MOTIVIC INVARIANTS OF ARTIN STACKS AND ‘STACK FUNCTIONS’

By D. Joyce. 2005

125 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a7d4719c3254fb540845af278044050e6cd07092

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Citations in the corpus

A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications (2013, methodology): “When K = C, a related result for coherent sheaves only, with U a complex manifold and f a holomorphic function, was proved by Joyce and Song [16, Th. 5.5] using gauge theory and transcendental complex methods.”
Constructible Functions on Artin Stacks (2004, background): “Each is associated with a notion of Euler characteristic χna, χstk, χorb of constructible sets in K-stacks.”
Hall algebras and curve-counting invariants (2010, unknown): “Joyce’s proof of this crucial result uses the full force of the technology developed in the long series of papers [11, 12, 13], including moduli spaces of configurations and the virtual projection operators Πk.”
An introduction to motivic Hall algebras (2010, background): “Joyce and Song [12] went on to use some of the ideas from [14] to prove an important property of Behrend functions (stated here as Theorem 5.3) and so incorporate such functions into Joyce’s Lie algebra integration map.”
A theory of generalized Donaldson–Thomas invariants (2008, background): “In [50] we study different kinds of stack function spaces with other choices of generators and relations, and operations on these spaces.”