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Let $G$ be a reductive algebraic group. Let $X$ be a $G$-variety and consider any closed subvariety $Z$ of $X$. Since any $g\in G$ acts as an automorphism, we know that $g.Z$ is again a closed subvariety of $X$. This yields an action of $G$ on the free module of cycles of $X$ which should induce an action of $G$ on the Chow ring of $X$. The invariants of this ring should be precisely the classes that correspond to linear combinations of $G$-orbits.

Has this action been studied before? Any kind of reference would be very welcome. Thanks!

Edit: It looks like my above idea is rather futile, so let me ask more broadly: Are there any techniques or results in intersection theory specifically on $G$-varieties? Could you name some references?

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The invariants are not the classes that correspond to linear combinations of $G$-orbits. The class of a point, or one point in each connected component of $X$, is invariant in the Chow ring, even if all the orbits are much larger than points. –  Will Sawin Oct 17 '12 at 16:19
There are also the papers of Edidin and Graham. –  Damian Rössler Oct 17 '12 at 21:36
@Damian Rössler: Could you give me the exact titles? Thanks a bunch! –  Jesko Hüttenhain Oct 18 '12 at 9:28
Edidin and Graham have several versions of their paper "Equivariant Intersection Theory" on the arXiv. You may prefer reading one of the earlier versions rather than the final published version. In fact, I suspect that is why they kept the earlier versions available. –  Michael Joyce Oct 22 '12 at 12:44

2 Answers 2

up vote 4 down vote accepted

If you are interested in intersection theory of varieties with $G$-actions, then you want to study equivariant intersection theory. This theory exploits the $G$-action in a way that leads to deeper invariants than ordinary intersection theory. The three references I would recommend if you are first learning the subject are:

Fulton's lectures notes on equivariant cohomology (compiled by Dave Anderson)

Equivariant Cohomology and Equivariant Intersection Theory by Michel Brion

Equivariant Chow Groups for Torus Actions by Michel Brion

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I assume that you are over an algebraically closed field. If $G$ is connect, the action is trivial, because any affine algebraic group is rational, so every point can be connected via a chain of open subsets of $\mathbb A^1$ to the identity.

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How unfortunate. Let me ask more broadly then: Are there any particular techniques or results in intersection theory on $G$-varieties that make use of the $G$-action? In fact, let me add that to my original question. –  Jesko Hüttenhain Oct 17 '12 at 16:43
To Jesko: yes, definitely. There is a whole lot of work on homogeneous spaces, toric varieties, wonderful compactifications, spherical varieties, and so on, in which one exploits the group action to get results on the intersection theory. –  Angelo Oct 17 '12 at 18:09

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