Tesi sul tema "Semistable"

Using the orbicell decomposition of the Deligne-Mumford compactification of the moduli space of Riemann surfaces studied before by the author, a chain complex based on semistable ribbon graphs is constructed which is an extension of the Konsevich’s graph homology.
En este trabajo mediante la descomposicion orbicelular de la compacticacion de Deligne-Mumford del espacio de moduli de supercies de Riemann (estudiada antes por el autor) construimos un complejo basado en grafos de cinta semiestables, lo cual constituye una extension de la homologa de grafos de Kontsevich.

Part I of this thesis concerns the relation, in p-adic Hodge theory, between the monodromy and the Hodge numbers of a filtered (φ, N)-module D. Studying the interaction of the Hodge and Newton polygons with N, we deduce that a monodromy operator of large rank forces the Hodge numbers of D to be large: this is the content of Theorem I.14.4. This result can then be applied to various Galois representations. For instance, starting with a Hilbert modular form f over a totally real field F , it is known that we can attach to it a global p-adic Galois representation pf: GF͢ GL2(E), for E/Qp some ënite extension. Choosing a prime p of F above p, we can then study the local p-adic Galois representation pf, p: GGFp GL2(E). Assuming that pf;p is semistable with matching Hodge–Tate weights, we can then use Fontaine–Dieudonné theory to obtain a filtered (φ, N)-module Dst (ρf, p). Applying Theorem I.14.4, we deduce that if the weights of f are too small, then ρf, p is in fact crystalline. We also present in section I.16.1 an example of a non-split semistable non-crystalline extension of crystalline characters which is not “trivial by cyclotomic”, even up to twists. In part II, we explore parallel results on the automorphic side of the Langlands correspondence. Concentrating on the case of Hilbert modular forms, the approach is to study the p-adic integrality properties of Hecke operators. This naturally leads to the study of integral models of Hilbert modular varieties and of their associated automorphic vector bundles. A careful study of these leads to the introduction of certain renormalisation factors for the action of Hecke operators (Proposition II.5.2). We then prove the integrality of these renormalised Hecke operators using the q-expansion principle (Proposition II.5.4). Finally, we use these integrality properties to arrive at conditions, dependent on the weight of a Hilbert modular form f , that guarantee that local components of f cannot be special; is the content of Τheorem II.6.2. For instance, in the case that there is a unique prime p above p in F, corresponding to the local component πf, p is a unique filtered ( φ,N, GFp)-module Df, p, and the results of Theorem I.14.4 and Theorem II.6.2 exactly match: if the weights ντ of f do not average at least 2, the former theorem shows that Df,p is potentially crystalline, while the latter shows that πf, p cannot be special. Other interesting behaviour can occur depending on the splitting behaviour of p in F , such as conditions on all pairs of weights of f as in the final example of section 6.2.1.

Dans cette thèse nous construisons des espaces de modules de faisceaux semi-stables sur une variété projective complexe lisse X, dotée d'une polarisation fixée sheaf{O}_X(1). Notre approche suit les idées de Le Potier et Jun Li, qui ont construit indépendamment des espaces de modules de faisceaux sans torsion, semi-stables par rapport à la pente sur des surfaces (projectives). Leurs espaces sont en relation, par la correspondance Kobayashi-Hitchin, avec la compactification de Donaldson-Uhlenbeck en théorie de jauge. Ici, cependant, nous sommes principalement intéressés par les aspects algébriques de leur travail. En particulier, cette thèse généralise leur construction au cas des faisceaux purs de dimension supérieure, dont le schéma de support peut être singulier. Nous introduisons d'abord une notion de stabilité pour les faisceaux cohérents purs de dimension d sur X, qui se situe entre la stabilité par rapport à la pente et la stabilité de Gieseker. Cette notion est définie par rapport au polynôme de Hilbert du faisceau, tronqué jusqu'à un certain degré. Nous l'appelons ell-(semi)stabilité, où ell marque le niveau de troncature. En particulier, on retrouve la notion classique de stabilité par rapport à la pente pour ell = 1 et de Gieseker-stabilité pour ell = d. Notre construction utilise comme ingrédient principal un théorème de restriction pour la (semi-)stabilité, disant que la restriction d'un faisceau ell-semistable (ou ell-stable) à un diviseur général D in |sheaf{O}_X(a)| de degré suffisamment grand dans X est à nouveau ell-semistable (respectivement ell-stable). À cet égard, dans le Chapitre 2, nous prouvons plusieurs théorèmes de restriction pour les faisceaux purs (voir les Théorèmes ef{thm:GiesekerRestriction},ef{thm:restrictionStable} etef{thm:ThmC}). Les méthodes utilisées dans la preuve nous permettent de donner des énoncés en caractéristique quelconque. De plus, nos résultats généralisent les théorèmes de restriction de Mehta et Ramanathan pour la (semi-)stabilité par rapport à la pente, et ils s'appliquent en particulier aux faisceaux Gieseker-semistables. Avant de donner la construction, nous faisons un bref détour pour généraliser la fibration d'Iitaka classique au cadre équivariant. Nous construisons alors des espaces de modules projectifs de faisceaux ell-semistables en dimensions supérieures, comme certaines fibrations d'Iitaka équivariantes (voir le Théorème~ef{thm:mainThm}). Notre construction est nouvelle dans la littérature lorsque 1 < ell < d ou lorsque ell=1 et d < dim(X). En particulier, dans le cas des faisceaux sans torsion, nous récupérons un résultat de Huybrechts-Lehn sur les surfaces et de Greb-Toma en dimensions supérieures. Enfin, nous décrivons en détail les points géométriques de ces espaces de modules (voir le Théorème~ef{thm:separation}). Comme application, nous montrons que dans le cas sans torsion, ils fournissent des compactifications différentes sur le lieu ouvert des fibrés vectoriels stables par rapport à la pente. Nous pouvons considérer ces espaces comme des compactifications intermédiaires entre la compactification de Gieseker et la compactification de Donaldson-Uhlenbeck
In this thesis we construct moduli spaces of semistable sheaves over a complex smooth projective variety X, endowed with a fixed polarization sheaf{O}_X(1). Our approach is based on ideas of Le Potier and Jun Li, who independently constructed moduli spaces of slope-semistable torsion-free sheaves over (projective) surfaces. Their spaces are closely related, via the Kobayashi-Hitchin correspondence, to the so-called Donaldson-Uhlenbeck compactification in gauge theory. Here, however, we are mainly interested in the algebraical aspects of their work. In a restrictive sense, this thesis generalizes their construction to higher dimensional pure sheaves, whose support scheme might be singular. First we introduce a notion of stability for pure coherent sheaves of dimension d on X, which lies between slope- and Gieseker-stability. This is defined with respect to the Hilbert polynomial of the sheaf, truncated down to a certain degree. We call it ell-(semi)stability, where ell marks the level of truncation. In particular, this recovers the classical notion of slope-stability for ell =1 and of Gieseker-stability for ell = d. Our construction uses as main ingredient a restriction theorem for (semi)stability, saying that the restriction of an ell-semistable (or ell-stable) sheaf to a general divisor D in |sheaf{O}_X(a)| of sufficiently large degree in X is again ell-semistable (respectively ell-stable). In this regard, in Chapter~ef{ch:RestrictionTheorems} we prove several restriction theorems for pure sheaves (see Theorems~ef{thm:GiesekerRestriction},ef{thm:restrictionStable} and ef{thm:ThmC}). The methods employed in the proofs permit us to give statements in arbitrary characteristic. Furthermore, our results generalize the restriction theorems of Mehta and Ramanathan for slope-(semi)stability, and they apply in particular to Gieseker-semistable sheaves. Before we give the construction, we take a short detour to generalize the classical Iitaka fibration to the equivariant setting. Given this, we construct projective moduli spaces of ell-semistable sheaves in higher dimensions as certain equivariant Iitaka fibrations (see Theorem~ef{thm:mainThm}). Our construction is new in the literature when 1

Object Tracking plays a very pivotal role in many computer vision applications such as video surveillance, human gesture recognition and object based video compressions such as MPEG-4. Automatic detection of any moving object and tracking its motion is always an important topic of computer vision and robotic fields. This thesis deals with the problem of detecting the presence of debris or any other unexpected objects in footage obtained during spacecraft launches, and this poses a challenge because of the non-stationary background. When the background is stationary, moving objects can be detected by frame differencing. Therefore there is a need for background stabilization before tracking any moving object in the scene. Here two problems are considered and in both footage from Space shuttle launch is considered with the objective to track any debris falling from the Shuttle. The proposed method registers two consecutive frames using FFT based image registration where the amount of transformation parameters (translation, rotation) is calculated automatically. This information is the next passed to a Kalman filtering stage which produces a mask image that is used to find high intensity areas which are of potential interest.
M.S.E.E.
Department of Electrical and Computer Engineering
Engineering and Computer Science
Electrical Engineering

La (semi)stabilité, introduite par Mumford en 1963, sert à la construction d'espaces de modules de fibrés vectoriels par les méthodes de GIT. Dans la frontière de l'espace de modules compactifié apparaissent des faisceaux non localement libres. La thèse vise à proposer un nouveau stock d'objets de frontière plus maniables, dans le cas de dimension 2 et de rang 2, qui sont des fibrés sur des arbres de bulles A ayant S comme racine. La motivation vient de la théorie de jauge et de l'étude par Nagaraj-Seshadri et Teixidor i Bigas des fibrés sur des courbes réductibles. La semistabilité sur A dépend d'une polarisation, c'est à dire, d'un fibré en droites ample. Le domaine des paramètres de la polarisation est bien plus petit et les fibrés semistables sont plus rares en dimension 2 que dans le cas de courbes. Pour certaines polarisations, on donne des critères de semistabilité des fibrés sur A en fonction de leurs restrictions aux composantes de A. Bien que les faisceaux étudiés sur A soient des fibrés, leur sous-faisceaux potentiellement déstabilisants peuvent être juste réflexifs. On entreprend alors la classification des faisceaux réflexifs sur des arbres de bulles, basée sur les travaux de Burban-Drozd. On étudie ensuite les déformations des fibrés arboriformes. Le résultat principal est qu'un fibré stable sur A, pour certaines polarisations, est toujours la limite de fibrés stables sur S. Enfin, on compare le stock des fibrés stables arboriformes, limites d'instantons de charge 2 sur le plan projectif, avec celui de Markushevich-Tikhomirov-Trautmann, obtenu par une autre approche
The (semi)stability, introduced by Mumford in 1963, was used for construction of moduli spaces of vector bundles by methods of GIT. In the boundary of the compactified moduli space appear non locally free sheaves. The thesis aims to propose a new stock of more manageable boundary objects, in the case of dimension 2 and rank 2, which are bundles on bubble trees A having S as root. Motivation comes from gauge theory and the study of bundles on reducible curves by Nagaraj-Seshadri and Teixidor i Bigas.The semistability on A depends on polarization, that is, on an ample line bundle. The domain of parameters of polarization is much smaller, and semistable bundles are more scarce in dimension 2 than in the case of curves. For certain polarizations, semistability criteria for bundles on A are given in terms of their restrictions to the components of A. Although the sheaves studied on A are bundles, their potentially destabilizing subsheaves can be just reflexive. Thence the classification of reflexive sheaves on bubble trees is undertaken, basing upon the work of Burban-Drozd. Next the deformations of tree-like bundles are studied. The main result is that a stable bundle on A, for certain polarizations, is always the limit of stable bundles on S. Finally, a comparison is made between the stock of stable tree-like bundles which are limits of instantons of charge 2 on the projective plane, and the one of Markushevich-Tikhomirov-Trautmann, obtained by a completely different approach

Let V be a complete discrete valuation ring of mixed characteristic (0,p), K be the fraction field and k be the residue field. We study p-adic differential equations on a semistable variety over V. We consider a proper semistable variety X over V and a relative normal crossing divisor D on it. We consider on X the open U defined by the complement of the divisor D and we call U_K and U_k the generic fiber and the special fiber respectively. In an analogous way we call D_K, X_K and D_k, X_k the generic and the special fiber of D, X. In the geometric situation described, we investigate the relations between algebraic differential equations on X_K and analytic differential equations on the rigid analytic space associated to the completion of X along its special fiber. The main result is the existence and the full faithfulness of an algebrization functor between the following categories: 1) the category of locally free overconvergent log isocrystals on the log pair (U_k,X_k), (where the log structure is defined by the divisor given by the union of X_k e D_k), with unipotent monodromy; 2) the category of modules with connection on U_K, regular along D_K, which admit an extension to modules with connection on X_K with nilpotent residue.
Sia V un anello di valutazione completo di caratteristica mista (0,p), sia K il campo delle frazioni e k il campo residuo. In questa tesi vengono studiate le equazioni differenziale p-adiche su una varieta' semistabile su V. Consideriamo una varieta' X propria e semistabile su V e un divisore D a incroci normali relativi, Denotiamo con U l'aperto di X definito dal complementare di D e indichiamo con U_K e U_k ripettivamente la fibra generica e la fibra speciale di U. Allo stesso modo chiamiamo X_K, D_K e X_k, D_k la fibra generica e la fibra speciale di X, D. In questa situazione geometrica studiamo le relazioni tra le equazioni differenziali algebriche su X_K e le equazioni differenziali analitiche definite sullo spazio analitico rigido associato al completamento di X lungo la sua fibra speciale. Il risultato principale di questa tesi e' l'esistenza e la piena fedelta' di un funtore tra le seguenti categorie: 1) la categoria dei log isocristalli localmente liberi surconvergenti definiti sulla log coppia (U_k,X_k), (dove la log e' definita dal divisore dato dall'unione di X_k e D_k), con monodromia unipotente; 2) la categoria dei moduli a connessione su U_K, regolari lungo D_K, che ammettono un' estensione a moduli a connessione su X_K con residuo nilpotente.

L'objet principal de cette thèse est de démontrer une inégalité d'Arakelov qui consiste à borner le degré d'un sous-faisceau inversible de l'image directe d'un faisceau relatif pluricanonique d'une famille semi-stable de courbes. Un problème naturel qui apparaît est la caractérisation des familles pour lesquelles sont satisfaites le cas d'égalité dans l'inégalité d'Arakelov, i.e. le cas d'égalité d'Arakelov. Peu d'exemples de telles familles sont connus. Dans cette thèse nous en proposons plusieurs en prouvant que le faisceau relatif bicanonique d'une famille semi-stable de courbes uniformisée par la boule unité et dont toutes les fibres singulières sont totalement géodésiques contient un sous-faisceau inversible qui satisfait l'égalité d'Arakelov
The main object of study in this thesis is an Arakelov inequality which bounds the degree of an invertible subsheaf of the direct image of the pluricanonical relative sheaf of a semistable family of curves. A natural problem that arises is the characterization of those families for which the equality is satisfied in that Arakelov inequality, i.e. the case of Arakelov equality. Few examples of such families are known. In this thesis we provide some examples by proving that the direct image of the bicanonical relative sheaf of a semistable family of curves uniformized by the unit ball, all whose singular fibers are totally geodesic, contains an invertible subsheaf which satisfies Arakelov equality

Dans cette thèse nous étudions deux problèmes dans le domaine de la géométrie arithmétique, concernant respectivement les points de torsion des variétés abéliennes et le polylogarithme motivique sur les schémas abéliens. La conjecture de Manin-Mumford (démontrée par Raynaud en 1983) affirme que si A est une variété abélienne et X est une sous-variété de A ne contenant aucune translatée d'une sous-variété abélienne de A, alors X ne contient qu'un nombre fini de points de torsion de A. En 1996, Buium présenta une forme effective de la conjecture dans le cas des courbes. Dans cette thèse, nous montrons que l'argument de Buium peut être utilisé aussi en dimension supérieure pour prouver une version quantitative de la conjecture pour une classe de sous-variétés avec fibré cotangent ample étudiée par Debarre. Nous généralisons aussi à toute dimension un résultat sur la dispersion des relèvements p-divisibles non ramifiés obtenu par Raynaud dans le cas des courbes. En 2014, Kings and Roessler ont montré que la réalisation en cohomologie de Deligne analytique de la part de degré zéro du polylogarithme motivique sur les schémas abéliens peut être reliée aux formes de torsion analytique de Bismut-Koehler du fibré de Poincaré. Dans cette thèse, nous utilisons la théorie de l'intersection arithmétique dans la version de Burgos pour raffiner ce résultat dans le cas où la base du schéma abélien est propre
In this thesis we approach two independent problems in the field of arithmetic geometry, one regarding the torsion points of abelian varieties and the other the motivic polylogarithm on abelian schemes. The Manin-Mumford conjecture (proved by Raynaud in 1983) states that if A is an abelian variety and X is a subvariety of A not containing any translate of an abelian subvariety of A, then X can only have a finite number of points that are of finite order in A. In 1996, Buium presented an effective form of the conjecture in the case of curves. In this thesis, we show that Buium's argument can be made applicable in higher dimensions to prove a quantitative version of the conjecture for a class of subvarieties with ample cotangent studied by Debarre. Our proof also generalizes to any dimension a result on the sparsity of p-divisible unramified liftings obtained by Raynaud in the case of curves. In 2014, Kings and Roessler showed that the realisation in analytic Deligne cohomology of the degree zero part of the motivic polylogarithm on abelian schemes can be described in terms of the Bismut-Koehler higher analytic torsion form of the Poincaré bundle. In this thesis, using the arithmetic intersection theory in the sense of Burgos, we give a refinement of Kings and Roessler's result in the case in which the base of the abelian scheme is proper

The context of this thesis is derived categories in algebraic geometry and geo- metric quotients. Specifically, we prove the embedding of the derived category of a smooth curve of genus greater than one into the derived category of the moduli space of semistable pairs over the curve. We also describe closed cover conditions under which the composition of a pullback and a pushforward induces a fully faithful functor. To prove our main result, we give an exposition of how to think of general Geometric Invariant Theory quotients as quotients by the multiplicative group.

碩士
國立臺灣大學
數學研究所
106
Abstract The aim of this article is to study Kieran G. O’Grady’s paper "Desingularized moduli spaces of sheaves on a K3" in 1998, where the author constructs the moduli space of rank two torsion-free semistable sheaves on a non-singular K3 surface with c1 = 0 and c2 = c a even number not less then 4. This moduli space is denoted by Mc, which is a G.I.T. quotient from the Quot-scheme and is singular. By using Kirwan’s method of successive blow ups of the strictly semistable loci with reductive stabilizer, one can obtain a desingularization Mcc of Mc. What’s surprising is that when c = 4, there is a Mori extremal divisorial contraction of Mc4 so that the outcome is a hyperk¨ahler manifold Mf4. Moreover, the natural map from Mf4 to M4 is a morphism and hence a simplectic desingularization of M4. The hyperk¨ahler manifold Mf4 is not birational/deformation equivalence to another two typical constructions of HK manifolds: the Hilbert schemes of points and Kummer varieties. Key words: moduli space of sheaves, semistable sheaves, geometric invariant theory, symplectic resolution, hyperk¨ahler variety.

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.