On the finite dimensionality of a K3 surface

Pedrini, Claudio

doi:10.1007/s00229-011-0483-4

Cited by 41 publications

(32 citation statements)

References 17 publications

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“…Remark 2.5. The following varieties have finite-dimensional motive: abelian varieties, varieties dominated by products of curves [32], K3 surfaces with Picard number 19 or 20 [38], surfaces not of general type with p g = 0 [23, Theorem 2.11], certain surfaces of general type with p g = 0 [23], [40], [55], Hilbert schemes of surfaces known to have finite-dimensional motive [13], generalized Kummer varieties [57, Remark 2.9(ii)], [21], threefolds with nef tangent bundle [27], [47,Example 3.16], fourfolds with nef tangent bundle [28], log-homogeneous varieties in the sense of [12] (this follows from [28,Theorem 4.4]), certain threefolds of general type [49,Section 8], varieties of dimension ≤ 3 rationally dominated by products of curves [47,Example 3.15], varieties X with A Clearly, if Y has finite-dimensional motive then also X = Y /G has finite-dimensional motive. The nilpotence theorem extends to this set-up: Proposition 2.8.…”

Section: Preliminarymentioning

confidence: 99%

Algebraic cycles on a very special EPW sextic

Laterveer

2018

Rend. Sem. Mat. Univ. Padova

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Section: Preliminarymentioning

confidence: 99%

Algebraic cycles on a very special EPW sextic

Laterveer

2018

Rend. Sem. Mat. Univ. Padova

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“…This means that any of the two associated K3 surfaces S has dim H 2 tr (S) ≤ 3, and so the Picard number of S is ≥ 19. Since K3 surfaces of Picard number 19 or 20 are either Kummer surfaces, or are related to Kummer surfaces via a Shioda-Inose structure, they have finite-dimensional motive [41]. In view of Theorem 4.8, this implies that X also has finite-dimensional motive.…”

Section: 5mentioning

confidence: 98%

Algebraic cycles and very special cubic fourfolds

Laterveer

2019

Indagationes Mathematicae

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This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial 1-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-Künneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds. 1 2 ROBERT LATERVEERCorollary (=Corollary 5.3). Let X be any Verra fourfold. The subalgebrainjects into cohomology via the cycle class map. (Here, c j (T X ) are the Chow-theoretic Chern classes.) (This can be generalized to self-products X m , cf. Corollary 5.6.) The construction of a multiplicative Chow-Künneth decomposition relies once more on the spread argument, by first showing (Theorem 4.8) that any Verra fourfold can be motivically related to a genus 2 K3 surface. This close link to K3 surfaces also has the following consequence:Corollary (=Corollary 5.9). Let X, X ′ be two Verra fourfolds that are isogenous (i.e., there exists an isomorphism of Q-vector spaces H 4 (X, Q) ∼ = H 4 (X ′ , Q) compatible with the Hodge structures and with cup product). Then there is an isomorphism of Chow motivesFurther consequences concern a multiplicative decomposition in the derived category (Corollary 5.12), the generalized Hodge conjecture for self-products, and finite-dimensionality of the motive of certain special Verra fourfolds (Corollary 5.14).Conventions. In this note, the word variety will refer to a reduced irreducible scheme of finite type over C. For any n-dimensional variety X, we will write A i (X) for the Chow group of dimension i cycles on X with Q-coefficients, and A j (X) for the operational Chow cohomology of [17] with Q-coefficients (this can be identified with A n−j (X) for smooth X).The notations A j hom (X) and A j AJ (X) will indicate the subgroups of homologically trivial (resp. Abel-Jacobi trivial) cycles.For a morphism between smooth varieties f : X → Y , we will write Γ f ∈ A * (X × Y ) for the graph of f , and t Γ f ∈ A * (Y × X) for the transpose correspondence.The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in [43], [39]) will be denoted M rat .We will write H * (X) = H * (X, Q) for singular cohomology with Q-coefficients. VERRA FOURFOLDSDefinition 2.1 ([50]). A Verra fourfold is a double cover of P 2 × P 2 branched along a smooth divisor of bidegree (2, 2). Remark 2.2. The moduli space of Verra fourfolds is 19-dimensional [25, Section 0.3]. Lemma 2.3. A Verra fourfold X is a Fano variety (i.e., the canonical bundle is anti-ample). In particular, A 4 (X) = Q and the Hodge conjecture is true for X. Proof. The canonical bundle formula shows that X is Fano. Fano varieties are rationally connected [13], [30], and so A 4 (X) = Q. The Hodge conjecture is known for fourfolds with trivial Chow group of 0-cycles [10].

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“…which is a projection onto a direct summand. Since q(S l ) = 0 the motive t 2 (S l ) splits as follows, see [Ped,Prop. 1]…”

Section: The Motive Of a Cubic Fourfoldmentioning

confidence: 99%

“…such that the induced map on Chow groups where, as in [Ped,Prop 1], t 2 (S) − is the direct summand of t 2 (S) where the involution σ acts as −1. The action of σ on t 2 (S) is defined via the homomorphism…”

Section: The Motive Of a Cubic Fourfoldmentioning

confidence: 99%

The transcendental motive of a cubic fourfold

Bolognesi

Pedrini

2020

Journal of Pure and Applied Algebra

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In this note we introduce the transcendental part t(X) of the motive of a cubic fourfold X and prove that it is isomorphic to the (twisted) transcendental part h tr 2 (F (X)) in a suitable Chow-Künneth decomposition for the motive of the Fano variety of lines F (X). Then we prove that t(X) is isomorphic to the Prym motive associated to the surface S l ⊂ F (X) of lines meeting a general line l. If X is a special cubic fourfold in the sense of Hodge theory, and F (X) ≃ S [2] , with S a K3, then we show that t(X) ≃ t 2 (S)(1), where t 2 (S) is the transcendental motive. Therefore the motive h(X) is finite dimensional if and only if S has a finite dimensional motive. If X is very general then t(X) cannot be isomorphic to the (twisted) transcendental motive of a surface. We relate the existence of an isomorphism t(X) ≃ t 2 (S)(1) to conjectures by Hassett and Kuznetsov on the rationality of a special cubic fourfold. Finally we consider the case of cubic fourfolds X admitting a fibration over P 2 , whose fibers are either quadrics or del Pezzo surfaces of degree 6, and prove the isomorphism t 2 (S)(1) ≃ t(X), with S a K3 surface.

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On the finite dimensionality of a K3 surface

Cited by 41 publications

References 17 publications

Algebraic cycles on a very special EPW sextic

Algebraic cycles on a very special EPW sextic

Algebraic cycles and very special cubic fourfolds

The transcendental motive of a cubic fourfold

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