我们研究了P[subscript g]=0 的复曲面x 上的ADE 向量丛和仿射ADE 向量丛。 / 首先，我们假设x 上有一个ADE 奇异点。这个奇异点在极小分解Y 中的例外轨迹是一条相应形式的ADE 曲线。利用这条ADE 曲线和向量丛的扩张，我们构造了Y 上的一个ADE 向量丛，而且这个向量丛可以下降到x上。此外，我们利用Y 上( -1)- 曲线的组合，描述了他们的极小表示向量丛。 / 其次，我们假设x 是一个椭圆曲面，而且x 上有一个仿射ADE 形式的奇异纤维。类似于以前，我们构造了X 上的一个仿射ADE 向量丛，而且这个向量丛在这条仿射ADE 曲线上的每一个不可约成分上都是平凡的。 / 然后，当X 是P²上突起n ≤9 个点时, x 上有一个典型的En 向量丛。我们详细的研究了x 的几何和这个E[subscript n] 向量丛的可变形性之间的关系。 / We study ADE and affine ADE bundles over complex surfaces X with P[subscript g] = 0. / First, we suppose X admits an ADE singularity. The exceptional locus of this singularity in the minimal resolution Y is an ADE curve of corresponding type. Using this ADE curve and bundle extensions, we construct an ADE bundle over Y which can descend to X. Furthermore, we describe their minuscule representation bundles in terms of configuration of (reducible) (-1)-curves. / Second, we assume X is an elliptic surface with a singular fiber of affine ADE type. Similar to above studies, we construct the affine ADE bundle over X which is trivial on each irreducible component of the affine ADE curve. / Third, when X is the blowup of P² at n ≤9 points, there is a canonical E[subscript n] bundle over it. We give a detailed study of the relationship between the geometry of X and the deformability of this bundle. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chen, Yunxia. / On t.p. "g" is subscript. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 84-87). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Chapter I --- ADE bundles --- p.9 / Chapter 1 --- ADE Lie algebra bundles --- p.10 / Chapter 1.1 --- ADE singularities --- p.10 / Chapter 1.2 --- ADE bundles --- p.12 / Chapter 2 --- Minuscule representations and ( -1)-curves --- p.16 / Chapter 2.1 --- Standard representations --- p.16 / Chapter 2.2 --- Minuscule representations --- p.17 / Chapter 2.3 --- Configurations of ( -1)-curves --- p.17 / Chapter 2.4 --- Minuscule representations from ( -1)-curves --- p.19 / Chapter 2.5 --- Bundles from ( -1)-curves --- p.21 / Chapter 2.6 --- Outline of Proofs for g ≠E₈ --- p.22 / Chapter 3 --- A[subscript n] case --- p.24 / Chapter 3.1 --- A[subscript n] standard representation bundle Lη^(An,Cn+1) --- p.24 / Chapter 3.2 --- An Lie algebra bundle Sη^(An) --- p.28 / Chapter 3.3 --- An minuscule representation bundle Lη^(An,^kCn+1) --- p.28 / Chapter 4 --- Dn case --- p.30 / Chapter 4.1 --- Dn standard representation bundle Lη^(Dn;C2n) --- p.30 / Chapter 4.2 --- Dn Lie algebra bundle Sη^(Dn) --- p.34 / Chapter 4.3 --- Dn spinor representation bundles Lη^(Dn;S±06) --- p.34 / Chapter 5 --- En case --- p.39 / Chapter 5.1 --- E₆ case --- p.39 / Chapter 5.2 --- E₇ case --- p.42 / Chapter 5.3 --- E₈ case --- p.44 / Chapter 6 --- Proof of Theorem 1.2.1 --- p.45 / Chapter II --- Affine ADE bundles --- p.50 / Chapter 7 --- Affine ADE Lie algebra bundles --- p.51 / Chapter 7.1 --- Affine ADE curves --- p.51 / Chapter 7.2 --- Affine ADE bundles --- p.53 / Chapter 8 --- Trivialization of E₀ gover Ci's after deformations --- p.57 / Chapter 8.1 --- Trivializations in loop ADE cases --- p.58 / Chapter 8.2 --- Trivializations in affine ADE cases --- p.60 / Chapter 8.3 --- Proof (except the loop E₈ case) --- p.60 / Chapter 8.4 --- Proof for the loop E₈ case --- p.62 / Chapter III --- Deformability --- p.65 / Chapter 9 --- En-bundle over Xn with n≤9 --- p.66 / Chapter 9.1 --- En-bundle over Xn with n ≤ 9 --- p.66 / Chapter 9.2 --- Deformability of such E₀E₈ --- p.68 / Chapter 9.3 --- Negative curves in X9 --- p.70 / Chapter 9.4 --- Proof of Theorems 9.2.1 and 9.2.2 --- p.75 / Chapter A --- Minuscule configurations --- p.78 / Chapter B --- A ffine Lie algebras --- p.80

Orientador: Adriano Adrega de Moura / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T09:20:36Z (GMT). No. of bitstreams: 1 Ferreira_GilmardeSousa_M.pdf: 480832 bytes, checksum: 42db64084fe99183a4fb95f90cd9a833 (MD5) Previous issue date: 2012 / Resumo: Nessa dissertação estudamos certas categorias de módulos graduados para uma classe de álgebras de Lie que inclui as álgebras de correntes. Em particular, estudamos diversas propriedades homológicas dessas categorias tais como resoluções projetivas e o espaço de extensões entre seus objetos simples. Em certas situações, os resultados levam ao estabelecimento de um relacionamento com álgebras de Koszul. O estudo é baseado em artigos recentes de Vyjayanthi Chari e seus co-autores / Abstract: In this dissertation, we study certain categories of graded modules for a class of Lie algebras which include current algebras. In particular, we study several homological properties of these categories such as projective resolutions and the space of extensions between two given simple objects. Under certain conditions, these results establish a relationship with Koszul algebras. The study is based on recent papers by Vyjayanthi Chari and her co-authors / Mestrado / Matematica / Mestre em Matemática