Let \(K\) be an algebraically closed field. In the study of isomorphism classes of quasi-projective varieties over \(K\), there are certain important invariants with values in a commutative ring \(A\), which satisfy the identities \(\gamma([S])= \gamma([Y])+ \gamma([X\setminus Y])\), whenever \(Y\) is a closed subvariety of \(X\), and \(\gamma([X\times Y)]= \gamma([X])\cdots\gamma([Y])\) for an arbitrary pair \((X,Y)\) of quasi-projective \(K\)-varieties. Examples of such an invariant \(\gamma\) include, for instance, the Euler characteristics, the virtual Hodge polynomials, and the virtual Poincaré polynomials. In general, any such invariant \(\gamma\) is called a motivic invariant of quasi-projective varieties, since constructions of invariants of this type have become particularly common in the context of the theory of motives and motivic integration.
In the paper under review, the author provides a significant extension of this conceptual framework of motivic invariants from quasi-projective varieties to Artin stacks. More precisely, the first goal of the current paper is to show that any motivic invariant \(\gamma\) for quasi-projective varities can be uniquely extended to a motivic invariant \(\gamma'\) of an Artin \(K\)-stack \({\mathcal F}\) of finite type such that \(\gamma'([X/G])= \gamma([X])/\gamma([G])\) when \(X\) is a \(K\)-variety, \(G\) a special \(K\)-group acting on \(X\), and \([X/G]\) is the quotient stack.
The second goal of the paper is to develop a general theory of stack functions on Artin stacks. These stack functions are to represent a universal generalization of the author’s concept of constructible functions on Artin stacks as developed in his foregoing paper devoted to this subject [cf.: D. Joyce, J. Lond. Math. Soc., II. Ser. 74, No. 3, 583–606 (2006; Zbl 1112.14004)].
The actual motivation for the generalizing constructions carried out in the present paper, together with a wider context regarding their concrete and far-reaching applications, is provided by the author’s recently launched extensive program of establishing a fundamental theory of what he calls “configurations in abelian categories”. The first four papers in this series of works [cf.: D. Joyce, Configurations in abelian categories. I: Adv. Math. 203, No. 1, 194–255 (2006; Zbl 1102.14009); II: Adv. Math. 210, No. 2, 635-706 (2007; Zbl 1119.14005); III: Adv. Math. 215, No. 1, 153–219 (2007; Zbl 1134.14007); IV: Adv. Math. 217, No. 1, 125–204 (2008; Zbl 1134.14008)] have been made available in 2006, partially through the math.AG electronic preprint service, and their aim is to built up a mathematically rigorous framework for properly counting \(D\)-branes as objects in certain triangulated categories.
With a view to this much wider context, the constructions established in the current paper, although being of independent interest and significance, may be understood as basic ingredients, conceptual fundamentals and methodological tools of the author’s above-mentioned developing theory of configurations in abelian categories.
As for the contents of the present article, the material is organized in six sections. After an instructive introduction to the main goals, \(K\)-groups, Artin stacks, and the author’s previous work on constructible functions on stacks are reviewed in Section 2. The basics on stack functions are developed in Section 3, whereas Section 4 is devoted to the construction of motivic invanants of stacks, on the one hand, and of further spaces of stack functions on the other. Sections 5 and 6 integrate the two main topics developed so far to produce refined versions of stack functions, according to various technical difficulties that have to be overcome.
As the author points out, some related, constructions, in particular a variant of the Grothendieck ring of Artin \(n\)-stacks, have also been obtained, simultaneously and independently, by B. Toën in his recent preprint “Anneaux de Grothendieck des \(n\)-champs d’Artin” [{\url math.AG/0509098}] published in 2005.
Altogether, the paper under review presents a wealth of new ideas, concepts, and refined constructions in a very comprehensive and detailed manner, the importance of which will become evident in the overall context of the author’s program around configurations in abelian categories as mentioned above.