For questions on vector bundles.

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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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12 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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21 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 ...
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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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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
71 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
52 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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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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31 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 ...
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1answer
34 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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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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18 views

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 $D \in T_p S$, ...
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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
210 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
51 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 ...
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1answer
27 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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32 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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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 ...
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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
81 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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0answers
25 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 ...
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1answer
46 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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96 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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49 views

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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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 ...
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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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43 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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55 views

$F$-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 ...
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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
32 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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33 views

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 ...
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1answer
28 views

punctured Mobius band in high dimension

Let $M^{n+1}$ be the non-trivial line bundle over sphere $S^n$. When $n=1$, $M^2$ is Mobius band and the punctured space $M^2\setminus *\simeq S^1$. How about $M^{n+1}\setminus *$ in general? Does ...
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1answer
43 views

Cannonical evaluation map

Let $C$ be a curve over $\mathbb{C}$, and $E$ be a vector bundle on $C$ such that $H^0 (C, E) \neq 0$. Everyone talks of the evaluation map $H^0 (C, E)\otimes O_C\longrightarrow E$. What is this map ...
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1answer
36 views

$k$-forms taking values on a line bundle.

Let $M$ be a manifold and let us consider an element $\Omega\in\Omega^{k}(M,L)$, namely a $k$-form that takes values on the flat line bundle $L\to M$. I want to know what this really means. Does it ...
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1answer
33 views

Semantics: basis of tangent vectors in Rn vs basis of position vectors in Rn

The tangent plane of any point in Rn (usual structure) is linearly isomorphic to Rn itself. What is clear is that position vectors and tangent vectors are not the same kind of object even in Rn, as ...
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1answer
28 views

Todd class of a hypersurface.

I would like to calculate $\mathrm{td}(X)$ for $X\in H^0(\mathbb P^n, \mathcal O(k))$. Since $$ 0\to TX\to T\mathbb P^n\to N\to 0, $$ we have $\mathrm{td}(X)=\mathrm{td}(\mathbb P^n)/\mathrm{td}(N)$ ...
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46 views

Vector bundle is homotopy equivalence [duplicate]

If $\pi:E\rightarrow B$ is a vector bundle (I allow it to be of nonconstant rank), then I want to prove that it is a homotopy equivalence. As a homotopy inverse I propose the zero section ...