What is the best introduction (textbook) to equivariant sheaves on algebraic varieties equipped with an action of an algebraic group?

  • Dear Alex, I don't know if there is such a text-book (at least, none comes to mind). You can learn the definition in various places, and the intuition is that the action of $G$ on the base extends to an action of $G$ on the sheaf. This is the kind of topic about which you would usually learn more by reading some research papers rather than a text book. Regards, – Matt E Dec 25 '12 at 3:19
  • @Matt Actually I first met equivariant sheaves a day before reading a research paper. I got the idea that equivariant sheaf in it simplest possible incarnation of equivariant vector bundle is a vector bundle whose fibres are isomorphic on orbits and isomorphisms are given by the action of the group. It looks like equivariant sheaves (action of $\mathbb{C}^*$) on the punctured spectrum of a graded ring is the same thing as sheaves on Proj of this ring. This was my main example, but I have a feeling that my explanation is incomplete. Do you know where I can find proof of this fact at least? – Alex Dec 25 '12 at 13:17
  • Section 5.1 (pages 231-243) of Chriss, Ginzburg "Representation Theory and Complex Geometry" looks nice. – evgeny Nov 8 '16 at 16:29
up vote 6 down vote accepted

If an algebraic group $G$ acts freely on a variety $X$, then $G$-equivariant sheaves on $X$ are the same as sheaves on the quotient $X/G$. (If the $G$-action is not free, the same statement is true if we instead talk about the quotient stack $[X/G]$, but it takes more work to give the statement meaning in this case.)

To see this, let $\pi: X \to X/G$ be the natural projection, let $g$ be an element of $G$, let $\alpha_g: X \to X$ be the automorphism of $X$ given by the $g$-action, and note that $\pi \circ g = \pi.$ Thus, if $\mathcal F$ is a sheaf on $X/G$, then there is anatural isomorphism $$\alpha_g^* \pi^* \mathcal F \cong (\pi \circ \alpha_g)^* \mathcal F = \pi^* \mathcal F.$$ This is the equivariant structure on $\pi^* \mathcal F$.

To see that any equivariant sheaf on $X$ arises as $\pi^*\mathcal F$ for some $\mathcal F$ in this way, note that the equivariant structure gives descent data on the equivariant sheaf for the map $\pi$, which allows us to descent the sheaf down to $X/G$.

In particular, if $R$ is a graded $\mathbb C$-algebra, say with $R_0 = \mathbb C$, so that $R$, and hence Spec $R$, is equipped with an action of $\mathbb C^{\times}$ (the action is being as follows: $z \in \mathbb C^{\times}$ acts on the $n$th graded piece of $R$ as mult. by $z^n$), then removing the point corresponding to the irrelevant ideal from Spec $R$, we get a free $\mathbb C^{\times}$-action, and equivariant sheaves for this action are the same as equivariant sheaves on the Proj.

As one example, if $R = \mathbb C[x_0,\ldots,x_N]$ and we consider the structure sheaf on Spec $R \setminus \{0\} = \mathbb A^{N+1} \setminus \{0\},$ this corresponds to the sheaf $\bigoplus_{n = 0}^{\infty} \mathcal O(n)$ on $\mathbb P^N$.

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