Littérature scientifique sur le sujet « Hyper-Kähler manifolds »

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

In this paper it is shown that a -dimensional almost symplectic manifold can be endowed with an almost paracomplex structure , , and an almost complex structure , , satisfying for , for and , if and only if the structure group of can be reduced from (or ) to . In the symplectic case such a manifold is called an almost hyper-para-Kähler manifold. Topological and metric properties of almost hyper-para-Kähler manifolds as well as integrability of are discussed. It is especially shown that the Pontrjagin classes of the eigenbundles of to the eigenvalues depend only on the symplectic structure and not on the choice of .

Let [Formula: see text] be a closed symplectic manifold of volume [Formula: see text]. We say that the symplectic packings of [Formula: see text] by balls are unobstructed if any collection of disjoint symplectic balls (of possibly different radii) of total volume less than [Formula: see text] admits a symplectic embedding to [Formula: see text]. In 1994, McDuff and Polterovich proved that symplectic packings of Kähler manifolds by balls can be characterized in terms of the Kähler cones of their blow-ups. When [Formula: see text] is a Kähler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple), these Kähler cones can be described explicitly using the Demailly and Paun structure theorem. We prove that for any Campana simple Kähler manifold, as well as for any manifold which is a limit of Campana simple manifolds in a smooth deformation, the symplectic packings by balls are unobstructed. This is used to show that the symplectic packings by balls of all even-dimensional tori equipped with Kähler symplectic forms and of all hyper-Kähler manifolds of maximal holonomy are unobstructed. This generalizes a previous result by Latschev–McDuff–Schlenk. We also consider symplectic packings by other shapes and show, using Ratner’s orbit closure theorem, that any even-dimensional torus equipped with a Kähler form whose cohomology class is not proportional to a rational one admits a full symplectic packing by any number of equal polydisks (and, in particular, by any number of equal cubes).

3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which appears to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that they are multiply foliated by four distinct fundamental foliations. The study of the transversal geometries with respect to these foliations allows us to link the 3-quasi-Sasakian manifolds to the more famous hyper-Kähler and quaternionic-Kähler geometries. Furthermore, we strongly improve the splitting results previously obtained; we prove that any 3-quasi-Sasakian manifold of rank 4l + 1 is 3-cosymplectic and any 3-quasi-Sasakian manifold of maximal rank is 3-α-Sasakian.

We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. "conformal hypercomplex" manifolds) and quaternionic manifolds of one dimension less. This map is related to a method for constructing supergravity theories using superconformal techniques. An explicit relation between the structure of these manifolds is presented, including curvatures and symmetries. An important role is played by "ξ transformations," relating connections on quaternionic manifolds, and a new type "[Formula: see text] transformations" relating complex structures on conformal hypercomplex manifolds. In this map, the subclass of conformal hyper-Kähler manifolds is mapped to quaternionic-Kähler manifolds.

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

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.