construction.

Matrix factorizations on a scheme X are generalizations of vector bundles. Their category depends on a choice of a global function on Xcalled a potential. We recall the definitions of the corresponding derived categories and of inverse and direct image functors.

Given an algebraic variety X with an an action of a reductive group G, we construct a certain category of equivariant matrix factorizations as follows. We recall the definition of the Grothendieck variety Grot(G). The moment map for G-action on X defines a function on the product of Grot(G) with the cotangent bundle to X. We consider the corresponding derived category of equivariant matrix factorizations.

Finally we recall the construction of Bezrukavnikov and Riche of certain correspondences on Grot(G). We prove that the same correspondences act on our category and define a braid group action on it.

Matrix factorizations on a scheme X are generalizations of vector bundles. Their category depends on a choice of a global function on Xcalled a potential. We recall the definitions of the corresponding derived categories and of inverse and direct image functors.

Given an algebraic variety X with an an action of a reductive group G, we construct a certain category of equivariant matrix factorizations as follows. We recall the definition of the Grothendieck variety Grot(G). The moment map for G-action on X defines a function on the product of Grot(G) with the cotangent bundle to X. We consider the corresponding derived category of equivariant matrix factorizations.

Finally we recall the construction of Bezrukavnikov and Riche of certain correspondences on Grot(G). We prove that the same correspondences act on our category and define a braid group action on it.

Organised by: QGM

Contact person: Jørgen Ellegaard Andersen