Dissertations / Theses on the topic 'Variétés de Calabi-Yau strictes'
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Bai, Chenyu. "Hodge Theory, Algebraic Cycles of Hyper-Kähler Manifolds." Electronic Thesis or Diss., Sorbonne université, 2024. http://www.theses.fr/2024SORUS081.
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
Dedieu, Thomas. "Auto-transformations et géométrie des variétés de Calabi-Yau." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2008. http://tel.archives-ouvertes.fr/tel-00358735.
Dans la première, je démontre que si certaines variétés de Severi universelles, qui paramètrent les courbes nodales de degré et de genre fixés existant sur une surface K3, sont irréductibles, alors une surface K3 projective générique ne possède pas d'endomorphisme rationnel de degré >1. J'établis également un certain nombre de contraintes numériques satisfaites par ces endomorphismes.
Voisin a modifié la pseudo-forme volume de Kobayashi en introduisant les K-correspondances holomorphes. Dans la seconde partie, j'étudie une version logarithmique de cette pseudo-forme volume. J'associe une pseudo-forme volume logarithmique intrinsèque à toute paire (X,D) constituée d'une variété complexe et d'un diviseur à croisements normaux et partie positive réduite. Je démontre qu'elle est génériquement non dégénérée si X est projective et K_X+D est ample. Je démontre d'autre part qu'elle s'annule pour une grande classe de paires à fibré canonique logarithmique trivial.
Bazhov, Ivan. "Zero-cycles and constant cycle subvarieties in Calabi-Yau and hyper-Kähler varieties." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066387/document.
We present in this thesis three results. In Chapter 2 we prove the existence of a canonical zero-cycle cX on a Calabi–Yau hypersurfacee X in a complex projective homogeneous variety. Namely, we show that the intersection of any n divisors on X , n = dim X is proportional to the class of a point on a rational curve in X. In Chapter 3 we give a new proof of the theorem of Beauville and Voisin about the decomposition of the small diagonal of a K3 surface S. Our proof is explicit and uses the degree 2g-2 embedding of S in projective space of dimension g. It is different from the one used by Beauville and Voisin, which employed the existence of one-parameters familie of elliptic curves. Chapter 4 is devoted to the study of similarities between the Fano varieties of lines on a cubic fourfold, a hyper-Kähler fourfold studied by Beauville and Donagi, and the hyper-Kähler fourfold constructed by Debarre and Voisin in [11]. We exhibit an analog of the notion of "triangle" for these varieties and prove that the 6-dimensional variety of "triangles" is a Lagrangian subvariety in the cube of the constructed hyper-Kähler fourfold
Vernier, Caroline. "Autour du programme de Calabi, méthodes de recollement." Thesis, Nantes, 2018. http://www.theses.fr/2018NANT4046/document.
We study the existence of metrics of constant Hermitian scalar curvature on almost-Kähler manifolds obtained as smoothings of a constant scalar curvature Kähler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphie vector fields, we show that an almost-Kähler smoothing (Mє, ωє) admits an almost-Kähler structure (Jє, gє) of constant Hermitian curvature. Moreover, we show that for є > O small enough, the (Mє, ωє) are all symplectically equivalent to a fixed symplectic manifold (M , ω) in which there is a surface S homologous to a 2-sphere, such that [S] is a vanishing cycle that admits a representant that is Hamiltonian stationary for gє
Amiot, Claire. "Sur les petites catégories triangulées." Paris 7, 2008. http://www.theses.fr/2008PA077068.
This thesis is divided in two mostly independent parts. The first one deals with the classification of the triangulated categories with finitely many indecomposable objects. We first compute the structure of the Ausiander-Reiten quiver of such categories. Then we prove that such categories which are algebraic and d-Calabi-Yau (where d is an integer greater than 2) are quotients of d-cluster categories associated with Dynkin quivers. In the second part, we generalize the notion of cluster category. To certain finite-dimensional algebras of global dimension smaller than 2, we associate a triangulated category which coincide to the cluster category when the global dimension is 1. Then we show that these new categories are 2-Calabi-Yau and endowed with a canonical cluster-tilting object
Benedetti, Vladimiro. "Sous-variétés spéciales des espaces homogènes." Thesis, Aix-Marseille, 2018. http://www.theses.fr/2018AIXM0224/document.
The aim of this thesis is to construct new interesting complex algebraic Fano varieties and varieties with trivial canonical bundle and to analyze their geometry. In the first part we construct special varieties as zero loci of homogeneous bundles inside generalized Grassmannians. We give a complete classification for varieties of small dimension when the bundle is completely reducible. Thus, we prove that the only fourfolds with trivial canonical bundle so constructed which are hyper-Kahler are the examples of Beauville-Donagi and Debarre-Voisin. The same holds in ordinary Grassmannians when the bundle is irreducible in any dimension. In the second part we use orbital degeneracy loci (ODL), which are a generalization of classical degeneracy loci, to construct new varieties. ODL are constructed from a model, which is usually an orbit closure inside a representation. We recall the fundamental properties of ODL. As an illustration of the construction, we construct three Hilbert schemes of two points on a K3 surface as ODL, and many examples of Calabi-Yau and Fano threefolds and fourfolds. Then we study orbit closures inside quiver representations, and we provide crepant Kempf collapsings for those of type A_n, D_4; this allows us to construct some special varieties as ODL.Finally we focus on a particular class of Fano varieties, namely bisymplectic Grassmannians. These varieties admit the action of a torus with a finite number of fixed points. We find the dimension of their moduli space. We then study the equivariant cohomology of symplectic Grassmannians, which turns out to help understanding better that of bisymplectic ones. We analyze in detail the case of dimension 6
Banos, Bertrand. "Opérateurs de Monge-Ampère symplectiques en dimensions 3 et 4." Angers, 2002. http://www.theses.fr/2002ANGE0041.
Tabuada, Gonçalo. "Théorie homotopique des DG-catégories." Paris 7, 2007. http://www.theses.fr/2007PA077060.
Differential graded categories (dg categories) enhance our understanding of triangulated categories appearing in representation theory and in (commutative and non-commutative) algebraic geometry. In this thesis we study them using the tools of Quillen's homotopical algebra and Heller-Grothendieck's derivators. Our main results are: (1) the category of dg categories admits a structure of model category whose weak equivalences are the Morita dg functors; (2) Quillen-Waldhausen's K-theory becomes corepresentable in the additive motivator of dg categories; (3) the internal Hom functor of the homotopy category of dg categories becomes a derived functor in the category of localization pairs of dg categories; (4) the category of alpha-compactly generated triangulated categories admits a Quillen enhancement; (5) every 2-Calabi-Yau category with a cluster tilting object is the stabilization of a 3-Calabi-Yau category endowed with a t structure
Palu, Yann. "Des catégories triangulées aux algèbres amassées." Paris 7, 2009. http://www.theses.fr/2009PA077054.
Cluster algebras were introduced by Fomin and Zelevinsky in order to study the semi-canonical basis of Lusztig, and total positivity. The theory of representations of quivers and finite dimensional algebras can be used to categorify some cluster algebras. In order to recover the cluster algebra from the category which categorifies it, one needs a map called a cluster character. In this thesis, we construct a cluster character associated with any cluster tilting object of a 2-Calabi-Yau, Hom-finite triangulated category. Under some additional constructibility hypothesis, we prove a multiplication formula for this cluster character (similar to that of Geiss-Leclerc-Schroer), thus generalizing the formulae of Caldero-Keller, Hubery, Xiao-Xu. We prove that this hypothesis is satisfied by stable categories of Hom-finite Frobenius categories, and by the generalized cluster categories of Amiot. We also study the Grothendieck group of the 2-Calabi-Yau, Hom-finite triangulated categories which admit cluster tilting subcategories. This allows us to give a K-theoretical interpretation of the mutation rule of Fomin and Zelevinsky
Jolany, Hassan. "Analytical log minimal model program via conical Kähler Ricci flow : Song-Tian program." Thesis, Lille 1, 2016. http://www.theses.fr/2016LIL10109.
Existence of canonical metric on a projective variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class(most of the manifolds do not have definite first Chern class). Thereis a program which is known as SongTian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. The main aim of this thesis is better undrestanding of SongTian program on pair (X;D). In this thesis, we apply SongTian program for pair (X;D) via Log Minimal Model Program where D is a simple normal crossing divisor on X with conic singularities. We investigate conical Kähler Ricci flow on holomorphic fiber spaces (X;D) -→B whose generic fibers are log Calabi Yau pairs (Xs;Ds), c1(KB) < 0, and D is a simple normal crossing divisor on X (we consider the cases c1(KB) = 0, and c1(KB) > 0 also). We show that there is a unique conical Kähler Einstein metric on (X;D) which is twisted by logarithmic Weil Petersson metric and an additional term which we will find it explicitly. We consider the semipositivity of fiberwise singular Kahler Einstein metric via SongTian program. We consider a twisted Kähler Einstein metric along Mori fibre space. Moreover, we give an analogue version of SongTian program for Sasakian manifolds. We give an arithmetic version of SongTian program for arithmetic varieties. Also we give a short proof of Tian’s formula for Kähler potential of logarithmic WeilPetersson metric on moduli space of log CalabiYau varieties (if such moduli space exists!)
Fu, Lie. "Sous-structures de Hodge, anneaux de Chow et action de certains automorphismes." Phd thesis, Ecole Normale Supérieure de Paris - ENS Paris, 2013. http://tel.archives-ouvertes.fr/tel-01001733.
Prins, Daniël. "On flux vacua, SU(n)-structures and generalised complex geometry." Thesis, Lyon 1, 2015. http://www.theses.fr/2015LYO10174/document.
Understanding supersymmetric flux vacua is essential in order to connect string theory to observable physics. In this thesis, flux vacua are studied by making use of two mathematical frameworks: SU(n)-structures and generalised complex geometry. Manifolds with $SU(n)$ structure are generalisations of Calabi-Yau manifolds. Generalised complex geometry is a geometrical framework that simultaneously generalises complex and symplectic geometry. Classes of flux vacua of type II supergravity and M-theory are given on manifolds with SU(4) structure. The N = (1,1) type IIA vacua uplift to N=1 M-theory vacua, with four-flux that need not be (2,2) and primitive. Explicit vacua are given on Stenzel space, a non-compact Calabi Yau. These are then generalised by constructing families of non-CY SU(4)-structures to find vacua on non-symplectic SU(4)-deformed Stenzel spaces. It is shown that the supersymmetry conditions for N = (2,0) type IIB can be rephrased in the language of generalised complex geometry, partially in terms of integrability conditions of generalised almost complex structures. This rephrasing for d=2 goes beyond the calibration equations, in contrast to d=4,6 where the calibration equations are equivalent to supersymmetry. Finally, Euclidean type II theory is examined on SU(5)-structure manifolds, where generalised equations are found which are necessary but not sufficient to satisfy the supersymmetry equations. Explicit classes of solutions are provided here as well. Contact with Lorentzian physics can be made by uplifting such solutions to d=1, N = 1 M-theory
Comparin, Paola. "Symétrie miroir et fibrations elliptiques spéciales sur les surfaces K3." Thesis, Poitiers, 2014. http://www.theses.fr/2014POIT2281/document.
A K3 surface is a complex compact projective surface X which is smooth and such that its canonical bundle is trivial and h0;1(X) = 0. In this thesis we study two different topics about K3 surfaces. First we consider K3 surfaces obtained as double covering of P2 branched on a sextic curve. For these surfaces we classify elliptic fibrations and their Mordell-Weil group, i.e. the group of sections. A 2-torsion section induces a symplectic involution of the surface, called van Geemen-Sarti involution. The classification of elliptic fibrations and 2-torsion sections allows us to classify all van Geemen-Sarti involutions on the class of K3 surfaces we are considering. Moreover, we give details in order to obtain equations for the elliptic fibrations and their quotient by the van Geemen-Sarti involutions. Then we focus on the mirror construction of Berglund-Hübsch-Chiodo-Ruan (BHCR). This construction starts from a polynomial in a weighted projective space together with a group of diagonal automorphisms (with some properties) and gives a pair of Calabi-Yau varieties which are mirror in the classical sense. The construction works for any dimension. We use this construction to obtain pairs of K3 surfaces which carry a non-symplectic automorphism of prime order p > 3. Dolgachev and Nikulin proposed another notion of mirror symmetry for K3 surfaces: the mirror symmetry for lattice polarized K3 surfaces (LPK3). In this thesis we show how to polarize the K3 surfaces obtained from the BHCR construction and we prove that these surfaces belong to LPK3 mirror families
Alexandrov, Sergey. "L'Approche Twistorielle aux Compactifications de la Théorie des Cordes." Habilitation à diriger des recherches, 2012. http://tel.archives-ouvertes.fr/tel-00682454.
Campling, Emily. "Fukaya categories of Lagrangian cobordisms and duality." Thèse, 2018. http://hdl.handle.net/1866/21746.