Chow group of a stack

In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For quotient stacks, it is closely related to the equivariant Chow group.

One difference from a usual Chow group is that a cycle may 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

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.

Virtual fundamental class

The notion originates in the Kuranishi theory in symplectic geometry.

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.

References

• 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.
• Vistoli, Angelo (1989), "Intersection theory on algebraic stacks and on their moduli spaces", Inventiones Mathematicae, 97 (3): 613–670, doi:10.1007/BF01388892, MR 1005008.