Littérature scientifique sur le sujet « Conjectures de Voisin »

AbstractGreen’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.

AbstractWe study a conjecture, due to Voisin, on 0-cycles on varieties with pg = 1. Using Kimura’s finite dimensional motives and recent results of Vial’s on the refined (Chow–)Künneth decomposition, we provide a general criterion for Calabi–Yau manifolds of dimension at most 5 to verify Voisin’s conjecture. We then check, using in most cases some cohomological computations on the mirror partners, that the criterion can be successfully applied to various examples in each dimension up to 5.

Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O’Grady’s filtration on the $\text{CH}_{0}$-group of the $K3$ surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the $\text{CH}_{0}$-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.

We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch–Ogus, Colliot-Thélène–Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch–Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel–Jacobi invariant has coniveau $1$ . This establishes a torsion version of a conjecture of Jannsen originally formulated $\otimes \mathbb {Q}$ . We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel–Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove $\ell$ -adic analogues of these results over any field $k$ which contains all $\ell$ -power roots of unity.

We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the canonical ribbon conjecture of Bayer and Eisenbud holds over a field of characteristic $0$ or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics the results of Voisin asserting that Green's conjecture holds for generic curves of each gonality.

We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov’s noncommutative $K3$ category associated to a nonsingular cubic 4-fold.By introducing a filtration on the $\text{CH}_{1}$-group of a cubic 4-fold $Y$, we conjecture a sheaf/cycle correspondence for the associated $K3$ category ${\mathcal{A}}_{Y}$. This is a noncommutative analog of O’Grady’s conjecture concerning derived categories of $K3$ surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sheaves supported on low degree rational curves.Our method provides systematic constructions of (a) the Beauville–Voisin filtration on the $\text{CH}_{0}$-group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli space of stable objects in ${\mathcal{A}}_{Y}$.

AbstractWe prove the integral Tate conjecture for cycles of codimension$2$on smooth cubic fourfolds over an algebraic closure of a field finitely generated over its prime subfield and of characteristic different from$2$or$3$. The proof relies on the Tate conjecture with rational coefficients, proved in that setting by the first author, and on an argument of Voisin coming from complex geometry.

AbstractFor a field k of cohomological dimension d we prove that the groups , (l, car.k) = 1, are birational invariants of smooth projective geometrically integral varieties over k of dimension n. Using the Kato conjecture, which has been recently established by Kerz and Saito [18], we obtain a similar result over a finite field for the groups . We relate one of these invariants with the cokernel of the l-adic cycle class map , which gives an analogue of a result of Colliot-Thélène and Voisin [5] 3.11 over ℂ for varieties over a finite field.

The influence of Chow groups on singular cohomology is motivated by classical results by Mumford and Roitman and has been investigated extensively. On the other hand, the converse influence is rather conjectural and it takes place in the framework of the ``philosophy of mixed motives'', which is mainly due to Grothendieck, Bloch and Beilinson. In the spirit of exploring this influence, Voisin formulated in 1996 a conjecture on 0--cycles on the self--product of surfaces of geometric genus one. There are few examples in which Voisin's conjecture has been verified, but it is still open for a general $K3$ surface. Our aim is to present a new example in which Voisin's conjecture is true, a family of Todorov surfaces. We give an explicit description of the family as quotient of complete intersection of four quadrics in $mathbb{P}^{6}$. We verify Voisin's conjecture for the family of Todorov surfaces of type $(2,12)$. Our main tool is Voisin's ``spreading of cycles'', we use it to establish a relation between 0--cycles on the Todorov surface and on the associated K3 surface. We give a motivic version of this result and some interesting motivic applications.

Cette thèse est consacrée à l'étude des cycles algébriques dans les variétés hyper-Kähleriennes projectives et les variétés de Calabi-Yau strictes. Elle contribue à la compréhension des conjectures de Beauville et de Voisin sur les anneaux de Chow des variétés hyper-kählériennes projectives et des variétés de Calabi-Yau strictes. Elle étudie également certains invariants birationnels des variétés hyper-kählériennes projectives.La première partie de la thèse, parue dans Mathematische Zeitschrift [C. Bai, On Abel-Jacobi maps of Lagrangian families, Math. Z. 304, 34 (2023)] et présentée dans le chapitre 2, étudie si les sous-variétés lagrangiennes dans une variété hyper-kählérienne partageant la même classe cohomologique ont également la même classe de Chow. Nous étudions la notion de familles lagrangiennes et ses applications aux applications d'Abel-Jacobi associées. Nous adoptons une approche infinitésimale pour donner un critère de trivialité de l'application d'Abel-Jacobi d'une famille lagrangienne, et utilisons ce critère pour donner une réponse négative à la question précédente, ajoutant aux subtilités d'une conjecture de Voisin. Nous explorons également comment la maximalité de la variation des structures de Hodge sur la cohomologie de degré 1 de la famille lagrangienne implique la trivialité de l'application d'Abel-Jacobi. La deuxième partie de la thèse, à paraître dans International Mathematics Research Notices [C. Bai, On some birational invariants of hyper-Kähler manifolds, ArXiv: 2210.12455, to appear in International Mathematics Research Notices, 2024] et présentée dans le chapitre 3, étudie le degré d'irrationalité, la gonalité fibrante et le genre fibrant des variétés hyper-kählériennes projectives. Nous commençons par donner une légère amélioration d'un résultat de Voisin sur la borne inférieure du degré d'irrationalité des variétés hyper-kählériennes générales de Mumford-Tate. Nous étudions ensuite la relation entre les trois invariants birationnels susmentionnés pour les surfaces K3 projectives de nombre de Picard 1, rajoutant la compréhension sur une conjecture de Bastianelli, De Poi, Ein, Lazarsfeld, Ullery sur le comportement asymptotique du degré d'irrationalité des surfaces K3 projectives très générales. La troisième partie de la thèse, présentée dans le chapitre 4, étudie les applications de Voisin de dimension supérieure sur les variétés de Calabi-Yau strictes. Voisin a construit des applications auto-rationnelles de variétés de Calabi-Yau obtenues comme des variétés de r-plans dans des hypersurfaces cubiques de dimension adéquate. Cette application a été largement étudiée dans le cas r=1, qui est le cas de Beauville-Donagi. Dans les cas de dimensions supérieures, nous étudions d'abord l'action de l'application de Voisin sur les formes holomorphes. Nous démontrons ensuite la conjecture de Bloch généralisée pour l'action des applications de Voisin sur les groupes de Chow dans le cas de r=2. Enfin, via l'étude de l'application de Voisin, nous apportons des éléments de preuve à une conjecture de Voisin sur l'existence d'un 0-cycle spécial sur les variétés de Calabi-Yau strictes
This thesis is devoted to the study of algebraic cycles in projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It contributes to the understanding of Beauville's and Voisin's conjectures on the Chow rings of projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It also studies some birational invariants of projective hyper-Kähler manifolds.The first part of the thesis, appeared in Mathematische Zeitschrift [C. Bai, On Abel-Jacobi maps of Lagrangian families, Math. Z. 304, 34 (2023)] and presented in Chapter 2, studies whether the Lagrangian subvarieties in a hyper-Kähler manifold sharing the same cohomological class have the same Chow class as well. We study the notion of Lagrangian families and its associated Abel-Jacobi maps. We take an infinitesimal approach to give a criterion for the triviality of the Abel-Jacobi map of a Lagrangian family, and use this criterion to give a negative answer to the above question, adding to the subtleties of a conjecture of Voisin. We also explore how the maximality of the variation of the Hodge structures on the degree 1 cohomology the Lagrangian family implies the triviality of the Abel-Jacobi map. The second part of the thesis, to appear in International Mathematics Research Notices [C. Bai, On some birational invariants of hyper-Kähler manifolds, ArXiv: 2210.12455, to appear in International Mathematics Research Notices, 2024] and presented in Chapter 3, studies the degree of irrationality, the fibering gonality and the fibering genus of projective hyper-Kähler manifolds, with emphasis on the K3 surfaces case, en mettant l'accent sur le cas des surfaces K3. We first give a slight improvement of a result of Voisin on the lower bound of the degree of irrationality of Mumford-Tate general hyper-Kähler manifolds. We then study the relation of the above three birational invariants for projective K3 surfaces of Picard number 1, adding the understandinf of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld, Ullery on the asymptotic behavior of the degree of irrationality of very general projective K3 surfaces. The third part of the thesis, presented in Chapter 4, studies the higher dimensional Voisin maps on strict Calabi-Yau manifolds. Voisin constructed self-rational maps of Calabi-Yau manifolds obtained as varieties of r-planes in cubic hypersurfaces of adequate dimension. This map has been thoroughly studied in the case r=1, which is the Beauville-Donagi case. For higher dimensional cases, we first study the action of the Voisin map on the holomorphic forms. We then prove the generalized Bloch conjecture for the action of the Voisin maps on Chow groups for the case of r=2. Finally, via the study of the Voisin map, we provide evidence for a conjecture of Voisin on the existence of a special 0-cycle on strict Calabi-Yau manifolds

The influence of Chow groups on singular cohomology is motivated by classical results by Mumford and Roitman and has been investigated extensively. On the other hand, the converse influence is rather conjectural and it takes place in the framework of the ``philosophy of mixed motives'', which is mainly due to Grothendieck, Bloch and Beilinson. In the spirit of exploring this influence, Voisin formulated in 1996 a conjecture on 0--cycles on the self--product of surfaces of geometric genus one. There are few examples in which Voisin's conjecture has been verified, but it is still open for a general $K3$ surface. Our aim is to present a new example in which Voisin's conjecture is true, a family of Todorov surfaces. We give an explicit description of the family as quotient of complete intersection of four quadrics in $mathbb{P}^{6}$. We verify Voisin's conjecture for the family of Todorov surfaces of type $(2,12)$. Our main tool is Voisin's ``spreading of cycles'', we use it to establish a relation between 0--cycles on the Todorov surface and on the associated K3 surface. We give a motivic version of this result and some interesting motivic applications.