For questions on vector bundles.

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24 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 ...
2
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0answers
21 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
24 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 ...
2
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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
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 ...
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?
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1answer
49 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 ...
3
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1answer
59 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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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
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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 ...
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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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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 ...
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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 ...
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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 ...
2
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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 $T_{\mathbb{P}^3|X}$?
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30 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
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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 ...
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 ...
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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 ...
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1answer
50 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 ...
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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 ...
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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 ...
4
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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 ...
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0answers
105 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 ...
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0answers
58 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 ...
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0answers
43 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 ...
5
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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 ...
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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 ...
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0answers
16 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 ...
4
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1answer
45 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
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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 $f: E(\xi) \to E(\xi)$ ...
5
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1answer
62 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
77 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
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1answer
56 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 ...
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1answer
33 views

Canonical Bundle Isomorphism $T_v(E)\cong \pi^* E$

Given a smooth vector bundle $\pi:E\to M$, the vertical bundle $T_v(E)$ is by definition $\ker T\pi$, which is a subbundle of $T(E)$. It is asserted in this answer that there is a canonical ...
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0answers
36 views

construction of $\Omega^r_B(E)$ and $\Omega^r_B( End E)$

I am trying to study these notes: https://www.dpmms.cam.ac.uk/~agk22/vb.pdf but at page 31 I have some problems in following the construction, starting from a vector bundle $(E,B,\pi)$, of the ...
2
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1answer
28 views

Why can one discriminate between the trivial $S^1$ line bundle and the Möbius strip by knowing the fibre transformation group?

Both $S^1\times\mathbb R$ and the Möbius strip can be regarded as line bundles over $S^1$. I have read that one can reconstruct a fibre bundle by knowing its base space, its fibre and the bundle ...
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1answer
61 views

Stably trivial bundle is trivial

I have a smooth embedding $f:S^2\to \mathbb{R}^4$ and would like to show that the normal bundle $\nu\to K$ of the image $K:=f(S^2)$ is trivial. I have already shown that it is stably trivial, i.e. $ ...
1
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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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2answers
50 views

Vector-valued differential forms

Given a smooth real manifold $M$, and a real vector space $V$, when we talk about a $V$-valued $k$-form on $M$, do we mean an element of $\Gamma(\wedge^{*k}M)\otimes V$, where $\Gamma$ denotes ...
4
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0answers
68 views

Splitting cotangent bundles over schemes

For smooth manifolds the followng is well known $T^*(M \times N) \cong p_1^*(T^*M)\oplus p_2^*(T^*N)$ as bundles over $M\times N$. Let $j: X \to M$ be an embedded submanifold and let $N^*X\subset ...
0
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1answer
66 views

Horrocks-Mumford bundle

Let $E$ be the Horrocks-Mumford bundle, which is a rank-2 vector bundle on $\mathbb P^4$ with $c_1(E)=5$ and $c_2(E)=10$, defined by some combinatorical construction (see Okonek, Schneider, Schindler, ...
3
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1answer
129 views

What kind of object is the push forward of a vector field?

I was actually not sure about asking this question since I think I know what the answer is, but here it goes: Let $M$ and $N$ be two smooth manifolds and $\mathbf{X}$ a vector field defined on $M$. ...
0
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1answer
48 views

Maximal trivial subspace in vector bundles

Let X is a locally compact Hausdorff space, given an vector bundle p: E$\to$X, a subspace Y of X is called trivial (for this bundle), if we restrict this bundle over Y, it is a trivial bundle. In ...
4
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1answer
69 views

Why doesn't this construction of Chern classes generalize to real bundles?

Can we mimic the construction of Chern classes using real or quaternionic bundles? If so, do we get anything interesting? My question concerns the construction of Chern classes in Bott and Tu's ...
2
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1answer
61 views

what does Whitney sum of vector bundles correspond to in the k-theory KO?

Let $X$ be a $CW$-complex and $\text{Vect}^n(X)$ the collection of $n$-dimensional real vector bundles over $X$. Let $$ \text{Vect}^*(X)=\bigoplus_{n=0}^\infty \text{Vect}^n(X) $$ with addition ...
4
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0answers
127 views

“Global” dimension for topological spaces // Geometric interpretation for global dimension of rings

The global dimension of a ring $R$ is the supremum of the projective dimensions of it's $R$-modules. $$\dim (R)=\sup \{\dim_\mathrm{proj}(M):M \in R\text{-mod} \}$$ I'd like to have some geometric ...
4
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1answer
49 views

$\mathbb{R}^1$-bundle $\xi$ possesses Euclidean metric iff $\xi$ represents an element of order $\le2$

The set of isomorphism classes of $1$-dimensional vector bundles over $B$ forms an abelian group with respect to the tensor product operation. How do I see that a given $\mathbb{R}^1$-bundle $\xi$ ...
3
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0answers
25 views

Transition function of line bundle

Let $X$ be a smooth algebraic curve over $\mathbb C$ and fix a closed point $p\in X$. I want to calculate the transition functions of the line bundle associated the sheaf $\mathcal O(p)$. I know ...