Parity sheaves (Q2922794)

The idea of parity sheaves resembles the definition of perverse sheaves [\textit{A. A. Beilinson} et al., Astérisque 100, 172 p. (1982; Zbl 0536.14011)]. For a given stratification of a complex algebraic variety \(X=\bigsqcup_{\lambda\in\Lambda}X_\lambda\) one considers pariversity, i.e., a function \(\Lambda\to\mathbb Z/2\). Let \(k\) be a complete local principal ideal domain. For a complex of sheaves of \(k\)-modules \(\mathcal F\) , or rather an element of \(D(X)\), the bounded constructible derived category of \(k\)-sheaves and a fixed pariversity there is defined a condition via vanishing of stalk and costalk cohomology of \(\mathcal F\) along strata. The equivariant context when a reductive group \(G\) is acting on \(X\) is also treated in parallel. Here in contrast to perversity condition only parity of the degree matters. The sheaves satisfying suitable conditions are called parity sheaves. The effects related to the torsion cohomology in this context might be better traced. For example, the Decomposition Theorem for a proper push forward is proven for a preferred pariversity. On the other hand parity sheaves with a given support might not exist in general. They do exist in many cases of varieties coming from the representation theory (Schubert varieties, nilpotent cones), also for toric varieties.