Timeline for K-equivalence ⇒ isomorphism of Chow motives?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 17, 2021 at 0:39 | history | edited | LSpice | CC BY-SA 4.0 |
Names of papers
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| May 16, 2021 at 20:42 | answer | added | Evgeny Shinder | timeline score: 4 | |
| Jan 26, 2021 at 17:05 | comment | added | Evgeny Shinder | @Mickhail Bondarko: I think we know $[M(X)] = [M(Y)]$ in $K_0(Chow) \otimes \mathbf{Q}$. | |
| Jan 26, 2021 at 15:38 | comment | added | Mikhail Bondarko | It appears that if we had a version of motivic integration with values in $K_0(Chow)$ then the D-equivalence of X and Y would imply that $M(X)\bigoplus Z\cong M(Y)\bigoplus Z$ for some Chow motif Z. Assuming some motivic conjecture one would obtain that $M(X)\cong M(Y)$ for motives with rational coefficients, but not with integral ones. | |
| Jan 22, 2021 at 15:02 | comment | added | crystalline | @abx: I agree with that, but my confidence in Evgeny's expertise on this topic led me to double check Huybrechts' statement more carefully. | |
| Jan 22, 2021 at 10:08 | history | edited | Evgeny Shinder | CC BY-SA 4.0 |
Example 2 added
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| Jan 22, 2021 at 9:19 | comment | added | Nico Berger | There is an upgrade of Huybrechts' result by Fu-Vial stating that the rational Chow motives of derived equivalent K3 surfaces are isomorphic as "Frobenius algebra objects", see arxiv.org/abs/1907.10868. Their Question 1 asks for the case of ihs manifolds. | |
| Jan 22, 2021 at 8:36 | comment | added | abx | @crystalline: Yes indeed, thanks — I was misled by the statement of Theorem 1.1 in Huybrechts' paper. Doesn't "Chow motive" usually refer to the integral ones? | |
| Jan 22, 2021 at 8:34 | comment | added | Nico Berger | Riess proved that K-equivalent (i.e. birational) irreducible holomorphic symplectic varieties have isomorphic integral Chow motives, see arxiv.org/abs/1304.4404. | |
| Jan 22, 2021 at 7:46 | comment | added | crystalline | @abx: Read carefully; Huybrechts considers rational Chow motives. | |
| Jan 22, 2021 at 6:13 | comment | added | abx | I am a bit puzzled by your last remark: for K3 surfaces, D-equivalence implies indeed isomorphism of Chow motives — this is a result of Huybrechts (Abh. Math. Semin. Univ. Hambg. 88 (2018), no. 1, 201–207). | |
| Jan 21, 2021 at 23:44 | history | edited | Evgeny Shinder | CC BY-SA 4.0 |
added 210 characters in body
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| Jan 21, 2021 at 23:11 | history | asked | Evgeny Shinder | CC BY-SA 4.0 |