Dissertations / Theses on the topic 'Positivité de fibrés en droites'
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Broustet, Amaël. "Positivité locale des fibrés en droites amples adjoints." Université Joseph Fourier (Grenoble), 2007. http://www.theses.fr/2007GRE10119.
This thesis is about local positivity ofample line bundles, mesured by the Seshadri constants ofthis line bundle. It is conjectured by Ein 1 and Lazarsfeld that Seshadri constants of a big and nef line bundle are bounded by below by 1 for each point in very general position. We show this conjecture:. For each ample line bundle on a smooth variety X of dimension 3 with -K_X nef,. For each big and nef adjoint line bundle on a variety X with canonical singularities of dimension 3 with K_X nef,. For each ample adjoint line bundle of « big volume» on a smooth variety X of dimension 3,. For each ample line bundle on a factorial almost Fano variety X, with terminal singularities, and of co-indice at most 3,. For each ample line bundle O(D) on a smooth Fano variety X of dimension 4 such that K_X - rD. The proof of this results use the existence of non-zero global sections for sorne line bundles. There is a conjecture of Kawamata about this and we prove it in dimension 3 for line bundles of « big volume ». The last chapter is about the uniruledness of the base loci of big and non-nef adjoint line bundles. Big and non-nef line bundles appear in a natural way when an ampleline bundle has a « small » Seshadri constant in a point
Denisi, Francesco Antonio. "Positivité sur les variétés irréductibles holomorphes symplectiques." Electronic Thesis or Diss., Université de Lorraine, 2023. http://www.theses.fr/2023LORR0162.
In this thesis, we study some aspects of the positivity of divisors on irreducible holomorphic symplectic (IHS) manifolds. Fix a projective IHS manifold X of complex dimension 2n. Inspired by the work of Bauer, Küronya, and Szemberg, we show that the big cone of X has a locally finite decomposition into locally rational polyhedral subcones, called Boucksom-Zariski chambers. These subcones have a geometric meaning: on any of them, the volume function is expressed by a homogeneous polynomial of degree 2n. Furthermore, in the interior of any Boucksom-Zariski chamber, the divisorial part of the augmented base locus of big divisors stays the same. After analyzing the big cone, we determine the structure of the pseudo-effective cone of X, generalizing a well-known result due to Kovács for K3 surfaces. In particular, we show that if the Picard number of X is at least 3, the pseudo-effective cone either is circular or does not contain circular parts and is equal to the closure of the cone generated by the prime exceptional divisor classes. From this result in convex geometry, we deduce some geometric properties of X and show the existence of rigid uniruled divisors on some singular symplectic varieties. We study the behaviour of the asymptotic base loci of big divisors on X, and we provide a numerical characterization for them. As a consequence of this numerical characterization, we obtain a description for the dual of the cones mathrm{Amp}_k(X), for any 1leq k leq 2n, where mathrm{Amp}_k(X) is the convex cone of big divisor classes having the augmented base locus of dimension strictly smaller than k. Using the divisorial Zariski decomposition, the Beauville-Bogomolov-Fujiki (BBF) form, and the decomposition of the big cone of X into Boucksom-Zariski chambers, we associate to any big divisor class alpha and a prime divisor E on X a polygon Delta_E(alpha) whose geometry is related to the variation of the divisorial Zariski decomposition of alpha in the big cone. Its euclidean volume is expressed in terms of the BBF form and is independent of the choice of E. We show that these polygons fit in a convex cone Delta_E(X) as slices, globalizing in this way the construction. To conclude, we show that under some hypothesis, the polygons Delta_E(alpha) can be expressed as a Minkowski sum of some polygons {Delta_E(Beta_i)}_{i in I}, for some big classes {Beta_i}_{_ iin I}. Remarkably, these polygons behave like the Newton-Okounkov bodies of big divisors on smooth projective surfaces
Claudon, Benoît. "Déformation de variétés kählériennes compactes : invariance de la $\Gamma$-dimension et extension de sections pluricanoniques." Phd thesis, Université Henri Poincaré - Nancy I, 2007. http://tel.archives-ouvertes.fr/tel-00199485.
Vallès, Jean. "Diviseurs inattendus de droites sauteuses : Fibrés de schwarzenberger." Paris 6, 1996. http://www.theses.fr/1996PA066420.
Plechinger, Valentin. "Espaces de modules de fibrés en droites affines." Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0367.
The study of fibre bundles is an important subject in complex geometry. This thesis considers the particular case of affine line bundles over complex spaces. Affine line bundles are a natural generalisation of line bundles. The first part of this thesis studies the classical moduli problem and the existence of fine moduli spaces. In analogy to the study of line bundles, an affine Picard functor is defined. It is shown that this moduli space will (unless trivial) not be Hausdorff which leads to the study of framed affine line bundles. An exact criterion for the existence of a moduli space for this problem is given. Since the existence of such moduli spaces is very rare, the modern approach of stacks is used in the second part. To give a simpler description of this stack, the theory of fibrewise split extensions is developed. This theory is very general and is of independent interest. For a complex projective variety X, this approach allows to identify the stack of affine line bundles with a quotient stack of linear fibre spaces over the Picard scheme Pic(X). As an application, the homotopy type of this stack is calculated
Dubouloz, Adrien. "Sur une classe de schémas avec actions de fibrés en droites." Université Joseph Fourier (Grenoble), 2004. https://tel.archives-ouvertes.fr/tel-00007733.
On a affine algebraic variety S defined over a field k of characteristic zer o, there exists a well-known correspondence between algebraic actions of the add itive group k+=(k,+) and locally nilpotent derivations of the algebra of regular functions on S. Here we extend this equivalence between actions and derivations to the following relative situation : p:S ? X is a scheme over a fixed base scheme X, which comes equipped with an algebraic action of a l ine bundle p:L?X over X. We study a special sub-class of schemes S as above, with the additional property that p:S ? X factors through the structural morphism p':S?Y of a principal homogeneous bundle under a line bundle p':L'?Y over an X-scheme d:Y?X, in such a way that the action of d*L on S factors through the one of L'. We call them Danielewski-Fieseler schemes. We give different procedures to construct these schemes. In particular, in case that the base scheme X is affine, we give an algorithm which produces explicit embeddings of these schemes in relatives affine spaces over X. Then we study the case that the base scheme X is isomorphic to the affine line over a field k of characteristic zero. In this case, we establish that a Danielewski-Fieseler scheme is uniquely determined by a combinatorial data consisting of a weighted rooted tree. We classify these schemes through their associated trees. Finally, we give a combinatorial characterization of those schemes which admit many actions of the additive group k+ with distinct gener al orbits
Mourougane, Christophe. "Notions de positivité et d'amplitude des fibrés vectoriels : théorèmes d'annulation sur les variétés kahlériennes." Université Joseph Fourier (Grenoble ; 1971-2015), 1997. http://www.theses.fr/1997GRE10028.
Păun, Mihai. "Fibrés en droites numériquement effectifs et variétés kahlériennes compactes à courbure de Ricci nef." Université Joseph Fourier (Grenoble ; 1971-2015), 1998. http://www.theses.fr/1998GRE10011.
Liu, Linyuan. "Cohomologie des fibrés en droites sur SL3/B en caractéristique positive : deux filtrations et conséquences." Thesis, Sorbonne université, 2019. http://www.theses.fr/2019SORUS229.
Let G be a semi-simple algebraic group over an algebraically closed field of positive characteristic. The cohomology of G-equivariant line bundles over G/B induced by a character of B are important objects in the representation theory of G. In this thesis, we concentrate on G = SL3. In the first chapter,we prove the existence of a two-step filtration of H1(μ) and H2(μ) when μ is in the closure of the Griffith region. In the second chapter, we prove the existence ofa p-Hi-D-filtration of Hi(μ) for all i and μ, which generalizes the p-filtration ofH0(μ) introduced by Jantzen. In the third chapter, we study and determine the structure of the modules appearing in the p-Hi-D-filtration. In the last chapter,we give an explicit and combinatorial description of H2(μ) for μ in the Griffith region and we generalize this description to Hd(G/B, μ) for G = SLd+1 and certain weights μ
Vallès, Jean. "Quelques contributions à la classification des fibrés vectoriels sur les espaces projectifs complexes." Habilitation à diriger des recherches, Université de Pau et des Pays de l'Adour, 2010. http://tel.archives-ouvertes.fr/tel-00512155.
Manivel, Laurent. "Théorèmes d'annulation pour la cohomologie des fibrés vectoriels amples." Grenoble 1, 1992. http://www.theses.fr/1992GRE10079.
Wang, Zhenjian. "Groupes projectifs et arrangements de droites." Thesis, Université Côte d'Azur (ComUE), 2017. http://www.theses.fr/2017AZUR4034/document.
The objective of this thesis is to investigate various questions about projective groups and line arrangements in the projective plane. A projective group is a group which is isomorphic to the fundamental group of a smooth complex projective variety. To study projective groups, sophisticated techniques in algebraic topology and algebraic geometry have been developed in the passed decades, for instance, the theory of cohomology jump loci, together with Hodge theory, has been proven a powerful tool. Line arrangements in the projective plane are of special interest in the study of projective groups. Indeed, there are many open questions related to projective groups, and the theory of hyperplane arrangements, and in particular that of line arrangements, which is quite an active area of research, may provide insights for these problems. Furthermore, problems concerning the fundamental groups of the complements of hyperplane arrangements can be reduced to the case of line arrangements, due to the celebrated Zariski theorem of Lefschetz type. Very often, in the study of projective groups or quasi-projective groups, one usually considers line arrangements first to get some intuitive ideas. In this thesis, we also prove some theorems that are of independent interest and can be used elsewhere, for instance, we prove properties concerning morphisms from products of projective spaces in Chapter 4, we show that some morphisms have generic connected fibers in Chapter 5 and we give criteria for a projective surface to be of general type in Chapter 7
Fang, Yanbo. "Study of positively metrized line bundles over a non-Archimedean field via holomorphic convexity." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7033.
This thesis is devoted to the study of semi-positively metrized line bundles in non-Archimedean analytic geometry, with the point of view of functional analysis over an ultra-metric field exploiting the geometry related to holomorphic convexity. The first chapter gathers some preliminaries about Banach algebras over ultra-metric fields and the geometry of their spectrum in the sense of V. Berkovich, which is the framework of our study. The second chapter present the basic construction, which encodes the related geometric information into some Banach algebra. We associate the normed algebra of sections of a metrized line bundle. We describe its spectrum, relating it with the dual unit disc bundle of this line bundle with respect to the envelope metric. We thus encode the metric positivity into the holomorphic convexity of the spectrum. The third chapter consists of two independent for the normed extension problem for restricted sections on a sub-variety. We obtain an upper bound for the asymptotic norm distorsion between the restricted section and the extended one, which is uniform with respect to the choice of restricted sections. We use a particular property of affinoid algebras to obtain this inequality. The fourth chapter treat the problem of regularity of the envelope metric. With a new look from the holomorphic analysis of several variables, we aime at showing that on ample line bundles, the envelop metric is continuous once the original metric is. We suggest a tentative approach based on a speculative analogue of Cartan-Thullen’s result in the non-Archimedean setting
Gillibert, Jean. "Invariants de classes pour les variétés abéliennes à réduction semi-stable." Phd thesis, Université de Caen, 2004. http://tel.archives-ouvertes.fr/tel-00011498.
Dans le chapitre I, nous étudions les propriétés fonctorielles de ces homomorphismes. Nous en déduisons une généralisation de résultats de Taylor, Srivastav, Agboola et Pappas concernant le noyau du class invariant homomorphism pour les variétés abéliennes ayant partout bonne réduction qui sont isogènes à un produit de courbes elliptiques.
Dans le chapitre II, nous donnons une lecture du class invariant homomorphism dans le langage des 1-motifs.
Dans le chapitre III, nous généralisons la construction du class invariant homomorphism pour un sous-groupe fini et plat d'un schéma en groupes semi-stable (sur un schéma de base intègre, normal et noethérien) dont la fibre générique est une variété abélienne. Nous étendons également les résultats de Taylor, Srivastav, Agboola et Pappas à cette situation.
Dans le chapitre IV, nous généralisons la construction du chapitre III en considérant un sous-groupe fermé, quasi-fini et plat du modèle de Néron d'une variété abélienne (la base étant un schéma de Dedekind). Ceci nous permet de généraliser un résultat arakélovien du à Agboola et Pappas.
Ancona, Michele. "Moments en géométrie algébrique réelle." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSE1274.
It is well known that the number of real roots of a real degree d polynomial is at most d. In the 90s, E. Kostlan proved that the average number of real roots equals the square root of d, once we equip the space of polynomials with some natural Gaussian measure. This result has a geometric interpretation, in which the real polynomials are sections of a line bundle over the Riemann sphere. We can extend this study in a more general case of a real Riemann surface equipped with ample line bundle and study the expected value of the number of real zeros of a random section. In this thesis, we compute all the central moments of these random variables. As an application, we prove that the measure of the space of real sections whose number of real zeros deviates from the expected one goes to zeros, as the degree of the line bundle goes to infinity.In a second part, we present analogues results in real Hurwitz theory, in which we study the real critical points of a random branched covering of the Riemann sphere. We compute the expected value of this number and also all the central moments.The techniques we use are of analytique nature (Bergman kernel, L^2 estimates) and gometric one (Olver multispaces, coarea formula)
Weimann, Martin. "La trace en géométrie projective et torique." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2006. http://tel.archives-ouvertes.fr/tel-00136109.
l'aide du calcul résiduel dans les cadres projectifs et toriques.
Dans la première partie, on obtient une caractérisation algébrique des formes traces sur une hypersurface analytique à l'aide du calcul résiduel élémentaire d'une variable. En conséquence, une version plus forte du théorème d'Abel-inverse de Henkin et Passare est prouvée. On montre que ce théorème est conséquence de la rigidité d'un système différentiel particulier lié à une équation de type ”onde de choc” et on établit le lien avec le théorème de Wood sur l'algébricité d'une famille de germes d'hypersurfaces analytiques. Enfin, on obtient une nouvelle méthode pour calculer la dimension de l'espace des formes abéliennes de degré maximal sur une hypersurface projective.
Dans la seconde partie, on caractérise de manière combinatoire les familles de fibrés en droites permettant de définir une notion intrinsèque de concavité dans une variété torique complète lisse et on étudie les ensembles analytiques dégénérés correspondants. On étend ainsi la notion de trace au cas torique. Courants résidus, résidus toriques et résultants donnent une borne optimale sur le degrés des traces en les différents paramètres. Si la variété torique est projective, on obtient finalement une version torique des théorèmes de Wood et d'Abel-inverse, permettant une description plus précise du support du polynôme construit dans le cas hypersurface.
Pelletier, Maxime. "Résultats de stabilité en théorie des représentations par des méthodes géométriques." Thesis, Lyon, 2017. http://www.theses.fr/2017LYSE1228/document.
The Kronecker coefficients, which are indexed by triples of partitions and describe how the tensor product of two irreducible representations of the symmetric group decomposes as a direct sum of such representations, were introduced by Francis Murnaghan in the 1930s. He notably noticed a remarkable behaviour of these coefficients: from any triple of partitions, one can construct a particular sequence of Kronecker coefficients which eventually stabilises.In order to generalise this property, John Stembridge introduced in 2014 a notion of stability for triples of partitions, as well as another notion -- of weakly stable triple -- about which he conjectured that it should be equivalent to the previous one. This conjecture was proven shortly after by Steven Sam and Andrew Snowden, with algebraic methods.In this thesis we especially give another proof -- this time geometric -- of this equivalence, using the classical expression of the Kronecker coefficients as dimensions of spaces of sections of line bundles on flag varieties. With these methods we can also be interested in more specific questions: since the stability which we discuss means that some sequences of coefficients stabilise, one can wonder at which point these sequences become constant.We then apply these techniques to other examples of branching coefficients, and are also interested in another problem: how can we produce stable triples of partitions? We thus generalise a result obtained independently by Laurent Manivel and Ernesto Vallejo on this subject
Darondeau, Lionel. "Sur la conjecture de Green-Griffiths logarithmique." Thesis, Paris 11, 2014. http://www.theses.fr/2014PA112134/document.
The topic of this memoir is the geometry of holomorphic entire curves with values in the complement of generic hypersurfaces of the complex projective space. The well-known conjectures of Kobayashi and of Green-Griffiths assert that for such hypersurfaces, having large degree, the images of these curves shall fulfill algebraic constraints. By adapting the jet techniques developed notably by Bloch, Green-Griffiths, Demailly, Siu, Diverio-Merker-Rousseau, in the case of curves with values in projective hypersurfaces (so-called compact case), we obtain the algebraic degeneracy of entire curves f : ℂ→Pⁿ∖Xd (so called logarithmic case), for generic hypersurfaces Xd in Pⁿ of degree d ≥ (5n)² nⁿ. As in the compact case, our proof essentially relies on the algebraic elimination of all derivatives in differential equations that are satisfied by every nonconstant entire curve. The existence of such differential equations is obtained thanks to the holomorphic Morse inequalities and a simplified variant of a residue formula firstly developed by Bérczi from the Atiyah-Bott equivariant localization formula. The effective lower bound d ≥ (5n)² nⁿ is obtained by radically simplifying a huge iterated residue computation. Next, the deformation of these differential equations by derivation along slanted vector fields, the existence of which is here generalized and clarified, allows us to generate sufficiently many new differential equations in order to realize the final algebraic elimination mentioned above
Shao, Guokuan. "Équidistribution des zéros de sections holomorphes aléatoires par rapport à des mesures modérées." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS141/document.
This thesis investigates the equidistributions of zeros of random holomorphic sections of line bundles for moderate measures. It consists of two parts. In the first part, we construct a large family of singular moderate measures on projective spaces. These measures are generated by quasi-plurisubharmonic functions with Holder potentials.The second part deals with an equidistribution property in general settings. We establish an equidistribution theorem in the case of several big line bundles endowed with singular metrics. A precise convergence speed for the equidistribution is obtained
DUBOULOZ, Adrien. "Sur une classe de schémas avec actions de fibrés en droites." Phd thesis, 2004. http://tel.archives-ouvertes.fr/tel-00007733.
Paris, Matthieu. "Quelques aspects de la positivité du fibré tangent des variétés projectives complexes." Phd thesis, 2010. http://tel.archives-ouvertes.fr/tel-00552308.
BOUCKSOM, Sébastien. "Cônes positifs des variétés complexes compactes." Phd thesis, 2002. http://tel.archives-ouvertes.fr/tel-00002268.
Jauffret, Colin. "Modules réflexifs de rang 1 sur les variétés nilpotentes." Thèse, 2016. http://hdl.handle.net/1866/19543.
Let G be a simple, connected, simply connected complex linear algebraic group with parabolic subgroup P G and nilpotent ideal n p. The proper collapsing map G x P n = Gn factors through the normal affine variety N := SpecC [G x P n] which is called a nilpotent variety. Assuming the collapsing is generically finite, we describe the equivariant divisor class group of N using rank 1 reflexive equivariant C[N]-modules. A representative of each class may be chosen as global sections of a line bundle over G x P' n' where G x P' n' = Gn' is a possibly distinct collapsing that factors through the same nilpotent variety. Assuming either G is of type A or the collapsing comes from specific weighted Dynkin diagrams,we showthat each representative arise from a weight that may be chosen dominant. Moreover, if the module represents a torsion element within the class group, then it is Cohen– Macaulay and we deduce a cohomological vanishing theorem.