[pt] Espaços de moduli de polígonos em R(3) com comprimento dos lados fixados é um exemplo amplamente estudado de variedade simplética. Esses espaços podem ser descritos como quociente simplético de um número finito de órbitas coadjuntas pelo grupo SU(2). Nesta tese esses espaços de moduli são identificados como folhas simpléticas de uma variedade de Poisson que pode ser construída como quociente. Essa construção é a seguir generalizada ao caso de um produto de um número finito de órbitas coadjuntas pelo grupo SU(n), e o resultado principal desse trabalho de tese descreve como esses espaços de moduli de polígonos generalizados formam uma folheação em folhas simpléticas de uma variedade de Poisson. / [en] Moduli spaces of polygons in R(3)with fixed sides length are a widely studied example of symplectic manifold that can be described as the symplectic quotient of a finite number of SU(2)−coadjoint orbits by the diagonal action of the group SU(2). In this thesis these spaces are identified as the symplectic leaves of a Poisson manifold, that can itself be obtained by a quotient procedure. The construction is then generalized to the case of the quotient of a product of finitely many SU(n)−coadjoint orbits by the diagonal action of SU(n), and the main result of this thesis describes how these moduli spaces of generalized polygons fit together as the symplectic leaves of a quotient Poisson manifold.

Nesta disertação estudamos algumas ferramentas básicas relacionadas aos espaços de Teichmüller. Introduzimos o espaço de Teichmüller de gênero g ≥ 1, denotado por Tg. O objetivo principal é construir as coordenadas de Fenchel-Nielsen ωG : Tg → R3g-3+ × R3g-3 para cada grafo trivalente marcado G. / In this dissertation we study some basic tools related to Teichmüller space. We introduce the Teichmüller space of genus g ≥ 1, denoted by Tg. The main goal is to construct the Fenchel-Nielsen coordinates ωG : Tg → R3g-3+ × R3g-3 to each marked cubic graph G.

Nesta disertação estudamos algumas ferramentas básicas relacionadas aos espaços de Teichmüller. Introduzimos o espaço de Teichmüller de gênero g ≥ 1, denotado por Tg. O objetivo principal é construir as coordenadas de Fenchel-Nielsen ωG : Tg → R3g-3+ × R3g-3 para cada grafo trivalente marcado G. / In this dissertation we study some basic tools related to Teichmüller space. We introduce the Teichmüller space of genus g ≥ 1, denoted by Tg. The main goal is to construct the Fenchel-Nielsen coordinates ωG : Tg → R3g-3+ × R3g-3 to each marked cubic graph G.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-26T14:32:26Z No. of bitstreams: 1 juanantoniopachecocruz.pdf: 674238 bytes, checksum: 5fbe428a7cb6ca56e7ceb6582082376f (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-26T15:14:36Z (GMT) No. of bitstreams: 1 juanantoniopachecocruz.pdf: 674238 bytes, checksum: 5fbe428a7cb6ca56e7ceb6582082376f (MD5) / Made available in DSpace on 2017-05-26T15:14:36Z (GMT). No. of bitstreams: 1 juanantoniopachecocruz.pdf: 674238 bytes, checksum: 5fbe428a7cb6ca56e7ceb6582082376f (MD5) Previous issue date: 2015-11-19 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Um complexo de retas quadrático, ou simplesmente um complexo quadrático, é um conjunto de retas do espaço projetivo Pn (n = 3, no nosso caso) que satisfazem uma equação quadrática. Um complexo quadrático também pode ser considerado como um feixe de quádricas e portanto tem um símbolo de Segre bem definido. Sabe-se que as retas de um dado complexo, passando por um ponto p ∈P3, formam em geral um cone quadrático. Os pontos nos quais esses cones são a união de dois planos formam uma superfície em P3, chamada Superfície Singular do complexo. O objetivo desse trabalho é, fixado um símbolo de Segre, construir o espaço de Moduli dos complexos quadráticos, o espaço de Moduli das superfícies singulares desses complexos e então estudar a relação entre esse espaços. / A quadratic line complex, or a quadratic complex, is by definition a set of lines in a projective space Pn (n = 3, in our case) which satisfy a given quadratic equation. A quadratic complex can also be considered as a pencil of quadrics. Hence, it has a well defined Segre symbol. It is a classical fact that lines of a given complex through any point p ∈P3 form in general a quadratic cone. The points such that theses cones break up into two planes form a surface, the Singular Surface of the complex. The objective of this work is, for a fixed Segre symbol, to construct the Moduli space of quadratic complex, the Moduli space of corresponding singular surfaces and to study the relation between them.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Rank 2 Drinfeld modular forms have been studied for more than 30 years, and while it is known that a higher rank theory could be possible, higher rank Drinfeld modular forms have only recently been de ned. In 1988 Gekeler published [Ge2] in which he studies the coe cients of rank 2 Drinfeld modular forms. The goal of this thesis is to perform a similar study of the coe cients of higher rank Drinfeld modular forms. The main results are that the coe cients themselves are (weak) Drinfeld modular forms, a product formula for the discriminant function, the rationality of certain naturally de ned modular forms, and the computation of some Hecke eigenforms and their eigenvalues. / AFRIKAANSE OPSOMMING: Drinfeld modulêre vorme van rang 2 word al vir meer as 30 jaar bestudeer en alhoewel dit lankal bekend is dat daar Drinfeld modulêre vorme van hoër rang moet bestaan, is die de nisie eers onlangs vasgepen. In 1988 het Gekeler die artikel [Ge2] gepubliseer waarin hy die koeffisiënte van Fourier reekse van rang 2 Drinfeld modulêre vorme bestudeer. Die doel van hierdie proefskrif is om dieselfde studie vir Drinfeld modulêre vorme van hoër rang uit te voer. Die hoofresultate is dat die koeffi siënte self (swak) Drinfeld modulêre vorme is, `n produk formule vir die diskriminant funksie, die feit dat sekere natuurlik gede finiëerde modulêre vorme rasionaal is, en die vasstelling van Hecke eievorme en hul eiewaardes.

Understanding the action of an autoequivalence on a triangulated category is generally a very difficult problem. If one can find a stability condition for which the autoequivalence is "compatible", one can explicitly write down the action of this autoequivalence. In turn, the now understood autoequivalence can provide ways of extracting geometric information from the stability condition. In this thesis, we elaborate on what it means for an autoequivalence and stability condition to be "compatibile" and derive a sufficiency criterion. / text

This thesis is on moduli spaces of complexes of sheaves and diagrams of such moduli spaces. The objects in these diagrams are constructed as geometric invariant theory quotients and the points in these quotients correspond to certain equivalence classes of complexes. The morphisms in these diagrams are constructed by taking direct sums with acyclic complexes. We then study the colimit of such a diagram and in particular are interested in studying the images of quasi-isomorphic complexes in the colimit. As part of this thesis we construct categorical quotients of a group action on unstable strata appearing in a stratification associated to a complex projective scheme with a reductive group action linearised by an ample line bundle. We study this stratification for a closed subscheme of a quot scheme parametrising quotient sheaves over a complex projective scheme and relate the Harder-Narasimhan types of unstable sheaves with the unstable strata in the associated stratification. We also study the stratification of a parameter space for complexes with respect to a linearisation determined by certain stability parameters and show that a similar result holds in this case. The objects in these diagrams are indexed by different Harder-Narasimhan types for complexes and are quotients of parameter schemes for complexes of this fixed Harder-Narasimhan type. This quotient is given by a choice of linearisation of the action and so the diagrams depend on these choices. We conjecture that these choices can be made so that for any quasi-isomorphism between complexes representing points in this diagram both complexes are identified in the colimit of this diagram.

In this thesis we investigate wall-crossing phenomena in the stability manifold of an irreducible principally polarized abelian surface for objects with the same invariants as (twists of) ideal sheaves of points. In particular, we construct a sequence of fine moduli spaces which are related by Mukai flops and observe that the stability of these objects is completely determined by the configuration of points. Finally, we use Fourier-Mukai theory to show that these moduli are projective.

The topic of this thesis is the moduli theory of (parabolic) sheaves on stable curves. Using geometric invariant theory (GIT), universal moduli spaces of semistable parabolic sheaves on stable marked curves are constructed: `universal' indicates that these are moduli spaces of pairs where the underlying marked curve may vary as well as the parabolic sheaf (as in the Pandharipande moduli space for pairs of stable curves and torsion-free sheaves without augmentations). As an intermediate step in this construction, we construct moduli spaces of semistable parabolic sheaves on flat families of arbitrary projective schemes (of any dimension or singularity type): this is the technical core of this thesis. These moduli spaces are projective, since they are constructed as GIT quotients of projective parameter spaces. The stability condition for parabolic sheaves depends on a choice of polarisation and is derived from the Hilbert-Mumford criterion. It is not quite the same as traditional stability with respect to parabolic Hilbert polynomials, but it is closely related to it, and the resulting moduli spaces are always compactifications of moduli of slope-stable parabolic sheaves. The construction works over algebraically closed fields of arbitrary characteristic.

This thesis is mainly concerned with the study of hypersymplectic structures in gauge theory. These structures arise via applications of the hypersymplectic quotient construction to the action of the gauge group on certain spaces of connections and Higgs fields. Motivated by Kobayashi-Hitchin correspondences in the case of hyperkähler moduli spaces, we first study the relationship between hypersymplectic, complex and paracomplex quotients in the spirit of Kirwan's work relating Kähler quotients to GIT quotients. We then study dimensional reductions of the ASD equations on $mathbb R^{2,2}$. We discuss a version of twistor theory for hypersymplectic manifolds, which we use to put the ASD equations into Lax form. Next, we study Schmid's equations from the viewpoint of hypersymplectic quotients and examine the local product structure of the moduli space. Then we turn towards the integrability aspects of this system. We deduce various properties of the spectral curve associated to a solution and provide explicit solutions with cyclic symmetry. Hitchin's harmonic map equations are the split signature analogue of the self-duality equations on a Riemann surface, in which case it is known that there is a smooth hyperkähler moduli space. In the case at hand, we cannot expect to obtain a globally well-behaved moduli space. However, we are able to construct a smooth open set of solutions with small Higgs field, on which we then analyse the hypersymplectic geometry. In particular, we exhibit the local product structures and the family of complex structures. This is done by interpreting the equations as describing certain geodesics on the moduli space of unitary connections. Using this picture we relate the degeneracy locus to geodesics with conjugate endpoints. Finally, we present a split signature version of the ADHM construction for so-called split signature instantons on $S^2 imes S^2$, which can be given an interpretation as a hypersymplectic quotient.