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Questions tagged [vector-bundles]

For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.

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Existence of Thom Class

In page 133, Theorem 8.5.5. (The Thom isomoprhism theorem) Let $\pi:V \rightarrow X$ be a complex vector bundle of rank $n$, over al locally comapct space $X$. Let $$ 0 \rightarrow \pi^* \...
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Quasicoherent sheaves on the groupoid of vector bundles on a surface

Consider the groupoid $Vect_n(S)$ of rank n-vector bundles over a projective surface $S$. What does it mean to have a sheaf $$\mathcal L\in QCoh(Vect_n(S))?$$ A notion of quasicoherent sheaf on a ...
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Relative $K$-theory, definition

On pg125 on proving that that the concordance classes quotiented by the acyclic ones froms a group, there is a lemma: Lemma 8.4.5 Let $(E,F,f)$ and $(E,F,g)$ be two $K$-cycles on $(X,Y)$. Assume ...
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Topology of decomposition of a space

In pg 121 of this notes, the author outlines a construction of gluing bundles. The scenario begins with Let $X= X_0 \cup X_1$ be union of two comapct spaces. $A = X_0 \cap X_1$ so that $X = X_0 \...
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Non-trivial explicit example of a flat connection

We all know that the exterior derivative on the trivial bundle forms an example of a flat connection. Can anyone provide an explicit example of a flat connection that is not just the exterior ...
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Inverse problem : Finding a vector bundle (resp. with connection), given its characteristic class (resp. differential character)

Given a rank $n$vector bundle $\alpha :E \to M$, and an element $u \in H^k(BG, \mathbb{Z})$, $G=GL(n,\mathbb{R})$we can define its characteristic class $u(\alpha) \in H^k(M, \mathbb{Z})$ as $f_\alpha^*...
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Partial differential operator is sum of order $0$ operators and composition of vector fields

Let $P:\Gamma(\Bbb R^n, E_0) \rightarrow \Gamma(\Bbb R^n, E_1)$ be a differential operator where $E_0$, $E_1$ are trivial vector bundles, with the standard bundle metric over $\Bbb R^n$. In page 28 ...
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Globally Generated Vector Bundle on a Riemann surface

This is a very vague question, in fact not really a question at all more of a search. I am studying some vector bundle theory on Riemann surfaces and would just like some non-trivial example of ...
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Every holomorphic vector bundle on a stein manifold is Nakano positive?

I encounter this statement on page 53 of Takeo Ohsawa's L2 Approaches in Several Complex Variables but I don't know how to prove it. Could anyone explain it to me or give me a reference for it?
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Intrinsic definition of Jacobian matrix on manifolds

For a vector field $X:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, the Jacobian matrix at $p\in\mathbb{R}^{3}$ is defined as $$\mathcal{J}_{p}X:=\begin{bmatrix}\left.\frac{\partial X^{i}}{\partial x^{j}}\...
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Cohomology of pushforwards of vector bundles over the projectivization of a vector bundle

Let $p:\mathcal E\to X$ be a vector bundle over a smooth projective variety $X$ and $p’:\mathbb P\mathcal E\to X$ its projectivization. We consider, then, two vector bundles: $\mathcal B$ over the ...
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Showing that a given vector bundle with connection is not trivial

Given the following exercise: where d is the trivial connection. We defined an isomorphism between two vector bundles with connection in the following way: I'm not sure what I have to show. Can I ...
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$W$ spin implies $\partial W$ spin

Let $M$ be a compact orientable manifold with the first two Stieffel-Whitney numbers equal to zero (this is my definition of SPIN manifold). Let $B$ be the boundary of $M$; I want to show that $B$ is ...
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Submanifold of codimension 1 orientable iff there exists unit normal vector field.

Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Where a unit normal vector field is a section $\nu$ of the normal bundle $ TM^{\bot} \to M$. So the fibers are all the ...
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Every submanifold is orientable (co-dimension 1)?

Suppose I have a submanifold $M \subset \mathbb{R}^{n}$, of dimension $n-1$. Apparently it's orientable if and only if there exists a unit normal vector field on $M$. Where a unit normal vector field ...
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On $\operatorname{End}(\mathcal F)$ of a locally free sheaf of finite rank

$\underline{Background}$:Let,$X$ be a scheme(for simplicity we can assume it to be noetherian).Let $\mathcal F$ be a locally free sheaf of rank $r$ on $X$.Let, us also assume that {$\operatorname{Spec}...
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Orientation for a vector bundle

After the definition of orientation for a vector bundle$^{(1)}$ in "Characteristic Classes" by Milnor/Stasheff (page 96), the author make this comment: The local compatibility condition implies ...
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Show that the Mobius bundle is a smooth real line bundle and it is non-trivial.

I'm reading John Lee's Introduction to Smooth manifolds 2nd edition. Consider the problem 10-1: Define an equivalence relation on $\mathbb{R}^2$ by $(x,y) \sim (x',y')$ if and only if $(x',y')=(x+n,...
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Operators on Sections of Vector Bundles

Let $(E,M,\pi)$ and $(F,M,\pi')$ be $C^{\infty}$ vector bundles. An operator $\alpha:\Gamma(E)\to \Gamma(F)$ is said to be local if whenever $s\in \Gamma(E)$ vanishes on an open set $U\subset M$ then $...
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The tangent bundle of a fibre bundle associated to a principal $G$-bundle.

I imagine this is fairly elementary, but I couldn't find a good reference. Let $G$ be a compact Lie group and $\pi:P\rightarrow M$ a principal $G$-bundle over a (compact) manifold $M$. Let $F$ be ...
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Least number of local trivializations of a vector bundle

Let $X$ be a compact Hausdorff space, and let $\pi: E \to X$ be a real or complex vector bundle over $X$ of rank $r$. Then there exist finitely many local trivializations of $E$, i.e. there exists an ...
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Normal Vector Bundle of RPn

From Hatcher: In the normal bundle $N S^n$ the identification $(x, v) ∼ (−x, −v)$ can be written as $(x,tx) ∼ (−x,t(−x))$. This identification yields the product bundle $\mathbb{RP}^n×\mathbb{R}$ ...
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Lin Independent Sections iff Trivial Bundle

This is from Hatcher: Let $(E,B,p)$ is a vector bundle. If one has $n$-linearly independent sections, the map $h:B \times \mathbb{R}^n \to E$ given by $h(b,t_1,···,t_n)= \sum_i t_i s_i(b)$ is a ...
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Classification theorem of topological vector bundles via presheaves

My issue is mainly set theoretical: In "Dale Husemoller: Fiber Bundles" it says on page 34 that there is a cofunctor $Vekt_{k}: P\rightarrow ens$, where $P$ is the category of paracompact spaces with ...
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Vector space structure on the fiber of a vector bundle

While trying to get accustomed to the very definitions of some usual objects in geometry/topology, a lot of questions I could find complete answers to came to my mind. I will try to be clear : Def (...
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Tangent Bundle Cotangent Bundle Isomorphism - Partitions Of Unity

Show that $TM$ is bundle isomorphic to $TM^{*}$ for a smooth manifold $M$ It is straight-forward to prove this isomorphism using a Riemannian metric, but suppose we did not have this tool. As a ...
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Complex structure : Existence of Local frame

We have a real vector bundle $\pi : E\rightarrow M$ of rank $2n$ over smooth manifold $M$ and $J\in \Gamma(E^*\otimes E)$(set of smooth sections of vector the bundle) is a complex structure on $E$. ...
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The pullback line bundle restricted on the exceptional divisor is trivial

Let $\sigma:\hat X\to X$ be the blow up of a point $x\in X$, denote the exceptional divisors $\sigma^{-1}(x)$ by $E$. $L\to X$ is a line bundle. Then we have a pullback line bundle $\sigma^*L\to\hat X$...
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Approximating the tangent vector in a phase space (or state space) reconstruction

I am investigating an application of differential geometry in experimental dynamical systems. Given a 1D time series (e.g., one that has been experimentally obtained), $x(t)$, I am considering the ...
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Existence of vector bundle isomorphism iff there exists a frame of sections

In class, we defined a vector bundle morphism in the following way: Definition: Let $\pi_i : E_i \rightarrow B_i$ be a vector bundle $(i = 1,2)$. A smooth map $F : E_1 \rightarrow E_2$ is a vector ...
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A question about pullback bundle and sheaf

Let $X$ be a compact complex manifold, $\sigma:\hat{X}\to X$ is the blow up of a point $x\in X$. Let $E:=\sigma^{-1}(x)$ and $L\to X$ be a line bundle, then how to give a rigorous proof to show that ...
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Showing linear independence of sections of a vector bundle

Let $\pi:E\to B$ be a smooth vector bundle and $B$ be a paracomapact manifold. Let $U\subset B$ be an open set and $(U,\phi)$ be a local trivialization. Then we know that the sections defined as $\...
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Extending a local smooth section to a global one

Let $\pi:E\to B$ be a smooth vector bundle and suppose $U$ is an open set of $B$. Let $\sigma:U\to E$ be a local smooth section of $\pi$. Can we extend $\sigma$ to global smooth section $\tilde{\sigma}...
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Vector bundle and disjoint union [closed]

My question is quite straightforward : given a $C^r$-manifold $X$ and a vector bundle $\pi : E \to X$ on $X$, is it true that $E \simeq \bigsqcup_{x \in} E_x$, where $E_x = \pi^{-1}(x)$? That is, can ...
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Classifying map for the hyperplane bundle

Let $G$ be a topological group and $X$ be a CW complex. Then principal $G$-bundles on $X$ is classified by the classifying space $BG$ in the sense that, given a principal $G$-bundle $P \to X$, there ...
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How to show the local section of a vector bundle $E\to M$ is continuous?

The zero section is the map $\zeta: M \to E$ defined by $\zeta(p)=0_p \in E_p$. Since $E \to M$ is a vector bundle, then for each $p \in M$ there exists a neighborhood $U$ of $p$ such that $\Phi: \pi^...
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Computing transition function problem

I'm studying vector bundles and I know that is a particular point of view in this subject, namely working with transition function. My problem is that I don't know how to compute transition function ...
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Prove check total space canonical bundle (vector bundle of projective space)

I have written a proof for the following question, but I'm not sure if I missed some subtilities or that I made some mistakes in my notation. Prove that $$K_\mathbb{R} = \left\{ \left( (y_0,\ldots, ...
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Prove the pullback bundle is a vector bundle

I'm stuck at proving that the pullback bundle is a vector bundle. The question is basically we have a smooth function $F$ from the smooth manifold $N$ to the smooth manifold $M$. And $(E, \pi, M)$ is ...
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Section of the Sheaf of Holomorphic Functions

I'm reading now the book "Vector Bundles on Complex Projective Spaces" from Okonek, Schneider and Spindler and I have an understanding problem with the interpretation of a section in following excerpt ...
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The cokernel of $H^0(\hat{X},\sigma^*L^k)\to H^0(E,\mathcal{O}_E)\otimes L^k(x)$

Let $X$ be a compact manifold. $\sigma:\hat{X}\to X$ is the blow up of $X$ of $x\in X$. Denote $\sigma^{-1}(x)$ by $E$. $L^k\to X$ is a very ample line bundle. $L^k(x)$ is a fiber. And by $L^k\to X$ ...
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Different definitions of positive line bundle

Let $E$ be a holomorphic line bundle over a compact, complex manifold $X$. Then $E$ is said to be a positive line bundle if and only if there exists a hermitian metric $h_X$ on $X$ and a hermitian ...
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The tangent bundle of a foliation is in fact a vector bundle

If $\mathcal{F}$ is a foliation on an $n$-manifold $M$, then the tangent bundle to $\mathcal{F}$, which I'll call $\mathrm{T}\mathcal{F}$, is a $k$-plane distribution on $M$, where $k$ is the ...
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Why is not every vectorbundle trivial and we only need local trivialization?

Why is should the trivialization only need to be local for vectorbundles? For example the tangent bundle. If the tangent space at a point p of a manifold M is identified with let's say $\mathbb{R}^n$ ,...
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Existence of isomorphism from the complex vector bundle to the dual

I know for a complex vector bundle $E\to X$, we have Chern class $c_i(E)=(-1)^ic_i(E^*)$. Therefore, in many cases, $E\to X$ and $E^*\to X$ are not isomorphic. I wonder if there exists some non-...
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Geometric intuition of vector bundles

Just a question about visualization of vector bundles. Imagine that we have a complex $3$-manifold $X$ and a vector bundle over $X$ of rank $4$. Suppose that we express the vector bundle as a sum of ...
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Why is there no guarantee that a linear combination of generic nondegenerate bilinear forms is still non degenerate?

Why is there no guarantee that a linear combination of generic nondegenerate bilinear forms is still non degenerate? Is there an example for that? From here there is a claim that a linear combination ...
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Very ample line bundle isomorphic to a restriction of $\mathcal{O}(-1)$ under an embedding?

I know this claim this wrong since a very ample line bundle isomorphic to a restriction of $\mathcal{O}(1)$ under an embedding. But I can still construct an isomorphism below: Let $L\to X$ be a very ...
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Definition of morphism between flat bundles

I'm reading this notes, and at the page 16 he defines a flat structure on a vector bundle as follows: A flat structure on a vector bundle $E$ is a family $\{\psi_U \}_{U∈\mathcal{U}}$ of ...
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The curvature class of the determinant bundle

I want to show that $\kappa(\bigwedge^k E)=\kappa(E)$, where $\kappa(E)=\frac{1}{2\pi i}[tr F_\nabla]$. $k$ denotes the rank of the bundle $E$. First, I have to define a connection $\nabla^\wedge$ ...