For questions on vector bundles.

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Vector space operations on fibres of associated bundles.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\text{ad}:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let ...
4
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1answer
42 views

Milnor's definition of bundle map in “Characteristic Classes”

In chapter 3 of "Characteristic Classes", Milnor defines bundle maps, requiring them to map fibers isomorphically onto fibers. Why not merely require homorphisms on fibers? (e.g., for the given ...
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1answer
28 views

Decomposition of $\pi\colon E\to\mathbb{P}^1_k$ as a direct sum of tensor powers of the tautological line bundle?

Suppose you have a vector bundle $\pi\colon E\to\mathbb{P}^1_k$, where $k$ is some field. Is it always possible to decompose the vector bundle into a direct sum of tensor powers of the tautological ...
2
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1answer
73 views

Examples with zero first Stiefel-Whitney class and nonzero second Stiefel-Whitney class

What's the simplest/most concrete vector bundle you can think of that has zero first Stiefel-Whitney class but non-zero second? That would be the simplest space that doesn't have spinors. (See Spin ...
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0answers
14 views

Morphism of modules of sections of pullback bundles

Suppose that we have a morphism $\theta: \Gamma(B,E_1) \to \Gamma(B,E_2)$ where $E_i$ are two vector bundles over $B$ and let $f:A \to B$ be a continuos map. Then we can define a pullback bundles ...
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1answer
10 views

Section of pullback bundle

Suppose that $E \to B$ is a vector bundle and $f:A \to B$ is continuos. If $s$ is a section of $E$ how to define a section of pullback bundle? On wikipedia they say that it induces the section of the ...
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0answers
20 views

Trivial sections of tautological line bundle for $\mathbb{P}_F(V)$.

I have a brief question which has been bothering me. Suppose you have a finite dimensional $F$-vector space, call it $V$. Is there a nice proof of why there only exist trivial sections of the ...
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13 views

Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
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1answer
27 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can ...
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0answers
22 views

Bianchi Identity - Gauge Theory

I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is, $h^*\Omega = 0$, where $\Omega$ is the curvature ...
5
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2answers
75 views

What are monomorphisms in the category of real vector bundles over a fixed base space $X$?

I have a (perhaps stupid) question dealing with vector bundles. In most textbooks, a morphism of real vector bundles $f:\xi\to \eta$ over a fixed base space $X$ is said to be a ...
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1answer
24 views

Gauge transformation on a principal bundle

I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by ...
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0answers
6 views

Reduction of the structure group

Given a vector bundle $V$ of rank $r$ over a curve. What proprieties should $V$ satisfies in order to admet a reduction of the structur group to $O_r(k)$ (resp. $SL_r(k)$, $Sp(r,k)$) Could you give ...
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1answer
73 views

Chern class of complex vector bundles

Let $\xi$ be an $n$-dimensional complex vector bundle. It is claimed that the Chern class of $\xi$ is $$ c(\xi)=(1+x_1)\cdots (1+x_n), $$ $|x_k|=2$, $c_j(\xi)$ is the $j$-th symmetric polynomial of ...
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1answer
53 views

Is the tensor product of a complex line bundle with itself trivial?

Let $\xi$ be a complex line bundle over a manifold $M$. Then $\xi\otimes \xi$ is a trivial complex line bundle. Is my statement right?
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0answers
12 views

Adjoint representation for matrix groups (Gauge theory)

This is a question in regards to an identity in Gauge theory. Let $\omega$ be the connection one form on a principal bundle $\pi:P\rightarrow M$ and let $A_{\alpha}:=s_{\alpha}^*\omega$ be the gauge ...
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22 views

Sections of associated bundles isomorphism between spaces

I am reading some lecture notes which can be found here . They say that sections of $P\times_G F$ are represented by the functions $f:P\rightarrow F$ satisfying $f(pg)=\rho(g^{-1})\circ f$. Or ...
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0answers
32 views

In which space does the Lie derivative of a vector field really live?

I'm slightly puzzled about the space in which the Lie derivatives of vector fields live. I get the worrying impression that it lives in the double tangent bundle, which I know is not true, so i need ...
2
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1answer
35 views

Stiefel-Whitney classes with Z-coefficients

Stiefel-Whitney classes are defined (for example, in Milnor's Characteristic classes) as elements of the cohomology groups with $\mathbb{Z}_2$-coefficients as follows. $$w_i(E)=\phi^{-1} Sq^i(u)$$ ...
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0answers
27 views

Compute a chern class $c(K^n)$ for a non-singular algebraic hypersurface $K^n$ of degree $d$ in $P^{n+1}(C)$.

This is Problem 16-D in Characteristic classes by John W. Milnor and James D. Stasheff. Problem 16-D) If the complex manifold $K^n$ is complex analytically embedded in $K^{n+1}$ with dual ...
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0answers
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Extending a vertical vector to a vertical vector field

Let's say $F: M \rightarrow N$ is a smooth submersion between manifolds. Then each fiber of $F$ is a properly embedded submanifold. If $S$ is such a fiber, and I take some derivation $D \in T_p S$, ...
5
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1answer
55 views

Is there a fibre bundle with fibre homeomorphic to $\mathbb R^k$ which cannot be given the structure of a vector bundle?

We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity ...
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4answers
212 views

Why can we always take the zero section of a vector bundle?

$\require{AMScd}$ As I understand it, a rank $k$ vector bundle is a pair of topological spaces with a map between them $$ E\xrightarrow{p}B $$ such that there exists an open cover $(U_\alpha)$ of $B$ ...
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1answer
39 views

Vector bundle over a compact, Hausdorff space is a summand of a trivial bundle.

I am trying understand the proof of the following (proposition 1.4 in Hatcher's book on Vector Bundle). For every vector bundle $E\overset{p}{\to} B$, with $B$ compact Hausdorff, there exists a ...
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1answer
52 views

Extension of Sections of Restricted Vector Bundles

Edit: Changing Question: There are two questions related questions: extending a smooth vector field extending a vector field defined on a closed submanifold I'm trying to answer a question which is ...
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0answers
20 views

Space of invariant sections

I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections ...
2
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1answer
28 views

Quick question: Tensoring a 2-torsion line bundle with a rank 2 nonsplit extension over a curve of genus 1

Everything is complex algebraic. Over a curve $C$ of genus $1$, let $V$ be a rank 2 vector bundle with $\deg \det(V)=1$, which is a nonsplit extension of $\mathcal{O}_C$ and $\mathcal{O}_C(p)$, where ...
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0answers
35 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
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0answers
16 views

Exterior Product

I am studying exterior product of vector bundle, and need some indication. Let $E$ be an algebraic vector bundle of rank $r$ and degree $d$, Then $\Lambda^2 E$ is of rank $r'=r(r-1)/2$, but is of ...
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1answer
53 views

Isometry of two Euclidean structures on the same vector bundle

I'm reading Milnor & Stasheff's Characteristic Classes, and I was struck by the following problem, which they call the Isometry Theorem: "Let $\mu$ and $\mu'$ be two different Euclidean metrics ...
3
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2answers
79 views

Vector bundle of dimension $\leqslant n$ on $n$-connected space is trivial

I wonder whether any vector bundle of dimension $\leqslant n$ on an $n$-connected CW-complex is trivial? It seems that, the complex can be given cellular structure with exactly one 0-dimensional ...
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1answer
39 views

Action of $Aut(X)$ on $Coh(X)$

I was reading about Bondal and Orlov reconstruction theorem. In particular that for a smooth variety with ample or anti-ample canonical bundle $\mathrm{Aut}(D^b(X)) \cong \mathbb{Z} \times ...
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0answers
83 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
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1answer
31 views

Determinant of this coherent sheaf on a surface $S$

If $C$ is a curve on a surface $S$, i.e. $i:C\subset S$, and $G$ is a line bundle on $C$, then $G|_U\cong \mathcal{O}_C$ where $U$ is an open subset of $S$, that is, $G$ is trivial on the complement ...
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0answers
19 views

Symmetric product

Let $E$ be a vector bundle of rank $r$ and degree $d$ over a smooth curve $X$. Is there any canonical exact sequence for $Sym^k(E)$? in particular what is the degree of $Sym^k(E)$? Suppose $E$ is ...
3
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1answer
47 views

Ways of thinking about vector-valued differential forms

I am trying to get a better intuition of vector-valued differential forms. Let $V$ be a vector space and $M$ a smooth manifold. Consider the space $\Omega^k(M;V)=\Gamma((M\times V)\otimes ...
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0answers
105 views

What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$. Suppose there is a surjection: $E\longrightarrow ...
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67 views

characteristic classes of $SO(3)$-bundles over $\mathbb{CP}^2$

Let us consider the complex line bundle $\xi$ over $\mathbb{CP}^2$ which is completely defined by its restriction on a complex projective line; this restriction is denoted by $\xi^{\prime}$ and the ...
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0answers
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Comparing normal bundles of embedded submanifolds and their sections.

Let $M,M'\subseteq \mathbb{R^n}$ two compact embedded submanifolds, which are abstractly diffeomorphic. Tangent and normal bundle of the two submanifolds inherit a metric from $\mathbb{R^n}$. By the ...
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0answers
42 views

Existence of Harder-Narasimhan filtration

I am trying to understand the proof of the existence of Harder-Narasimhan filtration from Huybrechts and Lehn. Let $X$ be a projective scheme with a fixed ample line bundle. Then the theorem says ...
3
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1answer
58 views

Pull-back line bundle under morphism of degree $d$

This question is partially related to Direct image of vector bundle Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a morphism of degree $d$. For $n>0$, how can we compute ...
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0answers
44 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let ...
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1answer
50 views

Endomorphism algebra of direct sum of two extensions of line bundle

Let $X$ be a smooth irreducible smooth projective curve. Let $L$ be a line bundle over X of degree $0$ such that $L^2\neq \mathcal{O}_X$. Let $V\in Ext(L,L^{-1})$ and $W\in Ext(L^{-1},L)$. i.e., ...
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1answer
36 views

What is the Chern class of the Kernel of a projection map after taking a blowup?

Let $S:= \mathbb{CP}^1 \times \mathbb{CP}^1 $ and $\tilde{S}$ the blowup of $S$ at one point. Let $a_1, a_2$ be generators for the cohomology $H^*(S, \mathbb{Z})$ and let $a_1, a_2$ and $E$ be the ...
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$F$-related vector fields

I'm having a bit of difficulty understanding $F$-related vector fields. I think I understand conceptually what's going on (taking a vector field on a manifold $M$ and getting a smooth vector field on ...
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1answer
83 views

Direct image of vector bundle

Let $f:X\to Y$ be a morphism of projective varieties and $\mathcal{E}$ be a vector bundle on $X$. How can I compute explicitly $f_*\mathcal{E}$ in various situations? For example, let ...
3
votes
1answer
56 views

Extensions of vector bundles

Let $F$ be a vector bundle over a smooth curve $X$, and consider $V$ be the extension : $$0\rightarrow F\rightarrow V \rightarrow \mathcal O_X\rightarrow 0$$ assosciated to an element $e\in H^1(F)$. ...
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0answers
33 views

A question on Grassmannian manifolds and universal line bundles

Problem. Fix an $ n \in \mathbb{N} $. Is it true that there exists a $ k \in \mathbb{N} $ such that every smooth complex line bundle over a smooth $ n $-dimensional real manifold is the pullback of ...
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2answers
57 views

Vector bundles and elementry transformation

Let $E$ be a vector bundle of rank $r$ and let $\phi:E\rightarrow \mathbb C_p$ non vanishing map to the skyscraper sheaf. consider the kernel $F$ of this sheaf which is a sub-bundle of $E$, every ...
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What is the poinacre dual of the projectivization of a line bundle inside the projectivization of the sum of two line bundles?

Let $S:= \mathbb{CP}^1 \times \mathbb{CP}^1$. Let $TS \approx L_1 \oplus L_2$, where $L_1$ and $L_2$ are the pullback of $T\mathbb{CP}^1$ wrt to the respective projection maps. Note that ...