Motivic invariants of Artin stacks and 'stack functions'

Abstract

An invariant I of quasiprojective K-varieties X with values in a commutative ring R is "motivic" if I(X)= I(Y)+I(X\Y) for Y closed in X, and I(X x Y)=I(X)I(Y). Examples include Euler characteristics chi and virtual Poincare and Hodge polynomials. We first define a unique extension I' of I to finite type Artin K-stacks F, which is motivic and satisfies I'([X/G])=I(X)/I(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 I(G) is invertible in R for all special K-groups G, which excludes I=chi as chi(K*)=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, as studied in the author's paper math.AG/0403305. There are several versions of the construction: the basic one SF(F), and variants SF(F,I,R),... "twisted" by motivic invariants. We associate a Q-vector space SF(F) or an R-module SF(F,I,R) to each Artin stack F, with functorial operations of multiplication, pullbacks phi^* and pushforwards phi_* under 1-morphisms phi : F --> G, and so on. They will be important tools in the author's series on "Configurations in abelian categories", math.AG/0312190, math.AG/0503029, math.AG/0410267 and math.AG/0410268.