What do you lose when passing to the motive?

Question

I contemplated about what information about a scheme we lose when passing to its motive. I came up with the following examples:

1. The projective bundle of a vector bundle does only depend on the rank of the vector bundle whatever it is twisted (this follows from the Mayer-Vietoris exact triangle).
2. The motive of the blow-up of a $\mathbf{P}^2$ in a point equals the motive of the quadric surface.

Are there further big classes of such phenomena? (Is there a class which 2. fits into?)

And conversely, what can we recover about the variety from its motive?

Answer 1

Both examples you consider have the property that the additive structure of the Chow groups are the same, but the multiplicative structures are different. In the first case multiplication depends on the Chern classes of the bundle, and in the second case intersection forms on $CH^1$ are $x^2 - y^2$ and $2uv$ which are different integrally. So if we capture multiplicative structure of the motive (which is $\Delta: M(X) \to M(X) \otimes M(X)$ I think) we'll be able to do better.

For your example 2, both varieties are cellular (glued from affine spaces), and the numbers of cells in each dimension are the same. Such varieties always have isomorphic (Tate) motives.

In general, I think there's no better answer to the question "What can we recover about the variety X from its motive?" rather than the trivial one: "We can recover all the reasonable cohomology theories evaluated on X". I am very curious what other people will say, though.

As an aside note, I remember reading somewhere that it is expected that the integral motive of a quadric determines the quadractic form itself.

Answer 2

Here the word "motive" will stand for Grothendieck pure motives modulo rational equivalence. Your point 1. is also true for Grassmann bundles. More precisely the following result holds :

Let $E\longrightarrow X$ be a vector bundle of rank $n$, $k\leq n$ and $Gr_k(E)\longrightarrow X$ the associated Grassmann bundle. Then $M(Gr_k(E))\simeq \coprod_{\lambda}M(X)[k(n-k)-\lambda]$, where $\lambda$ runs through all partitions $\lambda=(\lambda_1,...,\lambda_k)$ satisfying $n-k\geq \lambda_1\geq...\geq \lambda_k\geq 0$.

You can prove it in the same fashion as for the projective bundle theorem, as an application Yoneda type lemma for Chow groups.

We now know many things on the motives of quadrics. For example if a quadratic form $q$ is isotropic, the motive of the associated quadric $Q$ has a decomposition $\mathbb{Z} \oplus M(Q_1) \oplus \mathbb{Z}[\dim(Q)]$, where $Q_1$ is a quadric of dimension $\dim(Q)-2$ associated to a quadratic form $q_1$ Witt equivalent to $q$. Using it inductively you get the motivic decomposition of split quadrics and for example if $\dim(q)$ is odd and $q$ is split the motive of $Q$ is $\mathbb{Z}\oplus \mathbb{Z}[1]\oplus ... \oplus \mathbb{Z}[\dim(Q)]$. Another very important result is the Rost nilpotence theorem, which asserts that the kernel of the change of field functor on Chow groups of quadrics consists of nilpotents. This result is very fruitfull because it implies that the study of the motive of quadrics can be done over a field which splits the quadric, working with rational cycles in stead of cycles over the base field. Even though these motivic results give severe restrictions on the higher Witt indices of quadrics and have very important applications, the motive does not contain "everything" about the associated quadratic forms (even in terms of higher Witt indices).

Another interesting class of varieties to motivic computations are the cellular spaces, i.e. schemes $X$ endowed with a filtration by closed subschemes $\emptyset \subset X_0\subset ... \subset X_n= X$ and affine bundles $X_i\setminus X_{i-1}\rightarrow Y_i$. In this situation the motive of $X$ is isomorphic to the direct sum of (shifts) of the motives of the $Y_i$. For example the filtration of $\mathbb{P}^n$ given by $X_i=\mathbb{P}^i$ and those affine bundles are given by the structural morphism of $\mathbb{A}^{i}$ imply the motivic decomposition $M(\mathbb{P}^n)=\mathbb{Z}\oplus ... \oplus \mathbb{Z}[n]$, and as you can see this is the same motive as odd dimensional split quadrics, so you certainely loose information.

The situation is much more complicated replacing quadratic forms by projective homogeneous varieties, but still under some assumption you can recover some results such as Rost nilpotence theorem, and we now begin to have a good description of their motive. Under these assumption the motive of projective homogeneous varieties encodes informations about the underlying variety, such as the canonical dimension, with the example of the computation of those of generalized Severi-Brauer varieties. Some works have also been done to link motives in this case with the higher Tits indices of the underlying algebraic groups.

Just to cite a few mathematicians from who we owe these great results : V. Chernousov, N. Karpenko, A. Merkurjev, V.Petrov, M. Rost, N.Semenov, A. Vishik, K. Zainoulline and probably many others that i forgot to mention.

edit : to add more precision to the nice answers of Mr. Chandan Singh Dalawat and Mr. Evgeny Shinder, motives of (usual) Severi-Brauer varieties of split algebras are indeed the same as projective space and split quadrics (in odd dimension) but it is obvious that on the base field they're are not necessarily isomorphic since the Severi-Brauer variety is totally split as long as there is a rational point, whereas an isotropic quadratic form is not completely split.

Answer 3

I really don't know what motives are, but perhaps any smooth projective conic over $\mathbb{Q}$ has the same motive as $\mathbb{P}_1$. So you loose information about the places where the conic had bad reduction. The same applies to twisted forms of $\mathbb{P}_n$.

Also, perhaps any torsor under an abelian variety $A$ has the same motive as $A$, so you lose infomation about whether the torsor has a point or not.

Feel free to criticise or to correct me.

Bonus. A bug in MO is preventing me from asking questions. While Anton gets it fixed, let me give a free translation of the above "answer" into French :

J'ignore complètement ce que sont les motifs, mais il me semble que toute conique lisse projective sur $\mathbf{Q}$ a le même motif que $\mathbf{P}_1$. Le motif ne se souvient donc pas des places où la conique avait mauvaise réduction. On peut en dire autant des formes tordues de $\mathbf{P}_n$.

Aussi, un torseur sous une variété abélienne $A$ a probablement le même motif que $A$; il ne se souvient donc pas si le torseur avait un point rationnel ou non.

N'hésitez pas à critiquer ou corriger cette réponse, ainsi que le français.