A motivic global Torelli theorem for isogenous K3 surfaces

Fu L, Vial C (2021)
Advances in Mathematics 383: 107674.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torellitype theorem. We ask whether, more generally, twisted derived equivalent hyper-K & auml;hler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of & ldquo;Frobenius algebra object & rdquo; by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras. (c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Stichworte
Motives; K3 surfaces; Torelli theorems; Derived categories; Cohomology; ring
Erscheinungsjahr
2021
Zeitschriftentitel
Advances in Mathematics
Band
383
Art.-Nr.
107674
ISSN
0001-8708
eISSN
1090-2082
Page URI
https://pub.uni-bielefeld.de/record/2955095

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Fu L, Vial C. A motivic global Torelli theorem for isogenous K3 surfaces. Advances in Mathematics. 2021;383: 107674.
Fu, L., & Vial, C. (2021). A motivic global Torelli theorem for isogenous K3 surfaces. Advances in Mathematics, 383, 107674. https://doi.org/10.1016/j.aim.2021.107674
Fu, L., and Vial, C. (2021). A motivic global Torelli theorem for isogenous K3 surfaces. Advances in Mathematics 383:107674.
Fu, L., & Vial, C., 2021. A motivic global Torelli theorem for isogenous K3 surfaces. Advances in Mathematics, 383: 107674.
L. Fu and C. Vial, “A motivic global Torelli theorem for isogenous K3 surfaces”, Advances in Mathematics, vol. 383, 2021, : 107674.
Fu, L., Vial, C.: A motivic global Torelli theorem for isogenous K3 surfaces. Advances in Mathematics. 383, : 107674 (2021).
Fu, Lie, and Vial, Charles. “A motivic global Torelli theorem for isogenous K3 surfaces”. Advances in Mathematics 383 (2021): 107674.

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