Chow group of a stack explained

X = [Y/G] , the Chow group of X is the same as the G-equivariant Chow group of Y.

A key difference from the theory of Chow groups of a variety is that a cycle is allowed to carry non-trivial automorphisms and consequently intersection-theoretic operations must take this into account. For example, the degree of a 0-cycle on a stack need not be an integer but is a rational number (due to non-trivial stabilizers).

Definitions

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.

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. https://people.math.osu.edu/joshua.1/pub.html

Examples

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

A _{p} (E) = A _{p-n} (X) .

1. for each integral substack Z of dimension < p,

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

B G , the stack of principal G-bundles for a smooth linear algebraic group G. By definition, it is the quotient stack [*/G] , where * is viewed as the stack associated to * = Spec k. We approximate it as follows. Given an integer p, choose a representation G \to GL(V) such that there is a G-invariant open subset U of V on which G acts freely and the complement Z = V - U has codimension > - \operatorname{dim} G - p . Let * x _{G} V be the quotient of * x V by the action (x, v) ⋅ g = (xg, g ^{-1} v) . Note the action is free and so * x _{G} V is a vector bundle over BG . By Property 1 applied to this vector bundle,A _{p(BG)} = A _{p+\dim} (* x _{G} V). Then, since * x _{G} U = U/G , by Property 2,A _{p} (* x _{G} V) = A _{p} (U/G) since \dim[Z/G] = \dim Z - \dim G < \dim V + p .

As a concrete example, let

G = G _{m} and let it act on A ^{n} by scaling. Then G _{m} acts freely on U = A ^{n} - \{ 0 \} . By the above calculation, for each pair of integers n, p such that n + p \ge 0 ,A _{p(B} G _{m)} = A _{p+n} ( P ^{n-1} ). In particular, for every integer p ≥ 0, A _{p(B} G _{m)} = 0 . In general, A _{n-k} ( P ^{n)} = Q h ^{k} for the hyperplane class h, h ^{k} k-times self-intersection and h ^{k} =0 for negative k and soA _{p(B} G _{m)} = Q h ^{-1-p} where the right-hand side is independent of models used in the calculation (since different hs correspond under the projections between projective spaces.) For p = -1 = \dim B G _{m} , the class h ^{0} = [ P ^{n} ] , any n, may be thought of as the fundamental class of B G _{m} .

Similarly, we have

A ^{*(B} G _{m)} = Q [c] where 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 h ^{k} = c ⋅ h ^{k-1} , we have that A _{*(B} G _{m)} is the free Q [c] -module generated by h ^{0} .

Virtual fundamental class

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

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

[X] ^{vir} = s ^{![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]

See also

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

References

• • Ciocan-Fontanine . Ionuț . Kapranov . Mikhail . Virtual fundamental classes via dg–manifolds . . 13 . 3 . 2009 . 10.2140/gt.2009.13.1779 . 1779–1804 . math/0703214. 2496057. 1211344 .

External links

• 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

Notes and References

1. Fukaya . Kenji . Kenji Fukaya. Ono . Kaoru . Arnold conjecture and Gromov-Witten invariant . . 38 . 5 . 1999 . 10.1016/s0040-9383(98)00042-1 . 933–1048. 1688434. free .
2. Pardon. John. John Pardon. 2016-04-28. An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. Geometry & Topology. 20. 2. 779–1034. 10.2140/gt.2016.20.779. 1364-0380. 1309.2370. 119171219.
3. § 1.2.1. of Cisinski. Denis-Charles. Khan. Adeel A.. 2017-05-09. Brave new motivic homotopy theory II: Homotopy invariant K-theory. 1705.03340. math.AT.