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In this paper, we first generalize to any hyper-Kahler manifold X with $$b_3(X)\not =0$$ results proved by O’Grady for hyper-Kahler manifolds of generalized Kummer type. In the second part, we restrict to hyper-Kahler manifolds of generalized Kummer type and prove, using results of Markman, that their Kuga–Satake correspondence is algebraic.

Enseignement Cours et séminaires – Variétés hyper-kählériennes Les variétés hyper-kählériennes sont les généralisations naturelles des surfaces K3. Comme pour les tores complexes, ces variétés existent naturellement dans le cadre kählérien compact mais celles qui sont projectives et relèvent donc de la géométrie algébrique sont denses dans l’espace de modules. Si l’on se restreint aux variétés hyper-kählériennes projectives, leur étude est liée (par l’étude des espaces de modules via l’applic...

The Debarre-Voisin hyperk\"ahler 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., Comment: 47 pages. Introduction rewritten and various corrections made throughout the article

International audience; Titre en français. Anneau de Chow et gonalité des variétés abéliennes générales Titre court Gonality of abelian varieties

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 $\theta^4/4!$ 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., Comment: Final version to appear in JEMS

Let $X$ be a general cubic $4$-fold. It was observed by Donagi and Markman that the relative intermediate Jacobian fibration $\mathcal{J}_U/U$ (with $U=(\mathbb{P}^5)^\vee\setminus X^\vee$) associated with the family of smooth hyperplane sections of $X$ carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification $\overline{\mathcal{J}}$ of $\mathcal{J}_U$ with the property that the holomorphic symplectic form on $\mathcal{J}_U$ extends to a holomorphic symplectic form on $\overline{\mathcal{J}}$. In particular, $\overline{\mathcal{J}}$ is a $10$-dimensional compact hyper-Kähler manifold, which we show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Kuznetsov and Markushevich.

We give a rather detailed account of cohomological and Chow-theoretic methods in the study of the stable version of the Lroth problem, which ask how to distinguish (stably) rational varieties from general unirational varieties. In particular, we study the notion of Chow or cohomological decomposition of the diagonal, which is a necessary criterion for stable rationality. Having better stability properties than the previously known obstructions under specialization with mildly singular central fibers, it has been very useful in the recent study of rationality questions.

Claire Voisin Première mathématicienne à rentrer au Collège de France, Claire Voisin est titulaire de la chaire de géométrie algébrique depuis juin 2016. Elle a reçu le 14 décembre 2016 la médaille d’or du CNRS, plus haute récompense scientifique française, et s’est vue décerner le 23 mai 2017 le prix Shaw, équivalent du « prix Nobel asiatique » en mathématiques. Un prix qu’elle partage avec János Kollár, professeur au département de mathématiques de l’université de Princeton (États-Unis). ...

We describe recent progress on rationality and stable rationality questions. We discuss the cohomological or Chow decomposition of the diagonal, a very strong stably birationally invariant property which controls many of the previously defined stable birational invariants. On the other hand, it behaves very well under specialization and desingularization of mild singularities.

Let ϕ : X → B be a Lagrangian fibration on a projective irreducible hyper-Kahler manifold. Let M ∈Pic X be a line bundle whose restriction to the general fiber Xb of ϕ is topologically trivial. We prove that if the fibration is isotrivial or has maximal variation and X is of dimension ≤ 8, 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, providing further evidence for Beauville’s weak splitting conjecture.

This chapter turns to a number of birational invariants that can be defined using Z-coefficients. It is devoted in part to the study of the groups Z²ⁱ(X) measuring the failure of the Hodge conjecture with integral coefficients. Some vanishing and nonvanishing results are presented, together with a comparison of the group Z⁴(X) with the so-called unramified cohomology of X with torsion coefficients. This chapter also considers various forms of the existence of an integral cohomological decomposition of the diagonal of a threefold X with trivial CH₀ group. It shows that an affirmative answer to this question is equivalent to the vanishing of numerous birational invariants of X.

This chapter discusses some crucial notions to the interplay between cohomology and Chow groups, and also to the consequences, for the topology of a family of smooth projective varieties, of statements concerning Chow groups of the general or very general fiber. It surveys the main ideas and results presented throughout this volume. First, the chapter discusses the decomposition of the diagonal and spread. It then explains the generalized Bloch conjecture, the converse to the generalized decomposition of the diagonal. Next, the chapter turns to the decomposition of the small diagonal and its application to the topology of families. Finally, the chapter discusses integral coefficients and birational invariants before providing a brief overview of the following chapters.

This chapter first describes how to compute the Hodge coniveau of complete intersections. It then explains a strategy to attack the generalized Hodge conjecture for complete intersections of coniveau 2. The guiding idea is that although the powerful method of the decomposition of the diagonal suggests that computing Chow groups of small dimension is the right way to solve the generalized Hodge conjecture, it might be better to invert the logic and try to compute the geometric coniveau directly. And indeed, this chapter culminates with the proof of the fact that for very general complete intersections, the generalized Hodge conjecture implies the generalized Bloch conjecture.

This chapter explains the method initiated by Bloch and Srinivas, which leads to statements of the following: if a smooth projective variety has trivial Chow groups of k-cycles homologous to 0 for k ≤ c − 1, then its transcendental cohomology has geometric coniveau ≤ c. This result is a vast generalization of Mumford's theorem. A major open problem is the converse of this result. It turns out that statements of this kind are a consequence of a general spreading principle for rational equivalence. Consider a smooth projective family X → B and a cycle Z → B, everything defined over C; then, if at the very general point b ∈ B, the restricted cycle Z𝒳b ⊂ X𝒳b is rationally equivalent to 0, there exist a dense Zariski open set U ⊂ B and an integer N such that NZsubscript U is rationally equivalent to 0 on Xsubscript U.

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was discovered that they have a canonical 0-cycle o of degree 1 with the property that the product of two divisors of S is a multiple of o in CH₀(S). This result would later be extended to Calabi–Yau hypersurfaces in projective space. The chapter also considers a decomposition in CH(X × X × X)ℚ of the small diagonal Δ‎ ⊂ X × X × X that was established for K3 surfaces, and is partially extended to Calabi–Yau hypersurfaces. Finally, the chapter uses this decomposition and the spreading principle to show that for families π‎ : X → B of smooth projective K3 surfaces, there is a decomposition isomorphism that is multiplicative over a nonempty Zariski dense open set of B.

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.

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 $(\Sigma,H),\,H\in {\rm Pic}\,\Sigma$., Comment: Shortly after this paper was written, I was informed by Marian-Oprea-Pandharipande and Szenes-Vergne independently that they were able to check that the Lehn function satisfies the vanishing properties stated in Corollary 6, thus completing the proof of Lehn's conjecture. Final version, to appear in Algebraic Geometry

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., 23 pages; to appear in EPIGA Vol 1, paper no. 7

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. For example, using Hodge theory one nds that in each linear system there are at most

This paper proposes a conjectural picture for the structure of the Chow ring CH*(X) of a (projective) hyper-Kahler variety X, that seems to emerge from the recent papers [9], [24], [25], [26], with emphasis on the Chow group CH0(X) of 0-cycles (in this paper, Chow groups will be taken with ℚ-coefficients). Our motivation is Beauville’s conjecture (see [5]) that for such an X, the Bloch-Beilinson filtration has a natural, multiplicative, splitting.

The starting point of this note is our recent paper with Laza and Sacc\`a 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-K\"ahler manifolds.

The diet of Cattle Egret chicks in the lower Soummam valley, Algeria, was investigated in 1998 and 2006, when 145 and 23 regurgitates were collected, respectively. In 1998, 1 698 individual prey items were identified. Insects formed the bulk of the diet of chicks: 94% in terms of frequency, 71% in terms of mass. Orthopterans were dominant, especially in terms of biomass (22% by frequency, 58% by mass), whilst Dipterans were well represented in numbers (59% by frequency). Vertebrates, despite their lesser frequency (3%), contributed 23% by mass of the diet. In 2006, 441 prey items were identified. The contribution of insects decreased to 79% by frequency and 15% by mass (mostly Orthopterans) but vertebrates made up a higher frequency: 20% of prey items and 85% by mass of the diet. The availability of prey items was not correlated to their occurrence in the diet, suggesting that the Cattle Egret is a selective predator, with adults selecting prey species for their chicks. A drought in the spring of 2006 inc...

The decomposition theorem for smooth projective morphisms $\pi:\mathcal{X}\rightarrow B$ says that $R\pi_*\mathbb{Q}$ decomposes as $\oplus R^i\pi_*\mathbb{Q}[-i]$. 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.

These notes discuss Hodge theory in the algebraic and Kahler context. We introduce the notion of (polarized) Hodge structure on a cohomology algebra and show how to extract from it topological restrictions on compact Kahler manifolds, and stronger topological restrictions on projective complex manifolds. The second part of the notes is devoted to the discussion of the Hodge conjecture, showing in particular that there is no way to extend it to the Kahler context. We will also discuss algebraic de Rham cohomology which is specific to projective complex manifolds and allows to formulate a number of arithmetic questions related to the Hodge conjecture.

The purpose of this application, under Article 75.6 of the Code, is to conserve the accustomed usage of the name cornutus Temminck, 1822 for the Sumatran population of the southeast Asian species currently named Batrachostomus javensis (Horsfield, 1821) (Aves, podargidae). To this end, we ask that Temminck's action in proposing the replacement of Horsfield's name javensis with cornutus be set aside so that cornutus does not become a junior objective synonym of javensis (type locality in Java) and that the Sumatran specimen which Temminck actually described as cornutus be designated as its neotype. This would establish cornutus for the Sumatran form for which it has been in use for over 70 years as the subspecific name.

I will discuss positive and negative results on the Hodge conjecture. The negative aspects come on one side from the study of the Hodge conjecture for integral Hodge classes, and on the other side from the study of possible extensions of the conjecture to the general Kahler setting. The positive aspects come from algebraic geometry. They concern the structure of the so-called locus of Hodge classes, and of the Hodge loci.

The Kuga-Satake construction associates to a K3 type polarized weight 2 Hodge structure H an abelian variety A such that H is a quotient Hodge structure of H^2(A). The first step is to consider the Clifford algebra of H. It turns out that it is endowed with a weight 2 Hodge structure compatible with the algebra structure. We show more generally that a weight 2 polarized Hodge structure which carries a compatible (unitary, associative) algebra structure is a quotient of the H^2 of an abelian variety.

We show that the Chow group of 0-cycles on a K3 surface contains a class of degree 1 with remarkable properties: any product of divisors is proportional to this class, and so is the second Chern class c2.

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

International audience; 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.