Vector Bundles
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Connections on Bundles Md
Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II. -
Vector Bundles on Projective Space
Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X. -
Stable Isomorphism Vs Isomorphism of Vector Bundles: an Application to Quantum Systems
Alma Mater Studiorum · Universita` di Bologna Scuola di Scienze Corso di Laurea in Matematica STABLE ISOMORPHISM VS ISOMORPHISM OF VECTOR BUNDLES: AN APPLICATION TO QUANTUM SYSTEMS Tesi di Laurea in Geometria Differenziale Relatrice: Presentata da: Prof.ssa CLARA PUNZI ALESSIA CATTABRIGA Correlatore: Chiar.mo Prof. RALF MEYER VI Sessione Anno Accademico 2017/2018 Abstract La classificazione dei materiali sulla base delle fasi topologiche della materia porta allo studio di particolari fibrati vettoriali sul d-toro con alcune strut- ture aggiuntive. Solitamente, tale classificazione si fonda sulla nozione di isomorfismo tra fibrati vettoriali; tuttavia, quando il sistema soddisfa alcune assunzioni e ha dimensione abbastanza elevata, alcuni autori ritengono in- vece sufficiente utilizzare come relazione d0equivalenza quella meno fine di isomorfismo stabile. Scopo di questa tesi `efissare le condizioni per le quali la relazione di isomorfismo stabile pu`osostituire quella di isomorfismo senza generare inesattezze. Ci`onei particolari casi in cui il sistema fisico quantis- tico studiato non ha simmetrie oppure `edotato della simmetria discreta di inversione temporale. Contents Introduction 3 1 The background of the non-equivariant problem 5 1.1 CW-complexes . .5 1.2 Bundles . 11 1.3 Vector bundles . 15 2 Stability properties of vector bundles 21 2.1 Homotopy properties of vector bundles . 21 2.2 Stability . 23 3 Quantum mechanical systems 28 3.1 The single-particle model . 28 3.2 Topological phases and Bloch bundles . 32 4 The equivariant problem 36 4.1 Involution spaces and general G-spaces . 36 4.2 G-CW-complexes . 39 4.3 \Real" vector bundles . 41 4.4 \Quaternionic" vector bundles . -
LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The
LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class of fibrations given by fiber bundles. Fiber bundles play an important role in many geometric contexts. For example, the Grassmaniann varieties and certain fiber bundles associated to Stiefel varieties are central in the classification of vector bundles over (nice) spaces. The fact that fiber bundles are examples of Serre fibrations follows from Theorem ?? which states that being a Serre fibration is a local property. 1. Fiber bundles and principal bundles Definition 6.1. A fiber bundle with fiber F is a map p: E ! X with the following property: every ∼ −1 point x 2 X has a neighborhood U ⊆ X for which there is a homeomorphism φU : U × F = p (U) such that the following diagram commutes in which π1 : U × F ! U is the projection on the first factor: φ U × F U / p−1(U) ∼= π1 p * U t Remark 6.2. The projection X × F ! X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F . By definition, a fiber bundle is a map which is `locally' homeomorphic to a trivial bundle. The homeomorphism φU in the definition is a local trivialization of the bundle, or a trivialization over U. Let us begin with an interesting subclass. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R ! S1 is a covering projection with fiber Z. Suppose X is a space which is path-connected and locally simply connected (in fact, the weaker condition of being semi-locally simply connected would be enough for the following construction). -
Formal GAGA for Good Moduli Spaces
FORMAL GAGA FOR GOOD MODULI SPACES ANTON GERASCHENKO AND DAVID ZUREICK-BROWN Abstract. We prove formal GAGA for good moduli space morphisms under an assumption of “enough vector bundles” (which holds for instance for quotient stacks). This supports the philoso- phy that though they are non-separated, good moduli space morphisms largely behave like proper morphisms. 1. Introduction Good moduli space morphisms are a common generalization of good quotients by linearly re- ductive group schemes [GIT] and coarse moduli spaces of tame Artin stacks [AOV08, Definition 3.1]. Definition ([Alp13, Definition 4.1]). A quasi-compact and quasi-separated morphism of locally Noetherian algebraic stacks φ: X → Y is a good moduli space morphism if • (φ is Stein) the morphism OY → φ∗OX is an isomorphism, and • (φ is cohomologically affine) the functor φ∗ : QCoh(OX ) → QCoh(OY ) is exact. If φ: X → Y is such a morphism, then any morphism from X to an algebraic space factors through φ [Alp13, Theorem 6.6].1 In particular, if there exists a good moduli space morphism φ: X → X where X is an algebraic space, then X is determined up to unique isomorphism. In this case, X is said to be the good moduli space of X . If X = [U/G], this corresponds to X being a good quotient of U by G in the sense of [GIT] (e.g. for a linearly reductive G, [Spec R/G] → Spec RG is a good moduli space). In many respects, good moduli space morphisms behave like proper morphisms. They are uni- versally closed [Alp13, Theorem 4.16(ii)] and weakly separated [ASvdW10, Proposition 2.17], but since points of X can have non-proper stabilizer groups, good moduli space morphisms are gen- erally not separated (e.g. -
256B Algebraic Geometry
256B Algebraic Geometry David Nadler Notes by Qiaochu Yuan Spring 2013 1 Vector bundles on the projective line This semester we will be focusing on coherent sheaves on smooth projective complex varieties. The organizing framework for this class will be a 2-dimensional topological field theory called the B-model. Topics will include 1. Vector bundles and coherent sheaves 2. Cohomology, derived categories, and derived functors (in the differential graded setting) 3. Grothendieck-Serre duality 4. Reconstruction theorems (Bondal-Orlov, Tannaka, Gabriel) 5. Hochschild homology, Chern classes, Grothendieck-Riemann-Roch For now we'll introduce enough background to talk about vector bundles on P1. We'll regard varieties as subsets of PN for some N. Projective will mean that we look at closed subsets (with respect to the Zariski topology). The reason is that if p : X ! pt is the unique map from such a subset X to a point, then we can (derived) push forward a bounded complex of coherent sheaves M on X to a bounded complex of coherent sheaves on a point Rp∗(M). Smooth will mean the following. If x 2 X is a point, then locally x is cut out by 2 a maximal ideal mx of functions vanishing on x. Smooth means that dim mx=mx = dim X. (In general it may be bigger.) Intuitively it means that locally at x the variety X looks like a manifold, and one way to make this precise is that the completion of the local ring at x is isomorphic to a power series ring C[[x1; :::xn]]; this is the ring where Taylor series expansions live. -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
Differential Geometry of Complex Vector Bundles
DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES by Shoshichi Kobayashi This is re-typesetting of the book first published as PUBLICATIONS OF THE MATHEMATICAL SOCIETY OF JAPAN 15 DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES by Shoshichi Kobayashi Kan^oMemorial Lectures 5 Iwanami Shoten, Publishers and Princeton University Press 1987 The present work was typeset by AMS-LATEX, the TEX macro systems of the American Mathematical Society. TEX is the trademark of the American Mathematical Society. ⃝c 2013 by the Mathematical Society of Japan. All rights reserved. The Mathematical Society of Japan retains the copyright of the present work. No part of this work may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copy- right owner. Dedicated to Professor Kentaro Yano It was some 35 years ago that I learned from him Bochner's method of proving vanishing theorems, which plays a central role in this book. Preface In order to construct good moduli spaces for vector bundles over algebraic curves, Mumford introduced the concept of a stable vector bundle. This concept has been generalized to vector bundles and, more generally, coherent sheaves over algebraic manifolds by Takemoto, Bogomolov and Gieseker. As the dif- ferential geometric counterpart to the stability, I introduced the concept of an Einstein{Hermitian vector bundle. The main purpose of this book is to lay a foundation for the theory of Einstein{Hermitian vector bundles. We shall not give a detailed introduction here in this preface since the table of contents is fairly self-explanatory and, furthermore, each chapter is headed by a brief introduction. -
Vector Bundles. Characteristic Classes. Cobordism. Applications
Math 754 Chapter IV: Vector Bundles. Characteristic classes. Cobordism. Applications Laurenţiu Maxim Department of Mathematics University of Wisconsin [email protected] May 3, 2018 Contents 1 Chern classes of complex vector bundles 2 2 Chern classes of complex vector bundles 2 3 Stiefel-Whitney classes of real vector bundles 5 4 Stiefel-Whitney classes of manifolds and applications 5 4.1 The embedding problem . .6 4.2 Boundary Problem. .9 5 Pontrjagin classes 11 5.1 Applications to the embedding problem . 14 6 Oriented cobordism and Pontrjagin numbers 15 7 Signature as an oriented cobordism invariant 17 8 Exotic 7-spheres 19 9 Exercises 20 1 1 Chern classes of complex vector bundles 2 Chern classes of complex vector bundles We begin with the following Proposition 2.1. ∗ ∼ H (BU(n); Z) = Z [c1; ··· ; cn] ; with deg ci = 2i ∗ Proof. Recall that H (U(n); Z) is a free Z-algebra on odd degree generators x1; ··· ; x2n−1, with deg(xi) = i, i.e., ∗ ∼ H (U(n); Z) = ΛZ[x1; ··· ; x2n−1]: Then using the Leray-Serre spectral sequence for the universal U(n)-bundle, and using the fact that EU(n) is contractible, yields the desired result. Alternatively, the functoriality of the universal bundle construction yields that for any subgroup H < G of a topological group G, there is a fibration G=H ,! BH ! BG. In our A 0 case, consider U(n − 1) as a subgroup of U(n) via the identification A 7! . Hence, 0 1 there exists fibration U(n)=U(n − 1) ∼= S2n−1 ,! BU(n − 1) ! BU(n): Then the Leray-Serre spectral sequence and induction on n gives the desired result, where 1 ∗ 1 ∼ we use the fact that BU(1) ' CP and H (CP ; Z) = Z[c] with deg c = 2. -
Arxiv:1505.02430V1 [Math.CT] 10 May 2015 Bundle Functors and Fibrations
Bundle functors and fibrations Anders Kock Introduction The notions of bundle, and bundle functor, are useful and well exploited notions in topology and differential geometry, cf. e.g. [12], as well as in other branches of mathematics. The category theoretic set up relevant for these notions is that of fibred category, likewise a well exploited notion, but for certain considerations in the context of bundle functors, it can be carried further. In particular, we formalize and develop, in terms of fibred categories, some of the differential geometric con- structions: tangent- and cotangent bundles, (being examples of bundle functors, respectively star-bundle functors, as in [12]), as well as jet bundles (where the for- mulation of the functorality properties, in terms of fibered categories, is probably new). Part of the development in the present note was expounded in [11], and is repeated almost verbatim in the Sections 2 and 4 below. These sections may have interest as a piece of pure category theory, not referring to differential geometry. 1 Basics on Cartesian arrows We recall here some classical notions. arXiv:1505.02430v1 [math.CT] 10 May 2015 Let π : X → B be any functor. For α : A → B in B, and for objects X,Y ∈ X with π(X) = A and π(Y ) = B, let homα (X,Y) be the set of arrows h : X → Y in X with π(h) = α. The fibre over A ∈ B is the category, denoted XA, whose objects are the X ∈ X with π(X) = A, and whose arrows are arrows in X which by π map to 1A; such arrows are called vertical (over A). -
Lecture 3: Field Theories Over a Manifold, and Smooth Field Theories
Notre Dame Graduate Student Topology Seminar, Spring 2018 Lecture 3: Field theories over a manifold, and smooth field theories As mentioned last time, the goal is to show that n ∼ n 0j1-TFT (X) = Ωcl(X); n ∼ n 0j1-TFT [X] = HdR(X) Last time, we showed that: Ω∗(X) =∼ C1(ΠTX) ∼ 1 0j1 = C (SMan(R ;X) Today I want to introduce two other ingredients we'll need: • field theories over a manifold • smoothness of field theories via fibered categories 3.1 Field theories over a manifold Field theories over a manifold X were introduced by Segal. The idea is to give a family of field theories parametrized by X. Note that this is a general mathematical move that is familiar; for example, instead of just considering vector spaces, we consider families of vector spaces, i.e. vector bundles, which are families of vector spaces parametrized by a space, satisfying certain additional conditions. In the vector bundle example, we can recover a vector space from a vector bundle E over a space X by taking the fiber Ex over a point x 2 X. The first thing we will discuss is how to extract a field theory as the “fiber over a x 2 X" from a family of field theories parametrized by some manifold X. Recall that in Lecture 1, we discussed the non-linear sigma-model as giving a path integral motivation for the axioms of functorial field theories. The 1-dimensional non-linear sigma model with target M n did the following: σM : 1 − RBord −! V ect pt 7−! C1(M) [0; t] 7−! e−t∆M : C1(M) ! C1(M) The Feynman-Kan formula gave us a path integral description of this operator: for x 2 M, we had Z eS(φ)Dφ (e−t∆f)(x) = f(φ(0)) Z fφ:[0;t]!Xjφ(t)=xg If instead of having just one target manifold M, how does this change when one has a whole family of manifolds fMxg? 1 A first naive approach might be to simply take this X-family of Riemannian manifolds, fMxg. -
Surface Bundles in Topology, Algebraic Geometry, and Group Theory
Surface Bundles in Topology, Algebraic Geometry, and Group Theory Nick Salter and Bena Tshishiku Surface Bundles 푆 and whose structure group is the group Diff(푆) of dif- A surface is one of the most basic objects in topology, but feomorphisms of 푆. In particular, 퐵 is covered by open −1 the mathematics of surfaces spills out far beyond its source, sets {푈훼} on which the bundle is trivial 휋 (푈훼) ≅ 푈훼 × 푆, penetrating deeply into fields as diverse as algebraic ge- and local trivializations are glued by transition functions ometry, complex analysis, dynamics, hyperbolic geometry, 푈훼 ∩ 푈훽 → Diff(푆). geometric group theory, etc. In this article we focus on the Although the bundle is locally trivial, any nontrivial mathematics of families of surfaces: surface bundles. While bundle is globally twisted, similar in spirit to the Möbius the basics belong to the study of fiber bundles, we hope strip (Figure 1). This twisting is recorded in an invariant to illustrate how the theory of surface bundles comes into called the monodromy representation to be discussed in the close contact with a broad range of mathematical ideas. section “Monodromy.” We focus here on connections with three areas—algebraic topology, algebraic geometry, and geometric group theory—and see how the notion of a surface bundle pro- vides a meeting ground for these fields to interact in beau- tiful and unexpected ways. What is a surface bundle? A surface bundle is a fiber bun- dle 휋 ∶ 퐸 → 퐵 whose fiber is a 2-dimensional manifold Nick Salter is a Ritt Assistant Professor at Columbia University.