Motives of derived equivalent K3 surfaces

Abstract

Abstract. We observe that derived equivalent K3 surfaces have isomorphic Chow motives.There are many examples of non-isomorphic K3 surfaces S and S ′ (over a field k) with equivalent derived categories D b (S) ≃ D b (S ′ ) of coherent sheaves. It is known that in this case S ′ is isomorphic to a moduli space of slope-stable bundles on S and vice versa, cf. [10,20,22]. However, the precise relation between S and S ′ still eludes us and many basic questions are hard to answer. For example, one could ask whether S ′ contains infinitely many rational curves, expected for all K3 surfaces, if this is known already for S. Or, if k is a number field, does potential density for S imply potential density for S ′ ?As a direct geometric approach to these questions seems difficult, one may wonder whether S and S ′ can be compared at least motivically. This could mean various things, e.g. one could study their classes [S] and [S ′ ] in the Grothendieck ring of varieties K 0 (Var(k)) or their associated motives h(S) and h(S ′ ) in the category of Chow motives Mot(k). In the recent articles [7,14,16], examples in degree 8 and 12 have been studied for whichThe question whether the Chow motives h(S) and h(S ′ ) in Mot(k) are isomorphic was first addressed and answered affirmatively in special cases in [23]. Assuming finite-dimensionality of the motives the question was settled in [3].In this short note we point out that the available techniques in the theory of motives are enough to show that two derived equivalent K3 surfaces have indeed isomorphic Chow motives.Theorem 0.1. Let S and S ′ be K3 surfaces over an algebraically closed field k. Assume that there exists an exact k-linear equivalence D b (S) ≃ D b (S ′ ) between their bounded derived categories of coherent sheaves. Then there is an isomorphismin the category of Chow motives Mot(k).The assumption on the field k can be weakened, it suffices to assume that ρ(S) = ρ(Sk). We had originally expected that the invariance of the Beauville-Voisin ring as proved in [11,12] would be central to the argument However, it turns to out to have no bearing on the problem, but it implies that a distinguished decomposition of the motives in their algebraic and transcendental parts is preserved under derived equivalence.Acknowledgements: I am very grateful to Charles Vial for answering my questions and to him and Andrey Soldatenkov for comments on a first version and helpful suggestions.

Motives of isogenous K3 surfaces

“…In this section we show the following result which generalizes [13] from the case of fine moduli spaces of (complexes of) untwisted sheaves to the case of coarse(!) moduli spaces of (complexes of) twisted(!)…”

“…Hence, (2.2) v 2 (E) * : h 2 tr (S) / / h 2 tr (S ′ ) induces isomorphisms between the Chow groups of the motives. As this holds true after any base change, a version of Manin's identity principle then implies that (2.2) is an isomorphism, for details see [13].…”

“…2.2. The rest of the argument to prove Theorem 2.1 can be copied from [13]. Here is a rough outline: First, the motive of a K3 surface is decomposed into its algebraic and its transcendental part h(S) ≃ h 2 tr (S) ⊕ h alg (S), where h alg (S) ≃ L 0 ⊕ L ⊕ρ(S) ⊕ L 2 and the transcendental part h 2 tr (S), introduced in [15], has the property that CH * (h 2 tr (S)) = CH 2 (S) 0 ⊗ Q.…”

“…Combining Theorem 0.1 with the arguments in [13] generalized to the twisted case, one deduces a motivic interpretation of the notion of isogeneous K3 surfaces:…”

Stringy $E$-functions of Pfaffian�Grassmannian double mirrors

“…This, indeed, has been verified independently in [BC09] and [Kuz06] in the n = 7 case, thus providing a rare example of non-birational derived equivalent manifolds. For a sample of related problems in dimension 2; see [KS16,Huy18,ADM16].…”