Weil Conjectures for Grassmannians

To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?

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More generally, as Ben's answer notes, it's easy to do the point counts for varieties that have nice cell decompositions. So generalized flag varieties and toric varieties are other examples where there should be direct proofs. Maybe someone who knows Weil's original paper can comment: I thought that homogeneous spaces were one of his motivating examples? –  GS Jan 27 '10 at 10:03
Weil did this on the last paragraph of the paper where he states the Weil conjectures. –  Felipe Voloch Jan 27 '10 at 16:00
–  Felipe Voloch Jan 27 '10 at 16:02

The first result on the google search "zeta function of grassmannian" seems to contain quite a direct and not too long derivation of the zeta function for a grassmannian over a finite field:

From the zeta you see that it is rational, of course get the zeros (which are none), but you don't immediately get confirmation of the functional equation. Though, from the very simple combinatoric representation of the zeta function, it might be easy to prove directly, I will try with pen and paper later.

I'm glad I searched this, I didn't know the zeta was so simple in this case as well.

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A question that just occurred to me: There exist Grassamnians that are not projective varieties? if so, does the formulation of the Weil conjectures still trouble free? –  John McCarthy Nov 24 '09 at 14:05
@John: What do you mean? The usual Grassmanians (subspaces of a fixed dimension in a vector space of a given dimension) is always projective. –  Mariano Suárez-Alvarez Nov 24 '09 at 14:07
A projective variety is a closed subvariety of a projective space. –  David Speyer Nov 24 '09 at 14:14
It's not obvious, but it's true. Proof 1: Grassmannian's are proper, so the image of the Plucker embedding is closed. Proof 2: There are classic, explicit equations called the Plucker relations which cut out the Grassmannian. For a detailed and elementary proof that the Grassmannian is exactly the zeroes of the Plucker equations, see Ezra Miller and Bernd Sturmfels' book. –  David Speyer Nov 24 '09 at 14:17
@John: what Wikipedia means is that the simplest grassmanian which is not isomorphic to a projective space is $G(4,2)$, not that that is the simpler grassmanian which is not a projective variety (in which case we should correct it!) –  Mariano Suárez-Alvarez Nov 24 '09 at 18:19

Yes. Both cohomology and number of points are readily determined by looking at the Schubert cells (a cell of dimension k contributes one dimension to $H^k$, and $q^k$ to the point count) and they match.

In fact, it's very easy to check Weil's conjecture directly for any smooth variety which has a decomposition into cells.

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Cells in what sense? –  Kevin H. Lin Nov 24 '09 at 16:01
in the sense of a CW decomposition, I think, when you work over the complex numbers –  Sam Lichtenstein Nov 24 '09 at 16:47
But doesn't every smooth manifold have a CW decomposition by Morse theory? –  Kevin H. Lin Nov 24 '09 at 21:02
The cells are affine spaces inside the variety, which stratify it, not CW cells. For example $\mathbb{P}^n = \mathbb{C}^n \cup \mathbb{C}^{n-1} \cup \ldots \cup \mathbb{C}^0$. –  mdeland Nov 24 '09 at 21:34
Kevin- Think about an elliptic curve. –  Ben Webster Nov 24 '09 at 21:35