For questions on vector bundles.

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0
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0answers
22 views

tensor product of a line bundle with $\bigwedge^k T^*M$

I am reading a source that says: Let $L \to M$ be complex line bundle. Define $\Omega^k(M,L)=C^\infty(M,L \otimes\bigwedge^k T^*M)$. Can someone explain thoroughly what this means? What does it mean ...
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1answer
29 views

Does stereographic projection preserve or reverse orientation?

Let $S^n\subset\mathbb{R}^{n+1}$ denote the standard unit sphere with normal bundle $\nu$, let $N=(0,\dots,0,1)$ and $S=(0,\dots,0,-1)$. Then there are two sterographic projections $$\sigma_+\colon ...
3
votes
2answers
409 views

Triviality/non-triviality of line/circle bundle over $S^3$

I am interested in whether I can find non-trivial bundles in the case of a fiber bundle with base space $S^3$ and fiber either $\mathbb{R}$ or $S^1$. I know that in the case of a principal bundle ...
2
votes
1answer
29 views

Torsion in cotangent sheaf of the cusp

Let $X$ be the cuspidal curve $y^2 = x^3$. This is a singular curve with a unique singularity in the origin. One can easily see this using the Jacobi criterion. How does one show explicitly that the ...
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0answers
29 views

Is this complex vector bundle trivial?

Let $\Sigma$ be any Riemann surface, and let $L \rightarrow \Sigma$ be a complex line bundle (which is classified according to its degree). Then the vector bundle $L \oplus L^{-1} \rightarrow \Sigma$ ...
0
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1answer
17 views

Local Representation of Euclidean Connection

I'm trying to understand how connections are locally represented, and the definition I have to work with is this: Let $(x^1,\dots,x^n)$ be local coordinates defined in some chart $U \subset M$ ...
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0answers
19 views

Domnant morphism

Let $U=U(r,d)$ be the moduli space of stable vector bundles over some curve $X$, and $\Omega$ be the cotangent bundle on $U$. Let $W=\bigoplus _{i=1}^rH^0(X,K_X^i)$. The fiber of $\Omega$ over some ...
3
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1answer
60 views

Rank 2 vector bundle

$E$ is a rank $2$ vector bundle. Why is $E\simeq E^*\otimes \det E$? Any generalization (arbitrary rank, $E$ non locally free etc.)?
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4answers
631 views

Line bundles of the circle

Up to isomorphism, I think there exist only two line bundles of the circle: the trivial bundle (diffeomorphic to a cylinder) and a bundle that looks like to a Möbius band. Although it seems obvious ...
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0answers
30 views

Why the canonical bundle of a complex manifold is a line bundle?

I think I do not understand something in the definition of the line bundles. Line bundles have fibers of rank 1. That is isomorphic to $\mathbb{C}$. But I do not know how to connect this vector space ...
3
votes
3answers
212 views

How to show the existence and uniqueness of the pullback connection in vector bundles?

There is the following result: If $D$ is a connection on a vector bundle $E$ over $N$ and $φ$ is a smooth map from $M$ to $N$, then there is a pullback connection on $φ^*E$ over M, determined ...
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0answers
24 views

Hermitian Structure on $\mathcal{O}(1)$ line bundle over $\mathbb{P}^{n}$

I'm reading through Huybrecht's section on Vector Bundles. In general, he notes that for a (holomorphic) line bundle $L$ with $s_{1}, \ldots, s_{k}$ globally generating (holomorphic) sections, we can ...
2
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0answers
28 views

Wedge product of $k$-forms

I'm studying smooth manifolds with Lee's book. He defines a $k$-form on a manifold $M$ as a section $M \to \Lambda^k M$ (where $\Lambda^k M = \bigsqcup_{p\in M} \Lambda^k T_pM$ is the smooth vector ...
4
votes
1answer
84 views

Geometric intuition behind $V(M)=\operatorname{Spec}(\operatorname{Sym}(M))$?

In the equivalence between geometric vector bundles and locally free sheaves we assign to a locally free sheaf $M$ the bundle $V(M)=\operatorname{Spec}(\operatorname{Sym}(M))$. I don't doubt the ...
3
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0answers
42 views

Normal Space of an orbit at a point

Let $X$ be a curve of genus $g$. If $d>n(2g-1)$,then for any vector bundle $E$ of rank $n$ and degree $d$ over $X$ has $H^1(X,E)=0$ and $E$ is generated by its global sections. For such a vector ...
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0answers
44 views

Why are transition functions of an algebraic vector bundle are maps of algebraic varieties?

This is from Le Potier's Lectures on Vector Bundle Definition: A complex linear fibration (or just fibration) over an algebraic variety is a pair $(E,p)$ where E is an algebraic variety and ...
3
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0answers
34 views

$\mathbb{R}^n$ bundle over $B$ being compact and having finite covering dimension implies has finite type? [closed]

Let $\xi$ be a $\mathbb{R}^n$-bundle over $B$. If $B$ is paracompact and has finite covering dimension, does it follow that every $\xi$ over $B$ has finite type?
1
vote
1answer
50 views

What is a sympelctic bundle

What is a symplectic bundle? Is it a fibre bundle or a vector bundle? I am hoping for a not-very-technical answer because I'm not familiar with bundles in general. Sorry for that. PS: This symplectic ...
7
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0answers
111 views

Null-correlation and Tango bundles on $\mathbb{P}^3$

Let $V$ be a four-dimensional complex vector space and $\mathbb{P}^3=\mathbb{P}(V)$. There are two interesting bundles $N$ and $T$ on $\mathbb{P}^3$, both of rank 2, called respectively a ...
0
votes
1answer
52 views

What does “linear and injective on each fiber” really mean?

The question is about the proof of the following result: For a paracompact space $B$, the map $[B, \operatorname{Gr}_k] \to \operatorname{Vect}^k(B)$, $[f] \mapsto f^*(\gamma_k)$ is a bijection, ...
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vote
3answers
173 views

Elementary and purely topological proof of the non-triviality of tautological complex line bundle

I need some hints about the proof of the non triviality of the tautological complex line bundle, in a pure topological manner. Let $E$ be the t.c.l. bundle defined in this way $$ E= \{ (x,v) \in ...
3
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1answer
63 views

The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\mc}{\mathcal} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\pr}{pr} \newcommand{\at}{\big|} \DeclareMathOperator{\GL}{GL}$ Let $\pi:E\to N$ be a smooth vector bundle over a smooth ...
0
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0answers
19 views

Fiberwise isomorphism induces a bundle isomorphism

Given vector bundles $(\pi_1,E_1,M)$ and $(\pi_2,E_2,M)$ and a linear isomorphism defined in each fiber $f:E_p \rightarrow E_{f(p)}$, is it possible to define a $bundle$ isomorphism of the same vector ...
3
votes
1answer
35 views

Vector bundle morphisms and determinants

Let $F,E$ be two locally free sheaves of equal rank on a complex manifold $X$. Assume that we have an injection $0\to F\to E$ such that the induced map $0\to \det F\to \det E$ is an isomorphism. Is ...
3
votes
1answer
108 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 ...
5
votes
1answer
169 views

Extension of vector bundles on $\mathbb{CP}^1$

Let $\lambda\in\text{Ext}^1(\mathcal{O}_{\mathbb{P}^1}(2),\mathcal{O}_{\mathbb{P}^1}(-2))$ and $E_\lambda$ be a vector bundle on $\mathbb{CP}^1$ which is given by the exact sequence ...
0
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0answers
32 views

Computing the restriction $(\Lambda^2T_{\mathbb{P}^n})_{|L}$ for line in $\mathbb{P}^n$

From the Euler sequence $$0\to\mathcal{O}_{\mathbb{P}^n}\to V\otimes\mathcal{O}_{\mathbb{P}^n}(1)\to T_{\mathbb{P}^n}\to0$$ it is easy to deduce that ...
1
vote
1answer
23 views

1-dim Vector Bundle sufficient condition to be trivial

I'm a physics student studying differential geometry. I'm trying to understand how vector bundles work, I have the following exercise. Let be $ L $ a $1$-dim vector bundle on $M$. Prove that if ...
0
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1answer
50 views

Possible subbundles on $\mathbb{CP}^1$

I have the vector bundle $E:= \mathcal{O}(1)^{\oplus k}$ on the projective line, $\mathbb{CP}^1$. I want to know what are the possible subbundles of this, and why? We know that the ...
2
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1answer
51 views

Global generation of vector bundles by an exact sequence

Let $X$ be a smooth projective complex surface and $V$ a globally generated vector bundle on $X$. Suppose we have a vector bundle $E$ sitting in an exact sequence $$0\to V\to E\to O_X(C)\otimes A \to ...
2
votes
1answer
33 views

Computing the restriction $T_{\mathbb{P}^3|X}$ for twisted cubic in $\mathbb{P}^3$

Let $i:X=\mathbb{P}^1\to\mathbb{P}^3$ be a twisted cubic given by the embedding $(u:v)\mapsto(u^3: u^2v: uv^2: v^3)$. How to compute $T_{\mathbb{P}^3|X}$?
0
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0answers
31 views

Tensors in a vector bundle with infinite-dimensional fibers

If $\xi = (\pi,E,M)$ is a vector bundle such that each fiber has finite-dimension, and $V=\Gamma(M,E)$ is the space of smooth sections, then there is an isomorphism between $(\otimes^r ...
2
votes
1answer
42 views

Question on syzygies

It is hard to formulate a question, but I want to ask about a reference/recipe for computing syzygies in general. For example, on $\mathbb{P}^1_{(x:y)}$ there is an exact sequence $0\longrightarrow ...
5
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1answer
86 views

Tensor product for vector bundles is commutative, associative, and has an identity element?

How do I see that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element?
2
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0answers
28 views

Cohomologies of certain vector bundle on $\mathbb{P}^3$

Consider the collection of $m$ pairwise disjoint lines $L_1,\ldots,L_m$ in $\mathbb{P}^3$ and pose $Z=L_1\sqcup\cdots\sqcup L_m$. Consider the rank-$2$ vector bundle on $\mathbb{P}^3$ which is given ...
3
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0answers
49 views

Short exact sequence on $\mathbb{P}^1$

Let F be a torsion free sheaf of rank $n+4$ over $\mathbb{P}^1$ which fits in the SES $0\longrightarrow\mathcal{O}_\mathbb{P^1}(-3)^{n+2}\longrightarrow ...
0
votes
1answer
21 views

Natural map from cokernel of a monad

If I have a monad $$ U \stackrel{\alpha}{\longrightarrow} V \stackrel{\beta}{\longrightarrow}W $$ then there should be a natural map $$ \text{cokernel}(\alpha) \rightarrow W $$ but I can't think of ...
1
vote
1answer
56 views

A few questions about vector bundles on an algebraic variety

Let $X$ be a smooth projective complex variety and $E$ an algebraic vector bundle on $X$. (Q1) If $E$ is globally generated and $c_1(E)=0$ does it follows that $E$ is trivial? (Q2) If $E$ is ...
2
votes
1answer
67 views

Intuition behind a first Chern class computation

On a complex smooth algebraic surface $X$, say we have a vector bundle $F$ which fits in an exact sequence $$0\to F\to O_X^{r+1} \to A\to 0$$ with $A$ a torsion sheaf supported on a smooth curve ...
1
vote
1answer
156 views

Vector bundle $\gamma^1$ over infinite real projective space doesn't have finite type? [closed]

Using Steifel-Whitney classes, how do I see that the vector bundle $\gamma^1$ over $\textbf{RP}^\infty$ does not have finite type?
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vote
0answers
60 views

Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $Spec\mathbb{C}$, and the underlying set of $G$ is a ...
0
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0answers
44 views

The normal bundle of conic

Let $C\subset\mathbb{P}^2\subset\mathbb{P}^n$ be a smooth conic (everything is over the field $\mathbb{C}$). I want to compute $T_{\mathbb{P}^n|C}$ and $N_{C/\mathbb{P}^n}$. Let $z_0,z_1,...,z_n$ be ...
2
votes
1answer
31 views

What does $P\times_G V\to B$ mean?

Let $$\pi:P\to B$$ be a principal $G$-bundle and $$\rho:G\times V\to V$$ a continuous action of $G$ on the vector space $V$. What does the notation $P\times_G V\to B$ mean? It is supposed to be ...
0
votes
0answers
17 views

Differential subbundles of trivial bundle over a segment.

Let $F \subset (0,1)\times \mathbb R^n$ be a differential vector subbundle of rank $r$ of trivial vector bundle over the segment $(0,1)$. I'd like to prove that, if for every differential section ...
5
votes
1answer
75 views

Exists homeomorphism which carries each fiber isomorphically to itself, composition… make rigorous.

See here for a question I asked. Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see that there exists a homeomorphism $f: E(\xi) \to E(\xi)$ ...
4
votes
1answer
46 views

Redundancy in the definition of vector bundles?

In John Lee's classic Introduction to Smooth Manifolds, the following definition of vector bundle is given. Definition. Let $M$ be a topological space. A (real) vector bundle of rank $k$ over $M$ ...
5
votes
1answer
65 views

Interpretation for the curvature and monodromy of a connection - Reality check

Let $P \to M$ be a principal $G$-bundle with connection form $\omega \in \Omega^1(P,\mathfrak{g})$. Here are the statements I'm basing my viewpoint on: A connection is flat (vanishing ...
4
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1answer
78 views

Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite ...
3
votes
1answer
58 views

Which line bundle has transition function $\psi_{12}([z_1,z_2])=\frac{z_1/z_2}{|z_1/z_2|}$?

We know that the complex line bundles over $\Bbb{CP}^1$ are classified by the integers. Each is isomorphic to one of the bundles $\mathcal{O}(n)$ for $n\in\Bbb Z$, where $\mathcal{O}(-1)$ is the ...
4
votes
2answers
208 views

Which tangent bundles of real and complex projective spaces are trivial?

The tangent bundle of $S^m$ is known to be trivial only for $m=1,3,7$. Can this result be used to deduce that the tangent bundle of projective space $\mathbb{R}P^m$ is also trivial? (Clearly, for ...