Parity Sheaves

TL;DR

This paper introduces parity sheaves, a new class of objects in the derived category of sheaves on stratified varieties, with applications to representation theory and criteria for the Decomposition Theorem.

Contribution

It defines and proves the uniqueness of parity sheaves under cohomological parity conditions, and relates their properties to resolutions and intersection forms in stratified varieties.

Findings

Parity sheaves are uniquely determined objects in the derived category.

Decomposition of direct images into parity sheaves encodes intersection form data.

Framework applies to flag varieties, toric varieties, and nilpotent cones.

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.