MOTIVIC INVARIANTS OF ARTIN STACKS AND 'STACK FUNCTIONS'

An invariant ϒ of quasiprojective 𝕂-varieties X with values in a commutative ring Λ is motivic if ϒ(X) = ϒ(Y) + ϒ(X\ Y) for Y closed in X, and ϒ(X × Y) = ϒ(X)ϒ(Y). Examples include Euler characteristics χ and virtual Poincaré and Hodge polynomials. We first define a unique extension ϒ′ of ϒ to finite type Artin 𝕂-stacks $F$, which is motivic and satisfies ϒ′([X/G]) = ϒ(X)/ϒ(G) when X is a 𝕂-variety, G a special 𝕂-group acting on X, and [X/G] is the quotient stack. This only works if ϒ(G) is invertible in Λ for all special 𝕂-groups G, which excludes ϒ = χ as χ(𝔾_m) = 0. But we can extend the construction to get round this. Then we develop the theory of stack functions on Artin stacks. These are a universal generalization of constructible functions on Artin stacks. There are several versions of the construction: the basic one $SF(F)$, and variants $SF__(F,Υ,Λ),…$ ‘twisted’ by motivic invariants. We associate a ℚ-vector space $SF(F)$ or a Λ-module $SF__(F,Υ,Λ)$ to each Artin stack $F$, with functorial operations of multiplication, pullbacks ϕ* and pushforwards ϕ_* under 1-morphisms $ϕ:F→G$;, and so on. They will be important tools in the author's series on ‘Configurations in abelian categories’.