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The theory of motives is an attempt to cope with the fact that there are many reasonable cohomology theories of algebraic varieties. Now, sometimes your cohomology theory does not just give you a bunch of groups/vector spaces; it gives you a full-fledged (pro-)homotopy type (though based on limited responses to this question, I think there is no formal way to functorially produce homotopy types given a cohomology theory). The question is: what should be the extension of motives to homotopy types? What are some notable works in this direction? I have heard there is something called motivic homotopy theory, is it really relevant here or it just happens to have a similar name?

More specific questions:

  • First off, we need to show that usual motives are not good enough. This is probably obvious to anyone working in the field, but not everybody on this site is an expert on motives so an explicit example would be great. I think there is a rigorously constructed triangulated category of motives which is supposed but not known to be the derived category of some abelian category of motives. There is also a somewhat more pedestrian Grothendieck ring of varieties. I think the class in the Grothendieck ring does not determine the motive nor does the motive determine the class in the Grothendieck ring. Is there an example of two varieties with the same class in the Grothendieck ring which have different etale fundamental groups? An example of two varieties with the same motive in the triangulated category which have different etale fundamental groups?
  • Before you can say "every Weil cohomology theory factors through motives", you have to define a Weil cohomology theory. What should the definition be for homotopy types? Has somebody written down a manageable list of axioms definining exactly what we are interested in?
  • Before we can seriously talk about motivic homotopy types, we should understand on the categorical level what we expect the relevant category to be. In the case of motives, the relevant piece of category theory is Tannakian formalism, I believe (then we can say smart words like "a Weil cohomology theory is just a fiber functor blah blah"). What should the category theory look like for motivic homotopy types? A kind of non-abelian Tannakian formalism?

P.S.: yeah, I know the question is super naive, you are free to call me an idiot in the comments.

EDIT: I did not think about the coefficients when asking this. I believe Voevodsky's category makes sense with $\mathbb{Z}$-coefficients, so take that in the first question.

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    $\begingroup$ There is the $\mathbb{A}^1$-homotopy theory of schemes (Morel, Voevodsky). Voevodsky's triangulated category of motives $DM$ embeds into the $\mathbb{A}^1$-derived category $D_{\mathbb{A}^1}$ where the objects have "explicit" descriptions (essentially, complexes of presheaves on the category of smooth schemes). Is this in the spirit of what you are looking for? $\endgroup$ – François Brunault May 6 at 17:07
  • $\begingroup$ @FrançoisBrunault but does every "Weil homotopy theory" factor through it, like every Weil cohomology theory supposedly factors through the category of motives? Assuming we know what a Weil homotopy theory is. $\endgroup$ – user138661 May 6 at 17:14
  • $\begingroup$ It seems the category you want would map into the category DM and not the other way round, so I'm not sure my comment will be useful for your question. You could also look for the notion of motivic fundamental group, I don't know if this can be made into a functor from some natural category (just as motivic cohomology can be seen as a functor from DM). $\endgroup$ – François Brunault May 6 at 17:21
  • $\begingroup$ Relevant: arxiv.org/abs/0712.3291 $\endgroup$ – Denis Nardin May 6 at 17:47
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    $\begingroup$ Starting from categories of dualisable motives over $X$, a point in $X(k)$ gives you a fibre functor to $k$-motives, and Tannakian formalism then gives a form of homotopy type in $k$-motives. A Weil cohomology theory would give a fibre functor down to complexes, and the resulting thing would then be an arithmetic, rather than a geometric, homotopy type. For these homotopy types to behave, you want a $t$-structure (so the standard conjectures would help). For some thoughts about this, see arxiv.org/abs/1309.0637 $\endgroup$ – Jon Pridham May 7 at 13:08
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The projective plane and fake projective planes should have isomorphic motives in the triangulated category, at least with rational coefficients, but different etale fundamental groups. This is because they have isomorphic cohomology.

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  • $\begingroup$ Dr. Sawin, could you clarify "isomorphic cohomology"? Do you mean "isomorphic cohomology with respect to an arbitrary Weil cohomology theory"? If you mean that, how do I see that that is true? Then do we know that there is, in fact, a map of motives inducing these isomorphisms on cohomologies (like you know, two topological manifolds can have isomorphic homotopy groups in all degrees and still fail to be homotopy equivalent)? $\endgroup$ – user138661 May 16 at 18:14
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    $\begingroup$ @schematic_boi Any algebraic surface has an open set with a finite-to-one map to projective space. This gives a correspondence, giving a map of motives. For a fake projective plane, it should be easy to see that this correspondence gives an isomorphism on any reasonable cohomology theory - we just have to check that some degrees are nonzero. I don't know if it can be shown to give an isomorphism on an arbitrary Weil cohomology theory, except over finite fields. $\endgroup$ – Will Sawin May 16 at 18:49
  • $\begingroup$ @schematic_boi Under the Kunneth type standard conjecture and Conjecture d, a motive is just isomorphic to the sum of its cohomology groups as motives, and so two motives with isomorphic cohomology groups are isomorphic. Under the Hodge or Tate conjecture, the same is true when viewing the cohomology groups as Hodge structures / Galois representations. $\endgroup$ – Will Sawin May 16 at 18:52
  • $\begingroup$ An example in the spirit of this answer: a rational elliptic surface and an Enriques surface have the same motive with rational coefficients (1+10L+L^2), but have étale fundamental groups of order 1 and 2 respectively, similarly P^2 blow up in 8 points and a classical Godeaux surface have the same motives (1+9L+L^2) but étale fundamental groups of order 1 and 5 respectively, see pdfs.semanticscholar.org/7bbd/… and references there. $\endgroup$ – user25309 May 17 at 7:30
  • $\begingroup$ @user25309 Yes, good examples. If I remember correctly an Enriques surface can be obtained as an order $2$ logarithmic transformation of a rational elliptic surface (i.e. the unramified double cover of the Enriques is a ramified double cover of the rational elliptic surface, ramified at two elliptic fibers). The graph of this relation gives the relevant correspondence to give an explicit isomorphism of their motives. $\endgroup$ – Will Sawin May 17 at 11:45
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EDIT. Upon request of @schematic_boi, let me try to clarify. My original post only gives an example where the motive does not determine the variety, conditionally on a conjecture of Orlov. In fact, I realized that there are simpler and unconditional examples: if $S$ is any scheme, projective bundles of given rank over $S$ give rise to isomorphic motives (in fact just Tate motives) in the triangulated category $\mathrm{DM}(S,\mathbb{Z})$. This is the projective bundle formula, for this level of generality see Cisinski-Déglise, Triangulated categories of motives, 11.3.4. But I think this won't answer @schematic_boi's first question, because the topological fundamental group of a projective bundle should be isomorphic to that of the base. On the other hand, Schnell's article The fundamental group is not a derived invariant mentioned by @user25309, together with Orlov's conjecture, provides a conditional answer, at least with $\mathbb{Q}$-coefficients.

Original post. Orlov has conjectured [1] that two schemes having equivalent derived categories of quasi-coherent sheaves give rise to the same motive in the category $\mathrm{DM}$. In particular this would imply, due to a result of Lesieutre [2] that there are infinitely many non-isomorphic smooth projective 3-folds which give rise to the same motive.

References:

[1] Orlov, Derived categories of coherent sheaves, and motives. Russian Math. Surveys 60 (2005), no. 6, 1242–1244

[2] Lesieutre, Derived-equivalent rational threefolds. Int. Math. Res. Not. 2015, no. 15, 6011–6020.

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  • $\begingroup$ do they have different etale fundamental groups? Or are all of them simply-connected? $\endgroup$ – user138661 May 16 at 10:04
  • $\begingroup$ @schematic_boi: do you mean the examples of Lesieutre? They are simply-connected; they are blowups of $\mathbf P^3$ in sets of 8 points. $\endgroup$ – Bort May 16 at 10:30
  • $\begingroup$ but you see, the question was: "Is there an example of two varieties with the same class in the Grothendieck ring which have different etale fundamental groups? An example of two varieties with the same motive in the triangulated category which have different etale fundamental groups?" $\endgroup$ – user138661 May 16 at 10:30
  • $\begingroup$ Orlov's conjecture is about motives with rational coefficients. There are derived equivalent varieties with different étale fundamental groups ( arxiv.org/abs/1112.3586 ), but it seems an overkill: unless I am mistaken, étale covers induce isomorphisms of motives with rational coefficients and so there are obviously varieties with the same rational motive but different étale fundamental groups. I think the question of coefficient should be clarified in the original question. $\endgroup$ – user25309 May 16 at 12:00
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    $\begingroup$ @schematic_boi I was only giving an explicit example where the motive does not determine the variety. If you put Orlov's conjecture together with Schnell's article mentioned by user25309, then you get an answer to your first question, but it is conditional. $\endgroup$ – François Brunault May 16 at 18:07
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Some answer to one of the questions: two smooth projective varieties over $\mathbb{C}$ with the same class in the Grothendieck ring of varieties have isomorphic fundamental groups (and in particular étale fundamental group by taking the profinite completion).

Indeed, by Larsen and Lunts, https://arxiv.org/abs/math/0110255, the Grothendieck group of smooth projective complex varieties has, after localization of the Lefschetz motive, a linear basis indexed by stable birational equivalence classes.

On the other hand, the fundamental group of smooth projective varieties is invariant under stable birational equivalence (because the projective spaces are simply connected and birational equivalences only affect loci of codimension greater or equal to 2).

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