# Which varieties are sums of tensor powers of the Lefschetz motive?

Any smooth projective variety $$X$$ gives an object $$h(X)$$ in the category of pure Chow motives. If $$X$$ is a generalized flag variety, i.e. a quotient $$G/P$$ where $$G$$ is semisimple linear algebraic group and $$P$$ is a parabolic subgroup, I believe $$h(X)$$ is a direct sum of tensor powers of the Lefschetz motive, because $$X$$ can be decomposed into Schubert varieties which are copies of $$\mathbb{A}^n$$ for various $$n$$.

If this is correct, I'd like to know: which other smooth projective varieties give pure Chow motives that are direct sums of tensor powers of the Lefschetz motive?

My intuition is that any variety with something like a "Schubert decomposition" — roughly, a well-behaved way of expressing it as a disjoint union of copies of $$\mathbb{A}^n$$'s — will have this property. I feel there should be plenty. But I don't actually know any varieties with this property, apart from flag varieties!

Any variety $$X$$ of dimension $$d$$ over $$\mathbb{F}_p$$ having this property will have an associated polynomial $$N_X$$ of degree $$d$$ with natural number coefficients:

$$N_X(q) = \sum_{n = 0}^d a_n q^n$$

such that $$X$$ has $$N_X(q)$$ points over $$\mathbb{F}_q$$ when $$q$$ is any power of $$p$$.

Is the converse true? Is a smooth projective variety $$X$$ over $$\mathbb{F}_p$$ with a polynomial $$N_X$$ having this property always a direct sum of tensor powers of the Lefschetz motive?

• The key words are "paving" and maybe "Bialynicki-Birula", try searching those on here. I think good examples are quadrics and certain toric varieties. May 16, 2022 at 4:39
• Having a paving by affine spaces certainly suffices, but there are lots of examples of varieties with motives of this form which do not have affine pavings (in fact, they may even be of general type.) I believe that your converse is true assuming the Tate conjecture, but I may be misremembering. Another possibly mis-remembered statement: assuming the Hodge conjecture, a smooth projective variety over $\mathbb{C}$ has motive of this form if and only if all off-diagonal hodge numbers are zero.
– dhy
May 16, 2022 at 7:15
• A correction to my previous comment: This only holds if you consider the Chow motive rationally.
– dhy
May 16, 2022 at 7:36
• @dhy: This is certainly not known at the level of (rational) Chow motives even assuming the Hodge conjecture. For example, the statement would imply Bloch's conjecture about zero cycles on surfaces with $p_g = 0$. Similary, your statement about the Tate conjecture implying the converse is not true (though it follows from conjectures about Chow groups of varieties over finite fields).
– naf
May 16, 2022 at 7:45
• A very simple example without a decomposition is to take something with a decomposition, like $\mathbb P^2$ and blow up lots of points. A blow up adds Lefschetz motives, but doesn't extend a decomposition unless the point was already a stratum. You have to do more work to show no decomposition works. May 16, 2022 at 17:13

One class of examples is already indicated in the comments, and the question itself. I thought it would be good to include this in an official answer.

Proposition. Let $$X$$ be smooth projective variety over a field $$k$$, such that there exists a chain of closed sets $$X=X_n\supset X_{n-1}\supset \ldots X_{-1}=\emptyset$$ such that $$X_i-X_{i-1}$$ is a union of affine spaces or split tori (products of $$\mathbb{G}_m$$'s). Then the Chow motive is of Tate type, i.e. sum of powers of the Lefschetz motive.

Cor. The conclusion holds for flag varieties, and projective toric varieties.

The proposition follows from the distinguished triangles $$M_{gm}^c(X_{i-1})\to M_{gm}^c(X_{i})\to M_{gm}^c(X_{i}-X_{i-1})$$ where $$M_{gm}^c$$ is the motive with compact support in Voevodsky's category (cf. Voevodsky Triangulated categories of motives over a field and Mazza, Voevodsky, Weibel Lectures on motivic cohomology), induction, and the identification of $$M_{gm}^c(X)$$ with the Chow motive of $$X$$ (up to translation and twist).

If you allow quasi projective varieties, then complements of hyperplane arrangements are also mixed Tate.

A few additional comments. As Ben Wieland and Will Sawin have pointed out, you get more examples by blowing an example you have of Tate type along a subvariety with the same property. I can think of a few more examples off the top of my head, such that the Hilbert scheme of points on a rational surface. On the other hand, I hope it's clear that examples would have to be quite special. Over $$\mathbb{C}$$, the Hodge numbers $$h^{pq}$$ would vanish for $$p\not=q$$, so most varieties would not satisfy this condition.

• I think there are lots of elementary geometric constructions that produce varieties of this type - for example from a vector bundle on a toric variety one can form a bundle of flag varieties, and that will satisfy the hypothesis. One can also do blow-ups of a variety satisfying this condition. May 16, 2022 at 17:10
• @WillSawin Yes, I agreed. I meant that it's hard to produce examples which are covered by the prop. May 16, 2022 at 17:47
• I think fibrations (but not blow-ups, I agree) produce examples covered by the proposition. One can really get a lot of examples by taking projective space bundles on projective space bundles on projective space, or something like that. May 16, 2022 at 17:56