For questions on vector bundles.

learn more… | top users | synonyms

52
votes
2answers
2k 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 ...
14
votes
7answers
643 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\ ...
14
votes
5answers
1k 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 ...
14
votes
1answer
134 views

Exists homeomorphism which carries each fiber isomorphically to itself, composition?

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see that there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically ...
13
votes
2answers
1k 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, ...
12
votes
1answer
637 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 ...
12
votes
1answer
181 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 ...
12
votes
0answers
114 views

Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
11
votes
2answers
335 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 ...
11
votes
1answer
522 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 $\mathcal{O}$-...
11
votes
2answers
257 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 ...
11
votes
1answer
288 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 ...
11
votes
1answer
846 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 ...
10
votes
3answers
1k 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 $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ (...
10
votes
3answers
679 views

“Drawable” Examples of Vector Bundles

I'm looking for examples of vector bundles that can be easily drawn or "illustrated" on a whiteboard for a talk I am giving. I know of a couple simple examples that I could use: When our base ...
10
votes
2answers
278 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 ...
10
votes
1answer
230 views

Is paralellizability a topological invariant (Invariant under homemorphism)

This MO post is a motivation to ask: Is paralellizability a topological invariant (Invariant under homeomorphism)?
10
votes
2answers
673 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 ...
10
votes
1answer
1k views

Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
10
votes
3answers
450 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. ...
9
votes
2answers
1k views

Any example of manifold without global trivialization of tangent bundle

It is said for most manifolds, there does not exist a global trivialization of the tangent bundle. I am not quite clear about it. The tangent bundle is defined as $$TM=\bigsqcup_{p\in M}T_PM$$ So is ...
9
votes
4answers
779 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 ...
9
votes
3answers
1k views

Exterior power of a tensor product

Given 2 vector bundles $E$ and $F$ of ranks $r_1, r_2$, we can define $k$'th exterior power $\wedge^k (E \otimes F)$. Is there some simple way to decompose this into tensor products of various ...
9
votes
1answer
309 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 $\textbf P^1\times\textbf A^1$...
9
votes
2answers
152 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 $\nabla:\mathcal{...
9
votes
1answer
280 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 ...
9
votes
2answers
119 views

Properties of the category of smooth vector bundles over a smooth manifold

I am wondering if there are any sources that discuss the properties of the category of vector bundles over a smooth manifold. It seems that most differential geometry texts I've looked at avoid ...
9
votes
1answer
214 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 $\phi^*...
9
votes
1answer
192 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 reference,...
9
votes
1answer
115 views

Is a space with no nontrival vector bundles contractible?

Let $X$ be a "nice" space, say having the homotopy type of a CW complex. Suppose also that $X$ is connected. Suppose that all real vector bundles on $X$ are trivial. Does it follow that $X$ is ...
9
votes
1answer
423 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 ...
9
votes
0answers
97 views

What's Griffiths' ampleness in modern language?

When reading Griffiths' paper "Hermitian differential geometry, Chern classes and positive vector bundles", I found his definition of ampleness: Let $X$ be a compact complex manifold, $E\to X$ a ...
8
votes
1answer
2k 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, ...
8
votes
2answers
179 views

Regular sequence of sections of line bundles over a coherent sheaf

I am reading the first chapter from the book by Huybrechts and Lehn, where I encountered the following definition. I have the following doubts regarding this definition : What is the map $s:E\...
8
votes
2answers
270 views

Spin structures, frame bundles, and trivializations over the 2-skeleton

While reading an introduction to Spin- and Spin$^{\operatorname{c}}$ structures (found here), I encountered the following definition: Let $E\to X$ be an oriented $\mathbb{R}^n$-bundle over a CW ...
8
votes
1answer
328 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 ...
8
votes
2answers
89 views

Tensor product of vector bundles

So I have a problem that I encountered and I don't know what object I am looking at. So there are two manifolds $M_1$ and $M_2$ and a smooth map $f:M_1\rightarrow M_2$. We know that there is a vector ...
8
votes
1answer
187 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
912 views

Tangent bundle of a quotient by a proper action

Given a compact group $G$ acting freely on a manifold $X$, is there a "nice" way to describe the tangent bundle of the quotient $X/G$ (when it is a manifold)? In the case the group $G$ is finite, or ...
8
votes
0answers
146 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 null-...
7
votes
2answers
419 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
1answer
925 views

Classifying vector bundles

Given a manifold $M$, is there a way of classifying up to isomorphism all possible vector bundles over $M$ of a given rank? Some other questions on this site deal with specific cases, which all seem ...
7
votes
1answer
430 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?
7
votes
3answers
174 views

$2$-dimensional subbundle of tangent bundle of closed $3$-manifold integrable if and only if $\alpha \wedge d\alpha = 0$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. From here and here, I know that there is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any ...
7
votes
1answer
74 views

How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of tangent $2$-planes? [duplicate]

A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a subbundle of dimension $k$. How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of ...
7
votes
2answers
279 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 $\eta_{k}$. ...
7
votes
1answer
168 views

Is T($S^2 \times S^1$) trivial?

How would I find out if T($S^2 \times S^1$) is trivial or not? Using the hairy ball theorem I can show that T($S^2$) is not trivial, and it is straight forward to show that T($S^1$) is trivial. ...
7
votes
1answer
436 views

Reduction of a structure group.

Let $X$ be a smooth manifold and $\pi:E\rightarrow X$ a vector bundle of rank $k$ on X. If one manages to redefine $E$ by using a cocycle $\{g_{\alpha,\beta}\}$ whose values are all contained in a ...
7
votes
2answers
257 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
546 views

Comparing notions of degree of vector bundle

In this question, $X$ will be a smooth complex projective variety. This question will be about comparing two different ways of calculating the degree of a vector bundle on such an $X$. I understand ...