For questions on vector bundles.

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33
votes
2answers
907 views

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have ...
13
votes
7answers
594 views

An example of a triple $(E,\pi,M)$ which is not a vector bundle

What is an example of a pair of finite dimensional $C^{\infty}$ manifolds $E$ and $M$, and a smooth function $\pi:E\rightarrow M$ such that $\pi^{-1}(p)$ has a vector space structure for each $p\in M\ ...
11
votes
2answers
466 views

Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
11
votes
1answer
355 views

Second Stiefel-Whitney Class of a 3 Manifold

This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff. The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent ...
10
votes
2answers
206 views

Almost A Vector Bundle

I'm trying to get some intuition for vector bundles. Does anyone have good examples of constructions which are not vector bundles for some nontrivial reason. Ideally I want to test myself by seeing ...
10
votes
1answer
277 views

What is the line bundle $\mathcal{O}_{X}(k)$ intuitively?

I am always confused about how to understand the line bundle $\mathcal{O}_{X}(k)$ on a projective scheme $X=\mathrm{Proj}(\oplus_{n=0}^\infty A_{n})$. Of course this is by definition the ...
10
votes
1answer
142 views

Is there a characterization of injective $C(X)$-modules analogous to Serre-Swan?

The Serre-Swan theorem in topology says that if $X$ is compact Hausdorff and $C(X)$ the ring of continuous functions on $X$, then the category of finitely generated projective $C(X)$-modules is ...
9
votes
5answers
500 views

Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean. I am looking for a reference that ...
9
votes
2answers
145 views

Are there any simply connected parallelizable 4-manifolds?

On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's ...
9
votes
2answers
116 views

Explicit formula for the curvaure of a connection

Let $E$ be a vector bundle over $M$ and denote by $\mathcal{A}^k(E)$ the space of sections of $\Lambda^k (TM)^* \otimes E$, i.e. the space $k$-forms with values in $E$. A connection ...
9
votes
1answer
318 views

Vector Bundle Over Contractible Manifold

The problem comes from Liviu Nicolaescu's book Lectures on the Geometry of Manifolds. He asks the reader to prove that any vector bundle $E$ over $\mathbb{R}^n$ is trivializable. The idea he gives is ...
8
votes
2answers
221 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
8
votes
1answer
148 views

Tangent bundles of exotic manifolds

Consider a pair of homeomorphic but not diffeomorphic smooth manifolds $M_1$ and $M_2.$ Fix a homeomorphism $\phi\colon M_1\rightarrow M_2.$ If I understand correctly, two bundles $TM_1$ and ...
8
votes
1answer
162 views

Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)

It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
8
votes
1answer
142 views

Can the diagonal of a manifold be expressed as the zero set of a section of a vector bundle?

Let $\mathbb{C} \mathbb{P}^2$ be the two dimensional complex projective space and $$M:= \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2.$$ Let $ \gamma \rightarrow \mathbb{C} \mathbb{P}^2 $ be ...
8
votes
1answer
214 views

When is the pushforward / direct image of a reflexive sheaf locally free?

I have seen a number of theorems that guarantee the direct image of a reflexive sheaf to be reflexive again, or for the direct image of a locally-free sheaf to be locally free again. This makes me ...
8
votes
0answers
171 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
7
votes
3answers
697 views

Vector bundle transitions and Čech cohomology

I have read that transition maps $g_{\alpha\beta}:U_\alpha\cap U_\beta\to GL(n)$ of a vector bundle of rank $n$ are related to the Čech cohomology group $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ ...
7
votes
2answers
153 views

Adjunction for varieties with higher codimension

For $X \subseteq \mathbb{P}^n$ a smooth hypersurface, the canonical divisor $K_X$ can be computed as $$ K_X = (K_{\mathbb{P}^n} + X)|_X. $$ Is there a similar formula where $X$ is of higher ...
7
votes
2answers
204 views

Trivilisations of Vector Bundles

Let $\pi : E \to X$ be a smooth rank $k$ vector bundle on a manifold $X$ (I don't think my question depends upon the stipulations on the bundle, but I've just chosen smooth in case I'm incorrect). By ...
7
votes
1answer
183 views

Picard group of genus one curve

Is there a known example (or at least moral reason why such a thing should exist) of a genus $1$ curve $C/k$ over a field (assume perfect if you want) with no rational points such that ...
7
votes
1answer
159 views

Ways to think about vector bundle

I'm studying manifold theory and I've got to the point of discussing the definition of a vector bundle. The definition is quite long and a bit confusing and I was wondering if someone with a bit more ...
7
votes
1answer
496 views

Understanding the canonical line bundle $H$, and the fact that $(H \otimes H)\oplus 1 \simeq H \oplus H$

I am trying to understand Example 1.13 of Hatcher's book on vector bundles and K-Theory (page 24). The cannonical line bundle $H \to \mathbb{C}P^1$ satisfies the relation $(H \otimes H)\oplus 1 ...
7
votes
1answer
187 views

Alternate definition of vector bundle?

Recall the usual definition of a $k$-dimensional vector bundle (everything is assumed to be continuous/smooth/etc depending on the category): A $k$-dimensional vector bundle is a triple ...
7
votes
1answer
91 views

Line bundles over $\mathbb R P^2$

As in this post, I'm continuing studying line bundles. Now it's line bundle over $\mathbb R P^2$. I know that this bundle is not trivial. So I want list up to equivalence all bundles over $\mathbb R ...
7
votes
1answer
170 views

What's the intuition behind the tangent bundle?

Well, when we work with a smooth manifold $M$ we can associate with each point $p\in M$ a vector space $T_p M$ of all vectors at $p$ tangent to $M$: this is the space of linear functionals obeying ...
7
votes
2answers
218 views

Dual of a holomorphic vector bundle

Let $(E,\pi,M)$ be a holomorphic bundle, i.e. $(M,J)$ is a complex manifold and $\pi \colon E \to M$ is a complex bundle such that there exists a trivialization with holomorphic transition functions. ...
6
votes
3answers
203 views

Is there a way to establish a correspondence between the vector bundles over a torus and some kind of homotopy groups, just as we do to spheres

By introducing the "clutching function" one can relate the complex (real) vector bundles on a sphere with homotopy groups of $GL_n(\Bbb C)$ ($GL_n^+(\Bbb R)$ for oriented ones). Can we do a similar ...
6
votes
1answer
1k views

Tensor Product of Vector Bundles

In Hatcher's book on Vector Bundles he states (page 14) that It is routine to verify that the tensor product operation for vector bundles over a fixed base space is commutative, associative, ...
6
votes
4answers
228 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 ...
6
votes
1answer
122 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
6
votes
2answers
189 views

Vector bundles of rank $k$ with base $S_{1}$

I don't understand how to solve the following problem. Can you help me? Prove that there are only two vector bundles of rank $k$ with base $S^{1}$ $-$ trivial $1_{k}$ and non-trivial $\eta_{k}$. ...
6
votes
1answer
211 views

To prove a vector bundle $\xi$ is orientable $\iff w_1(\xi)=0$

I went about it this way: $\xi$ an $n$-rank vector bundle over a topological space $M$ is orientable. $\iff$ its top exterior power $\bigwedge^n\xi$ is trivial $\iff$ the 1-frame bundle ...
6
votes
2answers
91 views

Eigenbundle decomposition

Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action. It is known that we can decompose every $G$-equivariant vector bundle with respect to the action: ...
6
votes
1answer
229 views

Möbius strip and $\mathscr O(-1)$. Or $\mathscr O(1)$?

On the real $\textbf P^1$ we have these algebraic line bundles: $\mathscr O(1)$ and $\mathscr O(-1)$. Which one corresponds to the Möbius strip? (Both are $1$-twists of $\textbf P^1\times\textbf ...
6
votes
1answer
71 views

Is the space of smooth sections of a vector bundle finitely generated as a $C^\infty$-module?

Given a smooth finite-dimensional vector bundle $E\to M$ on a smooth manifold $M$, is the space of smooth global sections $\Gamma(E,M)$ always a finitely generated $C^\infty(M)$ module? This amounts ...
6
votes
1answer
148 views

Difference between $\mathfrak{X}(M)\otimes_{\mathbb{R}}\mathfrak{X}(M)$ and $\Gamma(TM\boxtimes TM)$

Let $M$ be a smooth manifold and $\mathfrak{X}(M)$ the real vector space consisting of vector fields over $M$. Let $TM\boxtimes TM$ be the vector bundle over $M\times M$ whose fiber at $(x,y)\in ...
6
votes
3answers
126 views

introductory reference for Hopf Fibrations

I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded, accessible book. ( I should probably say that I am just starting to ...
6
votes
1answer
108 views

Every vector bundle over $[0,1]^n$ is trivial

I would like to show the followoing result: Every vector bundle over $[0,1]^n$ is trivial First, I consider the case $n=1$, so let $E$ be a vector bundle over $[0,1]$. If $\nabla$ is a connexion ...
6
votes
1answer
388 views

Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle

Let $V$ be a finite dimensional vector space, and let $G_k(V)$ be the Grassmannian of $k$-dimensional subspaces of $V$. Let $T$ be the disjoint union of all these $k$-dimensional subspaces and ...
6
votes
1answer
64 views

exact sequence induced by restriction to closed subscheme

I'm somewhat embarrassed to be confused about this issue after a while studying algebraic geometry, but here goes. Let $\iota: Y \hookrightarrow X$ be the inclusion of a closed subscheme, $U$ is the ...
6
votes
1answer
181 views

Obstruction cocycle of Stiefel manifold

I was reading about the interpretation of Stiefel Whitney classes as obstructions from Milnor-Stasheff's book and I got stuck at a step. The context is the following. Let $E \to B$ be a vector bundle ...
6
votes
2answers
641 views

How to draw a complex line bundle?

The most basic example of a topologically non-trivial real line bundle is the well-known Möbius strip. Everyone who learns about vector bundles will be confronted by it, if only because it has the ...
6
votes
0answers
140 views

Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
6
votes
0answers
270 views

Orientability of the total space of a vector bundle over an oriented manifold

Let $M$ be a (smooth) manifold of dimension $n$, and let $\pi : E \to M$ be a (smooth) vector bundle of rank $r$. If I choose a connection on $E$, then I obtain a decomposition of $T E$ as $V E ...
5
votes
3answers
1k views

Chern Classes of a Trivial Bundle

Could someone explain to me why the chern classes of a trivial bundle are zero? (I'm studying it from Bott & Tu book) To be more specific I can't understand why, given the vector bundle $E$ on ...
5
votes
1answer
171 views

If a line bundle and its dual both have a section (on a projective variety) does this imply that the bundle is trivial?

Is there any reason that, on a projective variety X, if a line bundle L has a (non-zero) section and also its dual has a section then this implies that L is the trivial line bundle?
5
votes
1answer
136 views

Sheaves on $\mathbb{P}^n \times \mathbb{P}^m$, and a commutation relation for derived functors of global sections and tensor products on it.

I'll state my questions first and then provide some background. Question 3 is by far my most important one. We work over $k=\mathbb{C}$ whenever necessary. Is it true that $\text{Pic}(\mathbb{P}^n ...
5
votes
2answers
410 views

When does a line bundle have a meromorphic section?

Let $X$ be a scheme and $D$ be a Cartier divisor on $X$. Then $D$ determines a line bundle $\mathcal{O}(D)$ on $X$. Under which condition, is the converse true? That is, when does a line bundle come ...
5
votes
1answer
400 views

The Canonical Bundle over a Riemann Surface

I am trying to understand an example of a line bundle over a Riemann surface; as it is very terse and short, I have lots of trouble. It is written in block-quotes below, and I ask questions as I go. ...