Chow group of a stack

Angelo Vistoli (1989) develops the basic theory (mostly over Q) for the Chow group of a (separated) Deligne–Mumford stack. There, the Chow group is defined exactly as in the classical case: it is the free abelian group generated by integral closed substacks modulo rational equivalence.

If a stack X can be written as the quotient stack ${\displaystyle X=[Y/G]}$ for some quasi-projective variety Y with a linearized action of a linear algebraic group G, then the Chow group of X is defined as the G-equivariant Chow group of Y. This approach is introduced and developed by Dan Edidin and William A. Graham, as well as Burt Totaro. Later Andrew Kresch (1999) extended the theory to a stack admitting a stratification by quotient stacks.

For higher Chow groups (precursor of motivic homologies) of algebraic stacks, see Roy Joshua's Intersection Theory on Stacks:I and II.

The calculations depend on definitions. Thus, here, we proceed somehow axiomatically. Specifically, we assume: given an algebraic stack X locally of finite type over a base field k,

1. (homotopy-invariance) if E is a rank-n vector bundle on X, then ${\displaystyle A_{p}(E)=A_{p-n}(X)}$.
2. for each integral substack Z of dimension < p, ${\displaystyle A_{p}(X-Z)=A_{p}(X)}$, a corollary of a localization sequence.

These properties are valid if X is Deligne–Mumford and are expected to hold for any other reasonable theory.

We take X to be the classifying stack ${\displaystyle BG}$, the stack of principal G-bundles for a smooth linear algebraic group G. By definition, it is the quotient stack ${\displaystyle [*/G]}$, where * is viewed as the stack associated to * = Spec k. We approximate it as follows. Given an integer p, choose a representation ${\displaystyle G\to GL(V)}$ such that there is a G-invariant open subset U of V on which G acts freely and the complement ${\displaystyle Z=V-U}$ has codimension ${\displaystyle >-\operatorname {dim} G-p}$. Let ${\displaystyle *\times _{G}V}$ be the quotient of ${\displaystyle *\times V}$ by the action ${\displaystyle (x,v)\cdot g=(xg,g^{-1}v)}$. Note the action is free and so ${\displaystyle *\times _{G}V}$ is a vector bundle over ${\displaystyle BG}$. By Property 1 applied to this vector bundle,

${\displaystyle A_{p}(BG)=A_{p+\dim V}(*\times _{G}V).}$

Then, since ${\displaystyle *\times _{G}U=U/G}$, by Property 2,

${\displaystyle A_{p+\dim V}(*\times _{G}V)=A_{p+\dim V}(U/G)}$

since ${\displaystyle \dim[Z/G]=\dim Z-\dim G<\dim V+p}$.

As a concrete example, let ${\displaystyle G=\mathbb {G} _{m}}$ and let it act on ${\displaystyle \mathbb {A} ^{n}}$ by scaling. Then ${\displaystyle \mathbb {G} _{m}}$ acts freely on ${\displaystyle U=\mathbb {A} ^{n}-\{0\}}$. By the above calculation, for each pair of integers n, p such that ${\displaystyle n+p\geq 0}$,

${\displaystyle A_{p}(B\mathbb {G} _{m})=A_{p+n}(\mathbb {P} ^{n-1}).}$

In particular, for every integer p ≥ 0, ${\displaystyle A_{p}(B\mathbb {G} _{m})=0}$. In general, ${\displaystyle A_{n-k}(\mathbb {P} ^{n})=\mathbb {Q} h^{k}}$ for the hyperplane class h, ${\displaystyle h^{k}}$ k-times self-intersection and ${\displaystyle h^{k}=0}$ for negative k and so

${\displaystyle A_{p}(B\mathbb {G} _{m})=\mathbb {Q} h^{-1-p}}$

where the right-hand side is independent of models used in the calculation (since different h's correspond under the projections between projective spaces.) For ${\displaystyle p=-1=\dim B\mathbb {G} _{m}}$, the class ${\displaystyle h^{0}=[\mathbb {P} ^{n}]}$, any n, may be thought of as the fundamental class of ${\displaystyle B\mathbb {G} _{m}}$.

Similarly, we have

${\displaystyle A^{*}(B\mathbb {G} _{m})=\mathbb {Q} [c]}$

where ${\displaystyle c=c_{1}(h)}$ is the first Chern class of h (and c and h are identified when Chow groups and Chow rings of projective spaces are identified). Since ${\displaystyle h^{k}=c\cdot h^{k-1}}$, we have that ${\displaystyle A_{*}(B\mathbb {G} _{m})}$ is the free ${\displaystyle \mathbb {Q} [c]}$-module generated by ${\displaystyle h^{0}}$.

The notion originates in the Kuranishi theory in symplectic geometry.[1][2]

In § 2. of Behrend (2009), given a DM stack X and C_X the intrinsic normal cone to X, K. Behrend defines the virtual fundamental class of X as

${\displaystyle [X]^{\text{vir}}=s_{0}^{!}[C_{X}]}$

where s₀ is the zero-section of the cone determined by the perfect obstruction theory and s₀^! is the refined Gysin homomorphism defined just as in Fulton's "Intersection theory". The same paper shows that the degree of this class, morally the integration over it, is equal to the weighted Euler characteristic of the Behrend function of X.

More recent (circa 2017) approaches do this type of construction in the context of derived algebraic geometry.[3]

• Perfect obstruction theory
• Gromov–Witten invariant
• Cohomology of a stack

• Behrend, Kai (2009), "Donaldson-Thomas type invariants via microlocal geometry", Annals of Mathematics, 2nd Ser., 170 (3): 1307–1338, arXiv:math/0507523, doi:10.4007/annals.2009.170.1307, MR 2600874
• Ciocan-Fontanine, Ionuț; Kapranov, Mikhail (2009). "Virtual fundamental classes via dg–manifolds". Geometry & Topology. 13 (3): 1779–1804. arXiv:math/0703214. doi:10.2140/gt.2009.13.1779. MR 2496057. S2CID 1211344.
• Fantechi, Barbara, Virtual pullbacks on algebraic stacks (PDF)
• Kresch, Andrew (1999), "Cycle groups for Artin stacks", Inventiones Mathematicae, 138 (3): 495–536, arXiv:math/9810166, Bibcode:1999InMat.138..495K, doi:10.1007/s002220050351, S2CID 119617049
• Totaro, Burt (1999), "The Chow ring of a classifying space, Algebraic K-theory", Proc. Sympos. Pure Math, vol. 67, American Mathematical Society, pp. 249–281, MR 1743244, Zbl 0967.14005
• Vistoli, Angelo (1989), "Intersection theory on algebraic stacks and on their moduli spaces", Inventiones Mathematicae, 97 (3): 613–670, Bibcode:1989InMat..97..613V, doi:10.1007/BF01388892, MR 1005008, S2CID 122295050
• Nabijou, Navid (2015), Virtual Fundamental Classes in Gromov Witten Theory (PDF), archived from the original (PDF) on 2017-05-16, retrieved 2017-07-20
• Shen, Junliang (2014), Construction of the Virtual Fundamental Class and Applications (PDF)

• Virtual classes for the working mathematician
• The classical number 2875 of lines on the quintic, as a DT invariant
• What is the main failure in using Naive Chow group in Artin Stack
• Local model of virtual fundamental cycle
• https://ncatlab.org/nlab/show/virtual+fundamental+class
• On the Virtual Fundamental Class - a slide by Kai Behrend