arXiv subjects

Claire Voisin

Publications and source records attributed to Claire Voisin.

At least 37 records · Page 2Linked to original sources

arXiv2020

Hilbert squares of K3 surfaces and Debarre-Voisin varieties

The Debarre-Voisin hyperkähler fourfolds are built from alternating $3$-forms on a $10$-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate, in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general $1$-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface.

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arXiv2018

Triangle varieties and surface decomposition of hyper-Kähler manifolds

We introduce and study the notion of "surface decomposable" variety, and discuss the possibility that any projective hyper-Kähler manifold is surface decomposable, which would produce new evidence for Beauville's weak splitting conjecture. We show that surface decomposability relates to the Beauville-Fujiki relation, a constraint on the cohomology ring of the variety, and that general varieties with $h^{2,0}\not=0$ are not surface decomposable. We also formalize the notion of triangle variety that is useful to produce surface decomposition. We show the existence of these geometric structures on most explicitly constructed classes of projective hyper-Kähler manifolds of Picard number $1$.

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arXiv2018

Remarks on degenerations of hyper-Kähler manifolds

Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-Kähler setting, we can then deduce a finiteness statement for monodromy acting on $H^2$, once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-Kähler manifolds, we prove that a finite monodromy projective degeneration of hyper-Kähler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain geometric constructions of hyper-Kähler manifolds (e.g. Debarre--Voisin or Laza--Saccà--Voisin). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-Kähler manifolds.

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arXiv2018

Chow rings and gonality of general abelian varieties

We study the (covering) gonality of abelian varieties and their orbits of zero-cycles for rational equivalence. We show that any orbit for rational equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building on the work of Pirola, we show that very general abelian varieties of dimension $g$ have covering gonality $k\geq f(g)$ where $f(g)$ grows like ${\rm log}\,g$. This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B. Ullery. We also obtain results on the Chow ring of very general abelian varieties, eg. if $g\geq 2k-1$, for any divisor $D\in {\rm Pic}^0(A)$, $D^k$ is not a torsion cycle.

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arXiv2018

Torsion points of sections of Lagrangian torus fibrations and the Chow ring of hyper-Kähler manifolds

Let $ϕ:X\rightarrow B$ be a Lagrangian fibration on a projective irreducible hyper-Kähler manifold of dimension $\leq8$. Let $M\in {\rm Pic}\,X$ be a line bundle whose restriction to the general fiber $X_b$ of $ϕ$ is topologically trivial. We prove that if the fibration has maximal variation or is isotrivial, the set of points $b$ such that the restriction $M_{\mid X_b}$ is torsion is dense in $B$. We give an application to the Chow ring of $X$. We prove a similar result for elliptic fibrations which gives a toy model for the argument.

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arXiv2017

Anticanonical divisors and curve classes on Fano manifolds

It is well known that the Hodge conjecture with rational coefficients holds for degree 2n-2 classes on complex projective n-folds. In this paper we study the more precise question if on a rationally connected complex projective n-fold the integral Hodge classes of degree 2n-2 are generated over $\mathbb Z$ by classes of curves. We combine techniques from the theory of singularities of pairs on the one hand and infinitesimal variation of Hodge structures on the other hand to give an affirmative answer to this question for a large class of manifolds including Fano fourfolds. In the last case, one step in the proof is the following result of independent interest: There exist anticanonical divisors with isolated canonical singularities on a smooth Fano fourfold.

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arXiv2017

Segre classes of tautological bundles on Hilbert schemes of surfaces

We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande on top Segre classes of the tautological bundles on Hilbert schemes of $K3$ surfaces equipped with a line bundle. We then turn to the blow-up of $K3$ surface at one point and establish vanishing results for the corresponding top Segre classes in a certain range. This determines, at least theoretically, all top Segre classes of tautological bundles for any pair $(Σ,H),\,H\in {\rm Pic}\,Σ$.

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arXiv2017

Syzygies of Prym and paracanonical curves of genus 8

By analogy with Green's Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. The Prym-Green Conjecture is known to hold in odd genus for almost all levels. Probabilistic arguments strongly suggested that the conjecture might fail for level 2 and genus 8 or 16. In this paper, we present three geometric proofs of the surprising failure of the Prym-Green Conjecture in genus 8, hoping that the methods introduced here will shed light on all the exceptions to the Prym-Green Conjecture for genera with high divisibility by 2.

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arXiv2017

Variety of power sums and divisors in the moduli space of cubic fourfolds

We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics form a divisor in the moduli space of cubic fourfolds and that this divisor is not a Noether-Lefschetz divisor. We use this result to prove that there is no nontrivial Hodge correspondence between a very general cubic and its VSP.

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arXiv2017

A hyper-Kähler compactification of the Intermediate Jacobian fibration associated to a cubic fourfold

For a general cubic fourfold, it was observed by Donagi and Markman that the relative intermediate Jacobian fibration associated to the family of its hyperplane sections carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification of the intermediate Jacobian fibration giving a ten-dimensional compact hyper-Kähler manifold, which we then show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Markushevich.

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arXiv2016

Hyper-Kaehler compactification of the intermediate Jacobian fibration of a cubic fourfold : the twisted case

The starting point of this note is our recent paper with Laza and Saccà on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-Kähler manifolds.

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arXiv2015

(Stable) rationality is not deformation invariant

We prove the existence of a family $\mathcal{X}\rightarrow B$ of smooth projective fourfolds, such that the very general fiber $\mathcal{X}_t$ is not stably rational (a fortiori not rational), but some special fibers $\mathcal{X}_t$ are rational (a fortiori stably rational).

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arXiv2015

On the universal $CH_0$ group of cubic hypersurfaces

We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic hypersurface, or equivalently, the universal triviality of its ${\rm CH}_0$-group. We prove that for odd dimensional cubic hypersurfaces or for cubic fourfolds, this is equivalent to the existence of a cohomological decomposition of the diagonal, and we translate geometrically this last condition. For cubic threefolds $X$, this turns out to be equivalent to the algebraicity of the minimal class $θ^4/4!$ of the intermediate Jacobian $J(X)$. In dimension $4$, we show that a special cubic fourfold with discriminant not divisible by $4$ has universally trivial ${\rm CH}_0$ group.

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arXiv2015

On a conjecture of Matsushita

We prove a weak version of a conjecture of Matsushita saying that for a Lagrangian fibration on a hyper-Kaehler manifold $X$, the moduli map for the fibers is either generically of maximal rank or constant. Assuming the base is smooth and $b_2(X)-ρ(X)\geq5$, we prove the conjecture for a very general deformation of $X$ with constant Néron-Severi group.

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arXiv2015

Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties

This paper proposes a conjectural picture for the structure of the Chow ring of a (projective) hyper-Kähler variety, and the construction of a Beauville decomposition, with emphasis on the Chow group of $0$-cycles, which is endowed with a natural filtration of Brill-Noether type. Some of the conjectures are proved in the case of Hilbert schemes of K3 surfaces and Fano varieties of lines of cubic fourfolds.

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arXiv2014

Some new results on modified diagonals

O'Grady studied recently $m$-th modified diagonals for a smooth projective variety, generalizing the Gross-Schoen modified small diagonal. These cycles $Γ^m(X,a)$ depend on a choice of reference point $a\in X$ (or more generally a degree $1$ zero-cycle). We prove that for any $X,a$, the cycle $Γ^m(X,a)$ vanishes for large $m$. We also prove the following conjecture of O'Grady: if $X$ is a double cover of $Y$ and $Γ^m(Y,a)$ vanishes (where $a$ belongs to the branch locus), then $Γ^{2m-1}(X,a)$ vanishes, and we provide a generalization to higher degree finite covers. We finally prove the vanishing $Γ^{n+1}(X,o_X)=0$ when $X=S^{[m]}$, $S$ a $K3$ surface, and $n=2m$, which was conjectured by O'Grady and proved by him for $m=2,3$.

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arXiv2014

Unirational threefolds with no universal codimension 2 cycle

We prove that the general quartic double solid with $k\leq 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal ${\rm CH}_0$ group. The same holds if we replace in this statement "Chow theoretic" by "cohomological". In particular, it is not stably rational. We also prove that the general quartic double solid with seven nodes does not admit a universal codimension 2 cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibres of its Jacobian by a family of 1-cycles. This implies that its third unramified cohomology group is not universally trivial.

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arXiv2014

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

We prove an unconditional (but slightly weakened) version of the main result of our earlier paper with the same title, which was, starting from dimension $4$, conditional to the Lefschetz standard conjecture. Let $X$ be a variety with trivial Chow groups, (i.e. the cycle class map to cohomology is injective on $CH(X)_\mathbb{Q}$). We prove that if the cohomology of a general very ample hypersurface $Y$ in $X$ is ``parameterized by cycles of dimension $c$'', then the Chow groups $CH_{i}(Y)_\mathbb{Q}$ are trivial for $i\leq c-1$.

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