### Motives of Projective Bundles

Friday, December 14th, 2012 | Author:

Given a vector bundle E-->X of rank r+1 one can take the projective space of lines in each fiber, which results in a projective bundle P(E)-->X. A projective bundle formula for a functor F from spaces to rings tells us that F(P(E)) is a free F(X)-module of rank r.

In this post I look at some computations around projective bundle formulae for the Chow ring, the algebraic K-Theory and the (Chow) motive of some spaces, in particular flag varieties. We recover some results from the previous posts on cohomology, cycles & bundles and motive of projective space.

### Motivation and History

In the first section, I want to talk about (nice) topological spaces $X$ and their real vector bundles.

#### Classical topology

The idea of a projective bundle formula comes from classical topology.
Theorem (Projective Bundle formula for singular cohomology):
Given a vector bundle $\mathcal{E} \to X$ of rank $r+1$ the singular cohomology of its projectivization $\mathbb{P}(\mathcal{E}) \to X$ is a module over the singular cohomology of the base $X$, and there is a module homomorphism $H^\bullet(\mathbb{P}(\mathcal{E}),\mathbb{Z}) \simeq H^\bullet(X)[H]/(H^{n+1})$ where the $H$ stands for "hyperplane" and is in degree $2$.

A special case of the projective bundle formula is projective space $\mathbb{P}^n$ itself, the projectivization of a vector space $V = k^{n+1}$, considered as a vector bundle over a point. Since the cohomology of a point is concentrated in degree $0$ and there just $\mathbb{Z}$, we get $H^\bullet(\mathbb{P}^n)=\mathbb{Z}[H]/(H^{n+1})$.

One should pay attention to the fact that not every bundle with fibers projective spaces are projectivizations of vector bundles. The obstruction to this is a class in $H^2$ with values in $GL_1$, as one can see explicitly by a Cech resolution.

The projective bundle formula for singular cohomology can be seen as a special case of the Leray-Hirsch theorem, which states that a fiber bundle $F \to E \to B$ which has the property that $F \to E$ induces a surjection $H^\bullet(E) \to H^\bullet(F)$ has cohomology $H^\bullet(E) \simeq H^\bullet(B) \otimes H^\bullet(F)$, where the isomorphism is an isomorphism of $H^\bullet(B)$-modules and the tensor product is taken in the graded sense.
If the basis $B$ is a point (or just contractible, i.e. a point from the homotopy point of view) the theorem is trivial. If you take $E \to B$ to be a projective bundle of rank $r$, the fiber over any point is $F = \mathbb{P}^r$ and one can show that the classes $H^k$ that generate the cohomology of $\mathbb{P}^r$ are in the image of $H^\bullet(E) \to H^\bullet(F)$, hence Leray-Hirsch can be applied.
Leray-Hirsch follows from the Leray-Serre spectral sequence (which is, of course, just a special case of the Grothendieck spectral sequence for the composition of two functors and derivation), which is $H^p(B,\mathcal{H}^q(F)) \Rightarrow H^{p+q}(E)$.

Singular cohomology also satisfies a Künneth formula, which is $H^\bullet(X,\mathbb{Q}) \otimes H^\bullet(Y,\mathbb{Q}) \simeq H^\bullet(X \times Y,\mathbb{Q})$. If we try to do this with integral coefficients, there's not an isomorphism but a short exact sequence with a Tor-term which doesn't vanish in general. A Künneth formula would also give us a projective bundle formula for trivial projective bundles $\mathbb{P}^n \times X \to X$. In some sense, I like to think of bundle formulas as generalizations or versions of the Künneth formula (since bundles are locally products).

### The Formulas in Algebraic Geometry

Now we change our focus and switch to algebraic geometry. I've written about definitions and relations between Chow groups and algebraic K-Theory before.

#### Chow ring

Let $X$ be an algebraic scheme over a field for the rest of the article.

Chow groups don't satisfy a Künneth formula! They do satisfy a projective bundle formula. I want to start with a baby version, which I'd like to call projective Künneth formula:
Theorem: $CH^\bullet(X \times \mathbb{P}^n) \simeq CH^\bullet(X)[H]/(H^{n+1})$, where $H$ is a hyperplane class in degree $1$.

This formula, together with the Yoneda lemma (in a variant known as Manin Identity Principle), already gives a decomposition of the Chow motive of projective space into irreducibles.

The full projective bundle theorem can be deduced from a localization sequence for higher Chow groups.

Theorem (localization sequence):
For $Z \to X$ a closed immersion of pure codimension $c$ with open complement $U$ there is a long exact sequence
$\cdots \to CH^{i-c}(Z,n) \to CH^{i}(X,n) \to CH^{i}(U,n) \to CH^{i-c}(Z,n-1) \to \cdots$

Theorem (projective bundle formula):
Let $\mathcal{E} \to X$ be a vector bundle of rank $r+1$ and $\mathbb{P}(\mathcal{E}) \to X$ its projectivization. Then $CH^\bullet(\mathbb{P}(\mathcal{E}),\bullet)$ is a $CH^\bullet(X,\bullet)$-module isomorphic to $CH^\bullet(X,\bullet)[H]/(H^{n+1})$, with $H$ in degree $1$.

To prove this by induction on the dimension on $X$, we can just take an open subset $U \subset X$ such that $\mathcal{E}$ is trivial over $U$, then for $U$ we have a projective Künneth formula and for $Z$ we use the induction hypothesis.

If you take only the $n=0$ part of the localization sequence (which I think was known for a longer time), you can prove the projective bundle formula for ordinary Chow groups with this method.

#### Algebraic K-Theory

We start with a projective bundle formula for $K_0$, i.e. the K-Theory of coherent sheaves on an algebraic scheme.
Theorem
For a vector bundle $E \to X$ of rank $n+1$, $K_0(\mathbb{P}(E))$ is a $K_0(X)$-module which is module-isomorphic to $K_0(X)[H]/(H^{n+1})$.

For trivial bundles, this is quite easy to see:
By a Theorem of Serre, any coherent sheaf on $\mathbb{P}^n$ is a quotient of some $\mathcal{O}(k)^{\oplus m}$ (with $k,m \geq 0$), hence the $\mathcal{O}(k)$ generate $K_0(\mathbb{P}^n)$.
The Koszul complex on $\mathbb{A}^{n+1}$ is a $n+2$-term exact sequence which gives a relation in $K_0$ between any $\mathcal{O}(k+1),\dots,\mathcal{O}(k+n+2)$ by applying a shift and taking the coherent sheaf on $\mathbb{P}^n$ associated to a graded module. Therefore we see that $H := [\mathcal{O}(-1)]$ and $H^2,\dots,H^n$ and $H^0 = [\mathcal{O}]$ together form a generating set. There are no more relations between these generators.

The proof of the non-trivial case is given by Grothendieck and Berthelot in SGA6 Exposé VI. Such a bundle formula also holds for higher algebraic K-Theory (due to Quillen):

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, $K_\bullet(\mathbb{P}(E))$ is a $K_0(X)$-module which is module-isomorphic to $K_\bullet(X)[H]/(H^{n+1})$, with $H$ in degree $1$.

#### Chow Motive

If we're talking about bundles over a smooth projective (or smooth complete) base, we can talk about the Chow motive of the total space in relation to the base.

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, $h(\mathbb{P}(E)) = \bigoplus_{s=0}^n h(X)(s)[2s]$ ($=h(X)[H]/(H^{n+1})$, with $H = \mathbb{Z}(1)[2]$).

Manin's Identity Principle implies that a morphism of Chow motives $M\to N$ is an isomorphism iff the morphism of associated functors $\omega_M \to \omega_{N}$ is an isomorphism, where
$\omega_M : \mathcal{P}(k) \ni Y \mapsto M_\sim(h(Y),M(\ast)) := \bigoplus_{r \in \mathbb{Z}} M_\sim(h(Y),M(r))$

We guess the motive of a projective bundle $\mathbb{P}(E)$ of rank $n$ over a base $X$ to be $M := \bigoplus_{s=0}^n h(X)(s)[2s]$, then from the projective bundle formula for Chow groups we see that $\omega_M(\mathbb{P}(E)) \simeq \omega_{\mathbb{P}(E)}(\mathbb{P}(E))$, so the identity morphism on the right hand side yields a morphism $h(\mathbb{P}(E)) \to M$ which induces an isomorphism on the functors $\omega$, thus is an isomorphism.

#### Voevodsky Motive

If we look at an arbitrary smooth base (no properness assumption) we need Voevodsky's triangulated motives. I assume that the base is a $k$-scheme for $k$ a perfect field.

Theorem (projective bundle formula):
For a vector bundle $E \to X$ of rank $n+1$, the canonical morphism $\bigoplus_{s=0}^n \mathbb{Z}_{tr}(X)(s)[2s] \to \mathbb{Z}_{tr}(\mathbb{P}(E))$ is an isomorphism.

One can take a local trivialization of $E$ over a cover $\mathcal{U}$ of $X$, which allows (by Mayer-Vietoris triangles) to compute the motive of $\mathbb{P}(E)$ in terms of $\mathbb{P}(E)|_U$ for $U \in \mathcal{U}$. Thus one can look at the trivial case, where $\mathbb{P}(E) \simeq \mathbb{P}^n_X$ and the formula holds.

From this projective bundle formula one can deduce a Gysin triangle:
Gysin/localization triangle:
Let $X$ be a smooth scheme, $Z$ a smooth closed subscheme of codimension $c$. Then
$C_\ast \mathbb{Z}_{tr}(X \setminus Z) \to C_\ast \mathbb{Z}_{tr}(X) \to C_\ast \mathbb{Z}_{tr}(Z)(c)[2c] \to [1]$
is a distinguished triangle.

This can be proved by looking at the situation étale-locally, where $X= Y \times \mathbb{A}^c$ and $Z = Y \times 0$. Then the morphism $C_\ast \mathbb{Z}_{tr}(X) \to C_\ast \mathbb{Z}_{tr}(Z)(c)[2c]$ is just $C_\ast(\mathbb{Z}_{tr}(\mathbb{P}^n)/\mathbb{Z}_{tr}(\mathbb{P}^n\setminus 0)) \to \mathbb{Z}(n)[2n]$ from the projective bundle formula, tensored with $C_\ast\mathbb{Z}_{tr}(Y)$.

### More general bundle formulas

One could ask what happens for a general $G$-bundle $E \to X$, let's say for $G$ a (split semi-simple) reductive linear algebraic group, if we take $E/P \to X$ for any subgroup $P$ of $G$. Maybe in the case of a parabolic $P$ we can describe the motive of $E/P$ in terms of $X$. At least in the case of a certain maximal parabolic of $G=SL_{n+1}$ this yields $G/P = \mathbb{P}^n$, so the analogy should be clear.

Using localization techniques becomes more interesting, since a $G$-bundle which is étale-locally trivial needn't be Zariski- (or even Nisnevich-) locally trivial (though this is the case for $G=GL_n$ and $G=SL_n$). Of course, one can restrict attention to Zariski-locally trivial bundles first.

Köck has further developed the techniques used to prove the projective bundle formula for Chow motives, and computed the motive of $G/P$ itself.
Theorem:
Let $G$ be a split reductive group over a field $k$ and $P$ a parabolic subgroup, with $Y:=G/P$ the quotient homogeneous space. Denote by $Y_w := BwB/P$ the Bruhat cells in $Y = \bigcup_{w \in W} Y_w$. Then $h(Y) \simeq \bigoplus_{w \in W} \mathbb{Z}(-dim(Y_w))$.

Habibi and Rad have recently proved for a connected reductive group $G$ over a characteristics $0$ field $k$ that the motive of a Zariski-locally trivial $G$-bundle over an irreducible base with mixed Tate motive is itself a mixed Tate motive.

Is there more known about $G/P$-bundles? I would love to see a formula that takes the type of the parabolic (i.e. the corresponding subset of roots) and spits out a motivic decomposition. I have the impression that one should be able to prove this along the lines of the projective bundle formula for Voevodsky motives. Locally, one has just $G/P$ and there Köck described the motive. Alternatively, one could use the Gysin sequence.

### Example

To compute the Voevodsky motive of $\mathbb{P}^n$ from the bundle formula above would be cheating, since I did the proof by reduction to the motive of $\mathbb{P}^n$ (as it is done in the book of Mazza-Voevodsky-Weibel). I wrote about the Motive of $\mathbb{P}^n$ without a bundle formula before.

#### Flag varieties

One can see $\mathbb{P}^n$ as a special kind of partial flag variety which parametrizes flags of the form $0 = V_0 \subset V_1 \subset V_n = V$ with $V_1$ one-dimensional and $V$ an $n$-dimensional vector space (affine space).

Every partial flag variety $X$ parametrizing flags of the form $0 = V_0 \subset V_{i_1} \subset \cdots \subset V_{i_k} \subset V_n = V$ with $0 < i_1 < \cdots < i_k < n$ can be written as homogeneous space $X = GL_n/P(i_1,\dots,i_k)$, where $P(i_1,\dots,i_k)$ denotes a standard parabolic in $GL_n$. The $1$-flags in a vector space $V$ (of dimension $n+1$) form a variety $Fl_1$ which maps to a point, which we can consider as the space of $0$-flags $Fl_0$. This map $Fl_1 \to Fl_0$ is obviously just $\mathbb{P}^{n} \to \mathbb{P}^0$, hence we can compute the motive of $Fl_1$. Now $Fl_{1,2}$, the space of $1,2$-flags, fibers over $Fl_1$, since over each $1$-flag there is a projective space of $2$-flags containing this $1$-flag. This means $Fl_{1,2} \to Fl_1$ is a projective bundle, and we can compute the motive. And so on. Explicitly, this gives us $h(Fl_{1,2}) = \bigoplus_{s_2=0}^n \left(\bigoplus_{s_1=0}^n \mathbb{Z}(-s_1)[-2s_1] \right)(-s_2)[-2s_2]$ $= \bigoplus_{s_1,s_2=0}^n \mathbb{Z}(-s_1-s_2)[-2s_1-2s_2]$ and in general we have $h(Fl_{1,\dots,n}) = \bigoplus_{s_1,\dots,s_n=0}^n \mathbb{Z}(-(\sum s_i))[-2\sum(s_i)]$ reflecting the structure of the Bruhat cells of $Fl_{1,\dots,n}(V) \simeq GL(V)/B$ ($B$ a Borel subgroup). It would be nice if one could use a projective bundle formula to compute the motive of any homogeneous space $G/P$ for $P$ a parabolic, but if one tries to do that, the bundles one encounters are no longer projective. Do you know of other neat applications of the projective bundle formula?

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