Nice algebraic approximations of classifying spaces

Question

Let $G=GL_k(\mathbb C)$ be the complex linear group. Then the infinite Grassmannian is a model for the classifying space $BG$. We can write the infinte Grassmannian as a colimit of the finite Grassmannians $Gr(k,n)$, which are honest algebraic varieties, that have the following nice properties:

1) They admit an affine cell paving, the Bruhat stratification.

2) The intersection cohomology complexes of the strata have good vanishing properties: They are parity sheaves, meaning that their restriction to smaller strata have cohomology only in every second degree.

3) On top of that, when "interpreting" the intersection cohomology complexes as say mixed $\mathbb Q_l$ sheaves they have good purity properties: Their restrictions to smaller strata are still pure and even direct sums of shifted Tate twists.

Now my question is the following: If we replace $GL_k(\mathbb C)$ by another connected reductive affine complex algebraic group $G$, are there still algebraic varieties approximating $BG$ such that (some of) the above properties hold?

Answer 1

There is good hope for other groups in nice infinite families. $BGl_k$ is the classifying space of $k$-dimensional vector bundles. $Gr(k,n)$ is the classifying space of $k$-dimensional vector subbundles of an $n$-dimensional trivial vector bundle.

Similarly, $BSP_{2k}$ is the classifying space of $2k$-dimensional symplectic vector bundles. We can embed each such bundle into a trivial $2n$-dimensional symplectic vector bundle by the same trick one uses for the Grassmanians, and homotope two such embeddings to each other, again using an identical trick. This allows us to write $BSP_{2k}$ as a limit of flag varieties of $SP_n$.

We can do the same thing for $O_n,SL_n$, and $SO_n$. For $SL_n$ we might need to make it noncompact? I am not sure exactly how far one can extend this technique.

The first property is of course a consequence of being a flag variety. I am not so sure about the rest.

Answer 2

This beautiful paper of Totaro does much of what you ask. Namely, choose a representation $V$ of $G$ so that $G$ acts freely on $V$ outside of a codimension $\geq s$ subset $S$. Then Totaro shows that the Chow ring of $X=(V-S)/G$ (the geometric quotient) is independent of $V$ and $S$, in degree less than $s$. Furthermore, he shows that such pairs $(V, S)$ exist with $s$ arbitrarily large. So the set of such $(V-S)/G$ form a reasonable approximation to $BG$ (and in particular, give a reasonable definition of the Chow ring of $BG$)!

Let's see how these varieties stack up against your criteria.

1) They don't necessarily admit pavings by affines (unless $G$ is a so-called "special group"), but they do in some cases, e.g. if $G=GL_n$ or $SL_n$. Moreover, their "motives" admit a paving by affines, in the sense that $S$ can be chosen so that $V-S$ admits a paving by affines, as does $G$ in many cases (e.g. if it is split, affine, ...), so $(V-S)/G$ is morally $[X]/[G]$. (This can be made precise in the Grothendieck ring of varieties, tensored with $\mathbb{Q}$; the issue is that this quotient isn't necessarily a Zariski-fiber bundle. The question of whether such a quotient is even rational is, I think, pretty hard, and possibly open(?).)

(2+3) I have no idea about this (largely because I don't know many of the words you use).

That said, these have one additional virtue, mentioned by Totaro. Namely, in every case he computes, the Chow ring of $BG$ (defined in degree $< s$ by choosing $(V, S)$ as above with $S$ of codimension at least $s$), actually equals the ring $MU^*BG\otimes_{MU^*} \mathbb{Z}$. (This only makes sense over $\mathbb{C}$ of course.) These examples include $GL(n), Sp(2n), O(n), SO(2n+1),$ and $SO(4)$. So these spaces "really are" algebraic approximations to $BG$!

Hopefully this is in a similar spirit to your question!

Answer 3

I think there is some subtlety in the case of the symplectic group that has not yet been discussed in the two other answers. Embedding a symplectic bundle into a trivial symplectic bundle and constructing a classifying space from that actually means we are looking at the symplectic Grassmannian as classifying space. To recall what the symplectic Grassmannian is, let $V$ be a vector space of dimension $2n$, equipped with a symplectic form. The symplectic Grassmannian $\operatorname{HGr}(2r,2n)$ is the subset of the Grassmannian $\operatorname{Gr}(2r,2n)$ consisting of those $2r$-dimensional subspaces of $V$ on which the symplectic form has non-degenerate restriction. It can then be identified with the quotient $\operatorname{Sp}_{2n}/(\operatorname{Sp}_{2r}\times\operatorname{Sp}_{2n-2r})$. This, however, is not a flag variety, it is affine and not projective.

The geometry of these symplectic Grassmannians is more subtle than of the ordinary Grassmannians for the general linear group. In particular, they do not admit a paving with affines; they admit a paving with quasi-affines which are $\mathbb{A}^1$-contractible. This failure of the existence of a paving with affine spaces can already be seen at the level of motives for $\operatorname{HGr}(2,2n+2)$. The motives of these "quaternionic projective spaces" are $$ \operatorname{M}(\operatorname{HGr}(2,2n+2))\cong \bigoplus_{i=0}^n\mathbb{Q}(2i)[4i]. $$ The corresponding (but too complicated to write out here) statements for the other symplectic Grassmannians exactly fit the topological fact that Pontryagin classes live in dimensions which are multiples of $4$. If there was a paving with affines, some motives $\mathbb{Q}(i)[2i]$ for $i$ odd would have to appear. Somehow, geometrically, what prevents the paving to be by affine spaces (instead of just contractible quasi-affine varieties) in a motive as above is that the cells have to have a boundary which is of codimension $\geq 2$, which can not happen for affine spaces.

Anyway, all of the above can be found and is beautifully discussed in the paper by Panin and Walter on "Quaternionic Grassmannians and Pontryagin classes in algebraic geometry".

The result is that (1) is not true for symplectic Grassmannians. As far as I see, this does not really cause problems with (2) and (3); only the arguments have to be made a bit more carefully, because the stratification is not by affine spaces, but by more complicated (but still contractible) quasi-affine varieties.