On the Transcendental Part of the Motive of a Surface

Kahn, Bruno; Murre, Jacob P.; Pedrini, Claudio

doi:10.1017/cbo9781107325968.009

Cited by 74 publications

(106 citation statements)

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“…Following [15], the degree two part h 2 (S) can be further decomposed into its algebraic and transcendental parts…”

Section: Instead Of Producing An Isomorphism H(s)mentioning

confidence: 99%

Motives of derived equivalent K3 surfaces

Huybrechts

2017

Abh. Math. Semin. Univ. Hambg.

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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.

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“…Following [15], the degree two part h 2 (S) can be further decomposed into its algebraic and transcendental parts…”

Section: Instead Of Producing An Isomorphism H(s)mentioning

confidence: 99%

Motives of derived equivalent K3 surfaces

Huybrechts

2017

Abh. Math. Semin. Univ. Hambg.

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“…This result is hardly surprising: since the appearance of Kimura's landmark paper [21], where it is shown that finite-dimensionality implies the Bloch conjecture for surfaces, there have been a great many results attesting to the usefulness of finite-dimensionality [17], [1], [12], [18], [19], [10], [11], [34], [39], [24]. The present note is but one more instance of this general principle, illustrating how nicely finitedimensionality allows to bridge the abyss separating homological equivalence from rational equivalence.…”

Section: Then (P2) Implies (P1)mentioning

confidence: 99%

“…When X is a surface, (X, Π 2,0 ) is the transcendental part denoted t 2 (X) (and studied in detail) in [19].…”

Section: Remarkmentioning

confidence: 99%

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On a multiplicative version of Bloch’s conjecture

Laterveer

2016

Beitr Algebra Geom

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A theorem of Esnault, Srinivas and Viehweg asserts that if the Chow group of 0-cycles of a smooth complete complex variety decomposes, then the top-degree coherent cohomology group decomposes similarly. In this note, we prove that (a weak version of) the converse holds for varieties of dimension at most 5 that have finite-dimensional motive and satisfy the Lefschetz standard conjecture. The proof is based on Vial's construction of a refined Chow-Künneth decomposition for these varieties.

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“…Moreover, in characteristic 0, Kahn, Murre and Pedrini [11,Theorem 14.7.3] show that X also has a Lefschetz decomposition. Since the class of varieties having finite-dimensional motives includes abelian varieties and is closed under quotients, the above constructions of Acknowledgments.…”

Section: Introductionmentioning

confidence: 99%