arXiv subjects

Claire Voisin

Publications and source records attributed to Claire Voisin.

At least 55 records · Page 3Linked to original sources

arXiv2013

Curves and cycles on K3 surfaces

The notion of constant cycle curves on K3 surfaces is introduced. These are curves that do not contribute to the Chow group of the ambient K3 surface. Rational curves are the most prominent examples. We show that constant cycle curves behave in some respects like rational curves. E.g. using Hodge theory one finds that in each linear system there are at most finitely many such curves of bounded order. Over finite fields, any curve is expected to be a constant cycle curve, whereas over number fields this does not hold. The relation to the Bloch--Beilinson conjectures for K3 surfaces over global fields is discussed.

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arXiv2013

Rational equivalence of 0-cycles on $K3$ surfaces and conjectures of Huybrechts and O'Grady

We give a new interpretation of O'Grady's filtration on the $CH_0$ group of a $K3$ surface. In particular, we get a new characterization of the canonical 0-cycles $kc_X$ : this is the only 0-cycle on $X$ whose orbit under rational equivalence is of dimension $k$. Using this, we extend results of Huybrechts and O'Grady concerning Chern classes of simple vector bundles on $K3$ surfaces.

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arXiv2013

Infinitesimal invariants for cycles modulo algebraic equivalence and 1-cycles on Jacobians

We construct an infinitesimal invariant for cycles in a family with cohomology class in the total space lying in a given level of the Leray filtration. This infinitesimal invariant detects cycles modulo algebraic equivalence in the fibers. We apply this construction to the Ikeda family, which gives optimal results for the Beauville decomposition of the 1-cycle of a very general plane curve in its Jacobian.

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arXiv2013

Bloch's conjecture for Catanese and Barlow surfaces

Catanese surfaces are regular surfaces of general type with $p_g=0$. They specialize to double covers of Barlow surfaces. We prove that the $CH_0$ group of a Catanese surface is equal to $\mathbb{Z}$, which implies the same result for the Barlow surfaces. We will also give a conditional application (more precisely, assuming the variational Hodge conjecture) of the same method to the Chow motive of low degree $K3$ surfaces.

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arXiv2012

Symplectic involutions of K3 surfaces act trivially on CH_0

Symplectic involutions of a K3 surface are those involutions which leave the holomorphic 2-form invariant. We show, as predicted by Bloch's conjecture, that they act trivially on the CH_0 group of the K3 surface. This was recently proved by Huybrechts and Kemeny for one of the three types of symplectic involutions.

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arXiv2012

Remarks on curve classes on rationally connected varieties

We study for rationally connected varieties $X$ the group of degree 2 integral homology classes on $X$ modulo those which are algebraic. We show that the Tate conjecture for divisor classes on surfaces defined over finite fields implies that this group is trivial for any rationally connected variety.

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arXiv2012

Approximately rationally or elliptically connected varieties

We discuss a possible approach to the study of the vanishing of the Kobayashi pseudometric of a projective variety X, using chains of rational or elliptic curves contained in an arbitrarily small neighborhood of X in projective space for the Euclidean topology.

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arXiv2011

Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces

The decomposition theorem for smooth projective morphisms $π:\mathcal{X}\rightarrow B$ says that $Rπ_*\mathbb{Q}$ decomposes as $\oplus R^iπ_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a decomposition compatible with cup-product, even after restriction to Zariski dense open sets of $B$. We prove however that this is always possible for families of $K3$ surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author on the Chow ring of $K3$ surfaces $S$. We give two proofs of this result, the second one involving a certain decomposition of the small diagonal in $S^3$ also proved by Beauville and the author}. We prove an analogue of such a decomposition of the small diagonal in $X^3$ for Calabi-Yau hypersurfaces $X$ in $\mathbb{P}^n$, which in turn provides strong restrictions on their Chow ring.

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arXiv2011

Cohomologie non ramifiée et conjecture de Hodge entière

Building upon the Bloch-Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of the second-named author implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Ojanguren and the first named author implies that the integral Hodge conjecture in degree 4 fails for unirational varieties of dimension at least 6. For certain classes of threefolds fibered over a curve, we establish a relation between the integral Hodge conjecture and the computation of the index of the generic fibre. En nous appuyant sur la conjecture de Bloch-Kato en K-théorie de Milnor, nous établissons un lien général entre le défaut de la conjecture de Hodge entière pour la cohomologie de degré 4 et le troisième groupe de cohomologie non ramifié à coefficients Q/Z. Ceci permet de montrer que sur un solide uniréglé le troisième groupe de cohomologie non ramifié à coefficients Q/Z s'annule, ce que la K-théorie algébrique ne permet d'obtenir que dans certains cas. Ceci permet à l'inverse de déduire d'exemples ayant leur source en K-théorie que la conjecture de Hodge entière pour la cohomologie de degré 4 peut être en défaut pour les variétés rationnellement connexes. Pour certaines familles à un paramètre de surfaces, on établit un lien entre la conjecture de Hodge entière et l'indice de la fibre générique.

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arXiv2011

Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal

Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Markoushevich. We then conclude that the Hodge conjecture holds for degree 4 integral Hodge classes on fibrations into cubic threefolds over curves, with restriction on singular fibers.

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arXiv2011

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete intersection $Y$ of ample hypersurfaces in $X$ has geometric coniveau $\geq c$, then the Chow groups of cycles of dimension $\leq c-1$ of $Y$ are trivial. The generalized Bloch conjecture for $Y$ is this statement with "geometric coniveau" replaced by "Hodge coniveau".

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arXiv2010

Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal

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arXiv2010

Pseudoeffective and nef classes on abelian varieties

The cones of divisors and curves defined by various positivity conditions on a smooth projective variety have been the subject of a great deal of work in algebraic geometry, and by now they are quite well understood. However the analogous cones for cycles of higher codimension and dimension have started to come into focus only recently. The purpose of this paper is to explore some of the phenomena that can occur by working out the picture fairly completely in a couple of simple but non-trivial cases. Specifically, we study cycles of arbitrary codimension on the self-product of an elliptic curve with complex multiplication, as well as two dimensional cycles on the product of a very general abelian surface with itself. Already one finds various non-classical behavior, for instance nef cycles that fail to be pseudoeffective: this answers a question raised in 1964 by Grothendieck in correspondence with Mumford. We also propose a substantial number of open problems for further investigation.

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arXiv2009

Hyper-Kähler fourfolds and Grassmann geometry

We construct a new 20-dimensional family of algebraic hyper-Kaehler fourfolds and prove that they are deformation-equivalent to the second punctual Hilbert scheme of a K3 surface of degree 22.

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arXiv2008

Coniveau 2 complete intersections and effective cones

The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this effectiveness conjecture.

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arXiv2008

Rationally connected $3$-folds and symplectic geometry

We study the following question, asked to us By Pandharipande and Starr: Let $X$ be a rationally connected $3$-fold, and $Y$ be a compact Kaehler $3$-fold symplectically equivalent to it. Is $Y$ rationally connected? We show that the answer is positive if $X$ is Fano or $b_2(X)\leq2$.

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arXiv2007

Hodge structures on cohomology algebras and geometry

We study restrictions on cohomology algebras of Kaehler compact manifolds, not depending on the h^{p,q} numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the Lefschetz property but not having the cohomology algebra of a compact Kaehler manifold. We also prove the stability of these restrictions under products.

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arXiv2007

Potential density of rational points on the variety of lines of a cubic fourfold

We prove the potential density of rational points on the variety of lines of a sufficiently general cubic fourfold defined over a number field, where ``sufficiently general'' means that a condition of Terasoma type is satisfied. These varieties have trivial canonical bundle and have geometric Picard group equal to $\mathbb{Z}$.

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