On complete intersections in varieties with finite-dimensional motive

Laterveer, Robert; Nagel, Jan; Peters, Chris

doi:10.1093/qmath/hay038

Cited by 14 publications

(13 citation statements)

References 36 publications

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“…This observation can indeed be put to use as is shown in the examples of ; the present note sets up the proper theoretical framework.…”

Section: Introductionmentioning

confidence: 68%

On a motivic interpretation of primitive, variable and fixed cohomology

Peters

2018

Mathematische Nachrichten

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If M is a smooth projective variety whose motive is Kimura finite‐dimensional and for which the standard Lefschetz Conjecture B holds, then the motive of M splits off a primitive motive whose cohomology is the primitive cohomology. Under the same hypotheses on M, let X be a smooth complete intersection of ample divisors within M. Then the motive of X is the sum of a variable and a fixed motive inducing the corresponding splitting in cohomology. I also give variants with group actions.

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“…This observation can indeed be put to use as is shown in the examples of ; the present note sets up the proper theoretical framework.…”

Section: Introductionmentioning

confidence: 68%

On a motivic interpretation of primitive, variable and fixed cohomology

Peters

2018

Mathematische Nachrichten

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“…This is (essentially) the argument of [58, Proposition 1.6] (cf. also [36,Proposition 5.1] for a similar statement). Since the assumptions are different, we provide a stand-alone proof.…”

Section: 4mentioning

confidence: 81%

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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“…In case M = M is projective, this is essentially Voisin's 'spread' result [64, Proposition 1.6] (cf. also [35,Proposition 5.1] for a reformulation of Voisin's result). We give a proof which is somewhat different from [64].…”

Section: A Franchetta-type Resultsmentioning

confidence: 99%

“…Using the Hilbert schemes argument of [62, Proposition 3.7] (cf. also [35,Proposition A.1] for the precise form used here), the γ j b , V j b , W j b exist relatively, that is, one can find subvarieties V j , W j ⊂ X with codimV j + codimW j = dim X b − 1, and cycles γ j supported on…”

Section: Fano Varieties With Finite-dimensional Motivementioning

confidence: 99%

Motives and the Pfaffian–Grassmannian equivalence

Laterveer

2021

J. London Math. Soc.

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We consider the Pfaffian-Grassmannian equivalence from the motivic point of view. The main result is that under certain numerical conditions, both sides of the equivalence are related on the level of Chow motives. The consequences include a verification of Orlov's conjecture for Borisov's Calabi-Yau threefolds, and verifications of Kimura's finite-dimensionality conjecture, Voevodsky's smash conjecture and the Hodge conjecture for certain linear sections of Grassmannians. We also obtain new examples of Fano varieties with infinite-dimensional Griffiths group.

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On complete intersections in varieties with finite-dimensional motive

Cited by 14 publications

References 36 publications

On a motivic interpretation of primitive, variable and fixed cohomology

On a motivic interpretation of primitive, variable and fixed cohomology

Algebraic cycles and very special cubic fourfolds

Motives and the Pfaffian–Grassmannian equivalence

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