Parity Sheaves

Abstract

Given a stratified variety X with strata satisfying a cohomological parity-vanishing 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 semi-small 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 well-known 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.

55 pages, v3: shortened presentation (particularly introduction), fixed some mistakes, final version

References in corpus (8)

• Decomposition numbers for perverse sheaves
• Cohomology of Spaltenstein varieties
• Modular Koszul duality
• Equivariant Todd Classes for Toric Varieties
• Canonical basis, KLR-algebras and parity sheaves
• Cohomology of the minimal nilpotent orbit
• Modular representations of reductive groups and geometry of affine Grassmannians
• Sheaves on nilpotent cones, Fourier transform, and a geometric Ringel duality

Cited by in corpus (15)

• Modular Koszul duality
• Modular perverse sheaves on flag varieties II: Koszul duality and formality
• Modular generalized Springer correspondence III: exceptional groups
• The affine Grassmannian and the Springer resolution in positive characteristic
• Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture
• Kumar's criterion modulo p
• Mixed Motives and Geometric Representation Theory in Equal Characteristic
• Grassmannians and Koszul duality
• KLR and Schur algebras for curves and semi-cuspidal representations
• The geometric Satake equivalence for integral motives
• Modular equivariant formality
• Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology
• Torus actions and tensor products of intersection cohomology
• A Hecke action on $G_{1}T$-modules
• Block decomposition via the geometric Satake equivalence