In this thesis, I study two problems. First, I generalize a result by H-Y Chen to show that if $X$ is a smooth variety of general type and irregularity $q\geq 1$ that embeds into its Albanese variety as a smooth variety $Y$ of general type with codimension one or two, then $|Aut(X)|\leq |Aut(F_{min})||Aut(Y)|$ where $F_{min}$ is the minimal model of a general fiber. Then I describe a special type of fibration called a K-Fibration as a generalization to Kodaira Fibrations where we can compute its Chern numbers in dimensions 2 and 3. K-Fibrations act as an initial step in constructing examples of varieties that satisfy the generalization with the goal of computing their automorphism group explicitly.

Une surface K3 est une surface X complexe compacte projective lisse qui a fibré canonique trivial et h0;1(X) = 0. Dans cette thèse on s'intéresse à deux problèmes pour ces surfaces. D'abord on considère des surfaces K3 obtenues comme recouvrement double de P2 ramifié le long d'une sextique. On classifie les fibrations elliptiques sur ces surfaces et leur groupe de Mordell-Weil, c'est-à-dire le groupe des sections. Vu que une section de 2-torsion définit une involution de la surface (dite involution de van Geemen-Sarti), alors en classifiant les fibrations et les section de 2-torsion on obtient une classification complète des involutions de van Geemen-Sarti sur ce type de surfaces K3. On montre aussi comment calculer l'équation de la fibration et on étudie le quotient par l'involution de van Geemen-Sarti. Ensuite on montre la construction de Berglund-Hübsch-Chiodo-Ruan (BHCR): il s'agit d'une construction miroir qui part d'un polynôme dans un espace projectif à poids et d'un groupe d'automorphismes (avec certaines propriétés) et qui donne, en toute dimension, des paires de variétés Calabi-Yau. Ces deux variétés sont l'une miroir de l'autre en sense classique. On classifie toutes les paires de surfaces K3 obtenues avec cette construction qui aient en plus un automorphisme non{symplectique d'ordre premier p > 3. Pour les surfaces K3 une autre notion de symétrie miroir a été introduite par Dolgachev et Nikulin : la symétrie pour K3 polarisées (LPK3). On montre dans la thèse comment polariser les surfaces obtenues avec la construction BHCR et on preuve que deux surfaces miroir au sense BHCR, dûment polarisées, appartiennent à deux familles miroir LPK3. / 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.

Submitted by TAÍSA FERNANDA DE LIMA QUEMEL null (taisafernanda.10@hotmail.com) on 2016-03-10T20:25:22Z No. of bitstreams: 1 Versão final_Dissertação_Taísa Quemel.pdf: 674351 bytes, checksum: 3498053a8bb53e50ac3119a10d45a0c5 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-03-11T12:17:58Z (GMT) No. of bitstreams: 1 quemel_tfl_me_sjrp.pdf: 674351 bytes, checksum: 3498053a8bb53e50ac3119a10d45a0c5 (MD5) / Made available in DSpace on 2016-03-11T12:17:58Z (GMT). No. of bitstreams: 1 quemel_tfl_me_sjrp.pdf: 674351 bytes, checksum: 3498053a8bb53e50ac3119a10d45a0c5 (MD5) Previous issue date: 2016-02-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho é mostrar que πn(X) é sempre abeliano quando n ≥ 2 e que π1(X) é abeliano quando X for um H-espaço e por fim calcular alguns grupos de homotopia utilizando sequência exata de uma fibração. / The goal of this work is to show that πn(X) is always abelian when n ≥ 2 and that π1(X) is abelian when X is an H-space and finally calculate some homotopy groups using the exact sequence of a fibration.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: In this thesis we develop a self-dual categorical approach to some topics in non-abelian algebra, which is based on replacing the framework of a category with that of a category equipped with a functor to it. We also make some first steps towards a possible link between this theory and the theory of databases in computer science. Both of these theories are based around the study of Grothendieck bifibrations and their generalisations. The main results in this thesis concern correspondences between certain structures on a category which are relevant to the study of categories of non-abelian group-like structures, and functors over that category. An investigation of these correspondences leads to a system of dual axioms on a functor, which can be considered as a solution to the proposal of Mac Lane in his 1950 paper "Duality for Groups" that a self-dual setting for formulating and proving results for groups be found. The part of the thesis concerned with the theory of databases is based on a recent approach by Johnson and Rosebrugh to views of databases and the view update problem. / AFRIKAANSE OPSOMMING: In hierdie tesis word ’n self-duale kategoriese benadering tot verskeie onderwerpe in nie-abelse algebra ontwikkel, wat gebaseer is op die vervanging van die raamwerk van ’n kategorie met dié van ’n kategorie saam met ’n funktor tot die kategorie. Ons neem ook enkele eerste stappe in die rigting van ’n skakel tussen hierdie teorie and die teorie van databasisse in rekenaarwetenskap. Beide hierdie teorieë is gebaseer op die studie van Grothendieck bifibrasies en hul veralgemenings. Die hoof resultate in hierdie tesis het betrekking tot ooreenkomste tussen sekere strukture op ’n kategorie wat relevant tot die studie van nie-abelse groep-agtige strukture is, en funktore oor daardie kategorie. ’n Verdere ondersoek van hierdie ooreemkomste lei tot ’n sisteem van duale aksiomas op ’n funktor, wat beskou kan word as ’n oplossing tot die voorstel van Mac Lane in sy 1950 artikel “Duality for Groups” dat ’n self-duale konteks gevind word waarin resultate vir groepe geformuleer en bewys kan word. Die deel van hierdie tesis wat met die teorie van databasisse te doen het is gebaseer op ’n onlangse benadering deur Johnson en Rosebrugh tot aansigte van databasisse en die opdatering van hierdie aansigte.

[pt] Nesta tese introduzimos a noção de estrutura de Poisson fibrada em um fibrado localmente trivial. Isto é uma estrutura de Poisson no espaço total da fibração com condições naturais de compatibilidade com respeito as fibras e bases de Poisson dadas. Nosso resultado principal é uma receita para produzir estruturas de Poisson fibradas fora de apropriadas (pares de) ações de Poisson de grupos de Lie. Aplicamos este resultado para produzir estruturas de Poisson fibradas com fibra e base uma variedade tórica ou uma órbita coadjunta, aumentando assim a classe de variedades de Poisson compactas com um número finito de folhas simpléticas. / [en] In this thesis we introduce the notion of fibered Poisson structure on a locally trivial fiber bundle. This is a Poisson structure on the total space of the fibration with natural compatibility conditions with respect to the given Poisson base and fiber. Our main result is a recipe to produce fibered Poisson structures out of appropriate (pairs of) Poisson actions of Lie groups. We apply this result to produce fibered Poisson structures with fiber and base either a toric variety or a coadjoint orbit, thus enlarging the class of compact Poisson manifolds with a finite number of symplectic leaves.

Orientador: Elizabeth Terezinha Gasparim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T08:45:07Z (GMT). No. of bitstreams: 1 Callander_Brian_M.pdf: 1926930 bytes, checksum: 341dd0f9759ced382e138cd14fc4ae2c (MD5) Previous issue date: 2013 / Resumo: O propósito desta tese é estudar fibrações de Lefschetz simpléticas, nas quais os ciclos evanescentes são subvariedades Lagrangianas das fibras. Para a descrição da teoria de interseção dos ciclos evanescentes utilizamos cohomologia de Floer Lagrangiana, cujo conceito revemos nesta tese. Apresentamos três exemplos principais e de caráteres distintos: (1) twists de Dehn generalizados, (2) o "espelho" da reta projetiva, e (3) uma fibração numa órbita adjunta de sl(3,C). O terceiro destes exemplos é original e utiliza um teorema recente de Gasparim- Grama-San Martin / Abstract: The objective of this thesis is to study symplectic Lefschetz fibrations, in which the vanishing cycles are Lagrangian submanifolds of the fibres. In order to describe the intersection theory of vanishing cycles we use Lagrangian intersection Floer cohomology, which we review. We present three main examples of distinct characters: (1) generalized Dehn twists, (2) the "mirror" of the projective line, and (3) a fibration on an adjoint orbit of sl(3,C). The third of these examples is original and uses a recent theorem of Gasparim- Grama-San Martin / Mestrado / Matematica / Mestre em Matemática

En employant des méthodes de la théorie de Chern-Weil, Reznikov produit une condition suffisante qui assure la non-trivialité de la projectivisation \( \mathbb{P}(E) \) d'un fibré vectoriel complexe en tant que fibré Hamiltonien. Dans le contexte de la quantification géométrique, Savelyev et Shelukhin introduisent un nouvel invariant des fibrés Hamiltoniens avec valeurs dans la K-théorie et étendent le résultat de Reznikov. Cet invariant est donné par l'indice d'Atiyah-Singer d'une famille d'opérateurs \( \text{Spin}^{c} \) de Dirac. Dans ce mémoire, on s'intéresse à des fibrés Hamiltoniens résultant d'un produit fibré et d'un produit cartésien d'une collection de fibrés projectifs complexes \( \mathbb{P}(E_1), \cdots, \mathbb{P}(E_r) \). En usant des mêmes méthodes que Shelukhin et Savelyev, on définit une famille d'opérateurs \( \text{Spin}^{c} \) de Dirac qui agissent sur les sections d'un fibré de Dirac canonique à valeurs dans un fibré pré-quantique. L'indice de famille produit un invariant de fibrés Hamiltoniens avec fibres données par un produit d'espaces projectifs complexes et permet de construire des exemples de fibrés Hamiltoniens non-triviaux. / Using methods of Chern-Weil Theory, Reznikov provides a sufficient condition for the non-triviality of the projectivization \( \mathbb{P}(E) \) of a complex vector bundle \( E \) as a Hamiltonian fibration. In the setting of geometric quantization, Savelyev and Shelukhin introduce a new invariant of Hamiltonian fibrations and a K-theoretic lift of Reznikov's result. This invariant is given by the Atiyah-Singer index of a family of \( \text{Spin}^{c} \)-Dirac operators. In this thesis, we consider Hamiltonian fibrations given by the Cartesian product and the fiber product of a collection of complex projective bundles \( \mathbb{P}(E_1), \cdots, \mathbb{P}(E_r) \). Using the same methods as Savelyev and Shelukhin, we define a family of \( \text{Spin}^{c} \)-Dirac operators acting on sections of a canonical Dirac bundle with values in a suitable prequantum fibration. The family index gives then an invariant of Hamiltonian fibrations with fibers given by a product of complex projective spaces and allows to construct examples of non-trivial Hamiltonian fibrations.

Spectral sequences were developed during the mid-twentieth century as a way of computing (co)homology, and have wide uses in both algebraic topology and algebraic geometry.  Grothendieck introduced in his Tôhoku paper the Grothendieck spectral sequence, which given left exact functors $F$ and $G$ between abelian categories, uses the right-derived functors of $F$ and $G$ as initial data and converges to the right-derived functors of the composition $G\circ F.$  This thesis focuses on instead constructing a spectral sequence that uses the derived functors of $G$ and $G\circ F$ as initial data and converges to the derived functors of $F.$ Our approach takes inspiration from the construction of the Eilenberg-Moore spectral sequence, which given a fibration of topological spaces can calculate the singular cohomology of the fiber from the singular cohomology of the base space and total space. The Eilenberg-Moore spectral sequence can be constructed through the use of differential graded algebras and their bar construction, since this defines a double complex for which the column-wise filtration of the corresponding total complex induces the spectral sequence. The correct analogue of this with respect to composite functors is the bar construction for monads. Specifically, we let $G$ have an exact left adjoint $H$, which makes $G\circ H$ into a monad. Then, we extend our adjunction so that the derived functor $RG$ has left adjoint $RH$ in the corresponding derived categories, making $RG\circ RH$ into a monad. This allows us to apply the bar construction in the derived category, but we show that there emerge issues in obtaining a double complex and subsequent total complex from this construction.  Additionally, we present the essential theory of spectral sequences in general, and of the Serre, Eilenberg-Moore and Grothendieck spectral sequences in particular. / Spektralsekvenser utvecklades under mitten av 1900-talet som ett verktyg för att beräkna (ko)homologi, och har många användningsområden inom både algebraisk topologi och algebraisk geometri. Grothendieck introducerade i sin Tôhoku-artikel Grothendieck-spektralsekvensen, som givet vänsterexakta funktorer $F$ och $G$ mellan abelska kategorier använder de högerderiverade funktorerna av $F$ och $G$ som initialdata och som konvergerar till de högerderiverade funktorerna av kompositionen $G\circ F$. Denna masteruppsats fokuserar på att istället konstruera en spektralsekvens som använder de deriverade funktorerna av $G$ och $G\circ F$ som initialdata och konvergerar till de deriverade funktorerna av $F$. Vår metod tar inspiration från konstruktionen av Eilenberg-Moore-spektralsekvensen, som givet en fibrering av topologiska rum kan beräkna den singulära kohomologin av fibern från den singulära kohomologin av basrummet och totalrummet. Eilenberg-Moore spektralsekvensen kan konstrueras genom användningen av graderade differentialalgebror och deras bar-konstruktion, eftersom detta definierar ett dubbelkomplex vars kolumnvisa filtrering av det resulterande totalkomplexet inducerar spektralsekvensen. Vad gäller kompositioner av funktorer så är den korrekta analogin till detta bar-konstruktionen för monader. Specifikt så låter vi $G$ ha en exakt vänsteradjungerad funktor $H$, vilket gör $G\circ H$ till en monad. Sedan utvidgar vi denna adjunktion sådant att den deriverade funktorn $RG$ har vänsteradjunkt $RH$ i den deriverade kategorin, vilket gör $RG\circ RH$ till en monad. Detta ger oss möjligheten att använda bar-konstruktionen i den deriverade kategorin, men vi visar att det uppstår problem när vi ska definiera ett dubbelkomplex och resulterande totalkomplex från denna konstruktion. Utöver detta så innehåller denna uppsats en genomgång av den viktigaste teorin om spektralsekvenser i allmänhet, och om Serre-, Eilenberg-Moore- och Grothendieck-spektralsekvensen i synnerhet.

Neste projeto apresentamos algumas direções de pesquisa desenvolvidas no estudo da geometria/topologia da singularidade, no ambiente real e complexo, para funções e aplicações polinomiais. Para isso, utilizaremos as ferramentas da teoria de estratificação, técnicas de decomposição Open book, condições de regularidade no sentido Malgrange, t-regularidade, \'rho\'E-regularidade e trivialidade topológica no infinito / On this project we present some research lines developed in the study of the geometry/ topology of singularity, on the real and complex settings, for functions and polynomial mappings. For this, we use tools from stratification theory, techniques of Open Book decomposition, Malgrange regularity condition, t-regularity condition, \'rho\'E-regularity and topological triviality at infinity

Nos últimos anos, váarios pesquisadores tais como: A. Bodin, A. Dimca, A. Durfee, A. Jacquemard, A. Menegon Neto, A. Némethi, A. Pichon, A. Verjovsky, A. Zaharia, D. Siersma, H. A. Hamm, D. Massey, H. Aguilar-Cabrera, H. H. Vui, J. Cisneros, J. Seade, J. Snoussi, L. D. Tráng, L. Paunescu, L. R. Dias, M. A. S. Ruas, M. Oka, M. Tibar, N. Dutertre, R. N. Araújo dos Santos, S. A. Broughton, T. Gaffney, Y. Chen, entre outros, têm apresentado generalizações dos Teoremas de fibrações de Milnor no ambiente real e complexo (e do Teorema de Kurdyka-Orro-Simon, ver por exemplo [Di, KOS]), visando um melhor entendimento de propriedades topológicas locais e globais das singularidades. Nesta direção de pesquisa esses autores tem utilizado várias ferramentas e técnicas de diversas áreas da matemática. O que mostra a riqueza e a complexidade destes estudos e acrescenta, em nossa modesta opinião, um aspecto que é ao mesmo tempo interessante e desafiador. Neste trabalho, mostraremos como estender as fibrações de Milnor em esferas no caso local e global, real e complexo, para uma aplicação C2-semi-algébrica F = (f1, . . . , fp) : RN → Rp e uma variedade W ⊂ RN semi-algébrica com possível singularidade. Com tal objetivo, introduziremos as condições de Milnor (a) e (b) generalizadas\" e mostraremos como adaptar a técnica da decomposição open book superior com binding singular, introduzida em [AT, ACT1]. Nossos resultados sugerem que tal estrutura de fibração pode ser um caso particular de algum Teorema estrutural mais geral. Além do mais, considerando π : Rp → Rp-1 a projeção canônica na meta, mostraremos que se F satisfaz tais condições, então G = π o F : RN → Rp-1 também satisfaz e, consequentemente, G também induz em W uma fibração suave localmente trivial. Concluiremos mostrando que após as projeções as fibras destes fibrados são homotopicamente equivalentes e, em seguida, apresentando algumas fórmulas que relacionam a característica de Euler do \"link relativo\" W ∩ F-1 (0) com a característica de Euler das fibras. / In the last years, several researchers such as: A. Bodin, A. Dimca, A. Durfee, A. Jacquemard, A. Menegon Neto, A. Némethi, A. Pichon, A. Verjovsky, A. Zaharia, D. Siersma, H. A. Hamm, D. Massey, H. Aguilar-Cabrera, H. H. Vui, J. Cisneros, J. Seade, J. Snoussi, L. D. Trang, L. Paunescu, L. R. Dias, M. A. S. Ruas, M. Oka, M. Tibar, N. Dutertre, R. N. Araújo dos Santos, S. A. Broughton, T. Gaffney, Y. Chen, and others, have proven generalizations of Milnor fibrationss Theorems in the real and complex settings (and Kurdyka-Orro-Simons Theorem, see e.g. [Di, KOS]), aiming a better understanding of the local and global topological properties of singularity. In this research branch, these authors have used many different tools and techniques from several areas of Mathematics. This shows the richness and complexities of these studies and adds, in our modest opinion, an aspect that is simultaneously interesting and challenging. In this work, we introduce the generalized Milnors conditions (a) and (b) to show an extension of the Milnor fibration Theorems on spheres in the local and global cases, in the real and complex setting. For this, we consider a C2-semi-algebraic mapping F = (f1, . . . , fp) : RN → Rp , a possible singular semi-algebraic variety W ⊂ RN, and we show how to adapt the technique of Higher open book decomposition with singular binding, introduced by [AT, ACT1], to prove such extension. Our results suggest that such fibration structure may be a particular case of a more general fibration structure. Furthermore, considering : Rp Rp-1 the canonical projection on the target space, we show that if F satisfies the generalized Milnors conditions (a) and (b), then G = π o F : RN → Rp-1 also satisfies these conditions and, hence G also induces on W a smooth locally trivial fibration. Finally, we show that after the projections on the target space, the fibers of these fiber bundles are homotopically equivalent. We conclude by proving some formulae connecting the Euler characteristic of \"relative link\" W ∩ F-1 (0) with the Euler characteristic of the fibers. Key words and phrases: generalized open book decomposition, fibration structure on semi-algebraic sets, topology of singularity, real and complex Milnors fibrations and, local and global fibration.