arXiv subjects

Claire Voisin

Publications and source records attributed to Claire Voisin.

At least 19 recordsLinked to original sources

arXiv2026

On the Chow ring of very general abelian varieties and a question of Pirola

We prove that for a very general abelian variety of dimension $\geq 4$, a divisor $D\in {\rm CH}^1(A)$ that satisfies $D^2=0$ in ${\rm CH}^2(A)$ is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which states that any rational section of the Kummer fibration $K=J/\pm {\rm Id}\rightarrow \mathcal{M}_4$, where $J\rightarrow \mathcal{M}_4 $ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.

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arXiv2026

Unboundedness of zero-cycles on higher dimensional Fano manifolds

We show that, unlike del Pezzo surfaces, higher dimensional Fano manifolds do not satisfy in general boundedness properties for their ${\rm CH}_0$ group of $0$-cycles. For example, for quartic threefolds having a point of odd degree, there is no ``Coray type" uperbound on the minimal odd degrees of points. Also, the ${\rm CH}_0$-group of Fano hypersurfaces can be ``unbounded'' (a notion which is related to infinite dimensionality in the sense of Mumford), meaning that there is no integer $N$ such that $0$-cycles of degree at least $N$ are effective.

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arXiv2026

Rank 2 vector bundles and degrees of points of del Pezzo surfaces

We study points and 0-cycles on del Pezzo surfaces defined over a field K of characteristic 0, with emphasis on cubic surfaces. We prove that a cubic surface that admits a point defined over a field extension of K of degree coprime to 3 either has a K-point or has a point defined over a field extension of degree 4. This improves a result of Coray (who allowed also field extensions of degree 10). We also prove that 0-cycles of degree at least 18 on a cubic surface are effective and get similar results for degree 2 and degree 1 del Pezzo surfaces, improving results of Colliot-Thélène. In a different direction, we prove that the third symmetric product of a cubic hypersurface of dimension at least 2 is unirational over any field, and that in dimension 2 or 3, it is not stably rational in general.

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arXiv2025

Varieties with representable CH_0-group and a question of Colliot-Thélène

We continue our investigation of the geometry of the Albanese morphism on 0-cycles. We provide an example of a smooth projective variety with representable CH_0-group but with no universal 0-cycle, which answers a question asked by Colliot-Thélène. Our construction relies on a counterexample to the integral Hodge conjecture provided by Benoist and Ottem.

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arXiv2024

Universally defined cycles I

We introduce and study the notion of universally defined cycles of smooth varieties of dimension $d$, and prove that they are given by polynomials in the Chern classes. A similar result is proved for universally defined cycles on products of smooth varieties. We also state a conjectural explicit form for universally defined cycles on powers of smooth varieties, and provide some steps towards establishing it.

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arXiv2024

On the smoothability problem with rational coefficients

We consider the problem of smoothing algebraic cycles with rational coefficients on smooth projective complex varieties up to homological equivalence. We show that a solution to this problem would be incompatible with the validity of the Hartshorne conjecture on complete intersections in projective space. We also solve unconditionally a symplectic variant of this problem.

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arXiv2024

Flat pushforwards of Chern classes and the smoothability of cycles below the middle dimension

We prove in this paper the smoothability of cycles modulo rational equivalence in the Whitney range, that is, when the dimension is strictly smaller than the codimension. We introduce and study the class of cycles obtained as ``flat pushforwards of Chern classes" (or equivalently, flat pushforwards of products of divisors) and prove that they are smoothable in the Whitney range. Our main result is that all cycles (of any dimension) on a smooth projective variety are flat pushforwards of Chern classes. In the case of abelian varieties, one can even restrict to smooth pushforwards of Chern classes.

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arXiv2024

On Chern classes of Lagrangian fibered hyper-Kähler manifolds

We study the rank stratification for the differential of a Lagrangian fibration over a smooth basis. We also introduce and study the notion of Lagrangian morphism of vector bundles. As a consequence, we prove some of the vanishing, in the Chow groups of a Lagrangian fibered hyper-Kähler variety $X$, of certain polynomials in the Chern classes of $X$ and the Lagrangian divisor, predicted by the Beauville-Voisin conjecture. Under some natural assumptions on the dimensions of the rank strata, we also establish nonnegativity and positivity results for Chern classes.

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arXiv2023

Computing Riemann-Roch polynomials and classifying hyper-Kähler fourfolds

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3$^{[2]}$ deformation type. This proves in particular a conjecture of O'Grady stating that hyper-Kähler fourfolds of K3$^{[2]}$ numerical type are of K3$^{[2]}$ deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts-Riemann-Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3$^{[2]}$ hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.

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arXiv2023

Cycle classes on abelian varieties and the geometry of the Abel-Jacobi map

We discuss two properties of an abelian variety, namely, being a direct summand in a product of Jacobians and the weaker property of being "split". We relate the first property to the integral Hodge conjecture for curve classes on abelian varieties. We also relate both properties to the existence problem for universal zero-cycles on Brauer-Severi varieties over abelian varieties. A similar relation is established for the existence problem of a universal codimension 2 cycle on a cubic threefold.

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arXiv2023

Geometric representability of 1-cycles on rationally connected threefolds

We prove that for any rationally connected threefold $X$, there exists a smooth projective surface $S$ and a family of $1$-cycles on $X$ parameterized by $S$, inducing an Abel-Jacobi isomorphism ${\rm Alb}(S)\cong J^3(X)$. This statement was previously known for some classes of smooth Fano threefolds.

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arXiv2022

On fibrations and measures of irrationality of hyper-Kähler manifolds

We prove some results on the fibers and images of rational maps from a hyper-Kähler manifold. We study in particular the minimal genus of fibers of a fibration into curves. The last section of this paper is devoted to the study of the rational map defined by a linear system on a hyper-Kähler fourfold satisfying numerical conditions similar to those considered by O'Grady in his study of fourfolds numerically equivalent to $K3^{[2]}$. We extend his results to this more general context.

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arXiv2022

On the Lefschetz standard conjecture for Lagrangian covered hyper-Kähler varieties

We investigate the Lefschetz standard conjecture for degree $2$ cohomology of hyper-Kähler manifolds admitting a covering by Lagrangian subvarieties. In the case of a Lagrangian fibration, we show that the Lefschetz standard conjecture is implied by the SYZ conjecture characterizing classes of divisors associated with Lagrangian fibration. In dimension $4$, we consider the more general case of a Lagrangian covered fourfold $X$, and prove the Lefschetz standard conjecture in degree $2$, assuming $ρ(X)=1$ and $X$ is general in moduli. Finally we discuss various links between Lefschetz cycles and the study of the rational equivalence of points and Bloch-Beilinson type filtrations, giving a general interpretation of a recent intriguing result of Marian and Zhao.

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arXiv2021

Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds

We prove that the complex cobordism class of any hyper-Kähler manifold of dimension $2n$ is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of $K3$ surfaces. We also prove a similar result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles.

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arXiv2021

Footnotes to papers of O'Grady and Markman

In this paper, we first generalize to any hyper-Kähler manifold with nonzero third Betti number results proved by O'Grady for hyper-Kähler manifolds of generalized Kummer type. In the second part, we restrict to hyper-Kähler manifolds of generalized Kummer type and prove, using results of Markman, that their Kuga-Satake correspondence is algebraic.

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arXiv2021

On the coniveau of rationally connected threefolds

We prove that the integral cohomology modulo torsion of a rationally connected threefold comes from the integral cohomology of a smooth curve via the cylinder homomorphism associated to a family of $1$-cycles. Equivalently, it is of strong coniveau 1 in the sense of Benoist-Ottem.

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arXiv2021

Schiffer variations and the generic Torelli theorem for hypersurfaces

We show how to recover a general hypersurface in $\mathbb{P}^n$ of sufficiently large degree $d$ dividing $n+1$, from its finite order variation of Hodge structure. We also analyze the two other series of cases not covered by Donagi's generic Torelli theorem. Combined with Donagi's theorem, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.

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arXiv2021

Compact Kähler manifolds with no projective specialization

We show the existence of a compact Kähler manifold which does not fit in a proper flat family over an irreducible base with one projective (possibly singular) fiber. We also give a topological version of this statement. This strengthens our earlier counterexamples to the Kodaira algebraic approximation problem.

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