Parity Sheaves
Abstract
Given a stratified variety X with strata satisfying a cohomological parityvanishing condition, we define and show the uniqueness of "parity sheaves", which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If X admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semismall case. Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several wellknown complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig's conjecture.
 Publication:

arXiv eprints
 Pub Date:
 June 2009
 arXiv:
 arXiv:0906.2994
 Bibcode:
 2009arXiv0906.2994J
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Algebraic Geometry;
 55N33;
 20C20
 EPrint:
 55 pages, v3: shortened presentation (particularly introduction), fixed some mistakes, final version