Timeline for Motives and homotopy theories of algebraic varieties
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| May 17, 2019 at 7:15 | comment | added | user25309 | ok, my claim about finite étale covers was just obviously wrong, as pointed above. | |
| May 16, 2019 at 19:38 | history | edited | François Brunault | CC BY-SA 4.0 |
Clarified answer & comments
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| May 16, 2019 at 19:04 | comment | added | Will Sawin | @user25309 The claim about finite etale covers fails already for curves of higher genus, as the cover will have different genus than the basis. I think it should be possible to check Orlov's conjecture explicitly in the setting of Schnell's example by verifying that the obvious correspondence gives an isomorphism of rational Chow motives. | |
| May 16, 2019 at 18:11 | comment | added | user138661 | @FrançoisBrunault OK, I understand now, thank you. | |
| May 16, 2019 at 18:07 | comment | added | François Brunault | @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. | |
| May 16, 2019 at 17:19 | comment | added | user138661 | I am not still not quite sure which of the 3 sub-questions is this answer answering. Could you clarify? | |
| May 16, 2019 at 15:39 | comment | added | François Brunault | @user25309 An arithmetic example: any extension of number fields defines a finite etale cover, but the motives are not isomorphic as one can recover the degrees of the number fields (or even their Dedekind zeta functions). | |
| May 16, 2019 at 13:10 | comment | added | user25309 | @François Brunault : I was thinking about smooth projective varieties (example: isogenous elliptic curves) but I might still be wrong even in this case. | |
| May 16, 2019 at 12:25 | comment | added | François Brunault | The projective bundle formula (Cisinski-Déglise 11.3.4) says that projective bundles give rise to isomorphic motives, maybe this could give an example where the fundamental groups are different. | |
| May 16, 2019 at 12:22 | comment | added | François Brunault | @user25309 I'm not sure I understand your second statement: take an étale cover of curves e.g. $\mathbb{G}_m \setminus \mu_N \to \mathbb{G}_m \setminus \{1\}$ with the $N$-th power map, then the motives $H^1$ will not be isomorphic because they don't have the same rank. But maybe you are thinking about some other explicit example? | |
| May 16, 2019 at 12:00 | comment | added | user25309 | 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. | |
| May 16, 2019 at 10:30 | comment | added | user138661 | 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?" | |
| May 16, 2019 at 10:30 | comment | added | Bort | @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. | |
| May 16, 2019 at 10:04 | comment | added | user138661 | do they have different etale fundamental groups? Or are all of them simply-connected? | |
| May 16, 2019 at 9:54 | history | answered | François Brunault | CC BY-SA 4.0 |