For questions on vector bundles.

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4
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Are these “Stiefel-Whitney numbers” invariants of the (smooth)topology of the total space?

Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of ...
10
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1answer
172 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)?
2
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2answers
33 views

Mobius over the sphere is the sphere itself

The Mobius band can be thought as a line bundle over $S^1$ by giving the vector spaces half a twist at some point. Now, we can do the same kind of construction by considering a line bundle over the ...
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0answers
30 views

Subsheaves of locally free sheaves on a rational curve

Let $k$ be an algebraically closed field, $\mathcal{E}$ a locally free sheaf on $\mathbb{P}^1_k$ and $\mathcal{L} \subset \mathcal{E}$ a subsheaf of rank $1$. By Grothendieck's theorem, we know that ...
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0answers
14 views

Vector bundle morphism as section of a bundle?

Let $\xi:=(E, p, M)$ and $\eta:=(F, q, N)$ be two smooth real vector bundles. A vector bundle morphism from $\xi$ to $\eta$ is a pair of smooth maps $(f:E\longrightarrow F, g:M\longrightarrow N)$ ...
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0answers
47 views

Two simple counterexamples in algebraic geometry

Suppose we have a smooth complex algebraic variety $X$. Then in general, $K^a(X)\to K(X^{an})$ is not surjective. Could someone give an example of a topological vector bundle class which contains no ...
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0answers
45 views

Vector bundles on Hirzebruch surface $\mathbb{F}_2$

I would like to know a classification for all holomorphic vector bundles on the second Hirzebruch surface $\mathbb{F}_2$. Is this known? What is known? In particular, I'm looking for holomorphic ...
3
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2answers
111 views

“Basis extension theorem” for local smooth vector fields

Let $\pi: E \to M$ be a smooth vector bundle of rank $n$, and suppose $s_1, \ldots, s_m$ are independent smooth local sections over an open subset $U \subset M$. Can I prove the "basis extension ...
4
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2answers
65 views

Can $S^4$ be the cotangent bundle of a manifold?

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished $1$-form on $T^*V$. It seems that there is no such distinguished $1$-form on a ...
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0answers
24 views

Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
2
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0answers
49 views

$V$-bundles and vector bundles

I am looking for more information on $V$-bundles. They are hard to search for as either vector bundles come up or something like GL($V$)-bundles come up. I am looking for some nice expository ...
2
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1answer
44 views

What line bundle pulls back to the trivial line bundle

Let $X$ be an abelian surface. $C$ be a curve in $X$. Consider the projective bundle $\pi:\mathbb{P}^1_C\longrightarrow C$. This is a projective morphism. I have two questions : 1) Can we find an ...
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1answer
16 views

Sheaf of sections of vector bundle over a manifold is an $\mathcal O_M$-module

Section 13.1.2 of Ravi Vakil: "Fix a rank $n$ vector bundle $\pi:V\rightarrow M$. The sheaf of sections $F$ of $V$ is an $\mathcal O_M$-module - given any open set $U$, we can multiply a section over ...
3
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0answers
39 views

Vector bundles in Ravi Vakil's notes on quasicoherent sheaves

In chapter 13 (Quasicoherent and coherent sheaves) of Ravi Vakil's wonderful notes, the author starts by discussing vector bundles, supposedly for motivation. Having understood that each locally free ...
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0answers
56 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 - ...
3
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1answer
47 views

Möbius band as line bundle over $S^1$, starting from the cocycles

The professor asked us to construct a non-trivial line bundle over $S^1$ by giving an open cover of $S^1$ and the cocycles. My idea was to take as open cover $U_1:=S^1\setminus\{0\}$ and ...
0
votes
1answer
24 views

Obstruction to the splitness of an exact sequence of holomorphic vector bundles

This question was asked here, but I think there will not be an answer, so let me recopy the question on Mathoverflow. In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma ...
4
votes
2answers
60 views

Is the total space of a vector bundle over an irreducible scheme irreducible?

Let $X$ be an irreducible scheme over $\mathbb{C}$ and let $F$ be a locally free sheaf of rank $r$ on $X$. Is the total space $Y$ of the associated vector bundle to $F$, $Y=Spec(Sym(F^{\vee}))$, ...
1
vote
1answer
55 views

If a line bundle admits a non-vanishing section then it is trivial

Suppose $\pi:E\to B$ is a line bundle. Let $s:B\to E$ a non-vanishing section, i.e. for every $b\in B$ $s(b)\ne 0$ and $\pi\circ s=Id_B$. I have to prove that the line bundle above is trivial. Idea: ...
2
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0answers
63 views

What is the push forward of the canonical class?

Let $X$ be an abelian surface over $\mathbb{C}$. And let $i:X\longrightarrow X$ be the inverse map. $i$ is a degree 2 morphism. We consider $Y$ the quotient of $X$ by the action of $i$, that is, ...
2
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1answer
58 views

A fiber product is a fiber bundle

Let $F,B$ be topological spaces. A fiber bundle $E$ over the basis $B$ with fiber $F$ is a topological space $E$ endowed with a continuous surjection $\pi:E\to B$ such that there exists an open cover ...
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1answer
23 views

Are vector bundles special cases of étale bundles?

Is it possible to define vector bundles as particular instances of étale bundles? An étale bundle is a bundle $p:E\rightarrow X$ which is a local homeomorphism (as in Maclane-Moerdijk): every $e\in ...
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0answers
58 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
3
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0answers
55 views

Quick question: Determinant bundle is Cartier?

$X$ is an algebraic surface (i.e. compact complex, which embeds in a projective space). $V$ is a vector bundle of rank 2 over $X$. Why is $\det{V}=\mathcal{O}_X(D)$ for some divisor $D$? Is there a ...
1
vote
1answer
83 views

Lie Derivative of a section on a vector bundle

I'm still trying to figure out how to do the Lie derivative of a Jacobian. (c.f. earlier unanswered post). If I know how how to do Lie derivatives on section of vector bundles, that would be ...
1
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1answer
51 views

Two definitions of conormal bundle

Suppose $X$ is a smooth manifold and $f: Z \to X$ is an immersion with transverse self-intersection. I've seen (e.g. here) the conormal bundle to $Z$ in $T^* X$ defined as Definition A: $\quad L_Z := ...
2
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2answers
53 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. $$w_i(E)=\phi^{-1} Sq^i(u)$$ ...
2
votes
1answer
118 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
3
votes
1answer
92 views

Compactifying $\mathcal{O}_{\mathbb{P}^1} (-2)$

I have the total space of $\mathcal{O}_{\mathbb{P}^1} (-2)$ and I see that a "standard" way to compactify is to add the trivial line bundle, $\mathcal{O}_{\mathbb{P}^1}$, and then projectivize. That ...
3
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1answer
55 views

Boundedness of the operator $[\Lambda, \Theta]$.

I am reading Griffiths-Harris book on algebraic geometry and in "Theorem B", where he proves (the analogue of Serre's vanishing theorem) that Let $M$ be a compact, complex manifold, $L\rightarrow ...
0
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1answer
34 views

Rank $n$ vector bundle with $n$ pointwise linearly independent sections is trivial

I have an $n$-dimensional vector bundle $E \to X$ and sections $s_1, \dots, s_n : X \to E$ such that for any $x \in X$, the elements $s_1(x), \dots, s_n(x)$ are linearly independent in the vector ...
3
votes
1answer
40 views

Symbol of the differential operator on vector bundles

Suppose that we have a differential operator $D:C^{\infty}(\mathbb{R}^n) \to C^{\infty}(\mathbb{R}^n)$ of the form $(Df)(x)=\sum_{|\alpha| \leq k}a_{\alpha}(x)\frac{\partial^{|\alpha|}f}{\partial ...
0
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2answers
71 views

Triviality of a vector bundle is an open condition

Is the following statement true (and if not, are there additional assumptions that make it true?) A vector bundle on a variety which is trivial if restricted to a closed subvariety is trivial on ...
4
votes
1answer
58 views

$w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$

How do I show that $w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$?
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0answers
11 views

Defining an Ehresmann Connection Using Linear Connection

In some places that I have seen, given the covariant derivative $\nabla$ of a linear connection on a vector bundle $\pi: E\rightarrow M$ (dim(M)=k), an Ehresmann connection is defined in the following ...
0
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1answer
46 views

injective morphism between line bundles on curves

Let $X$ be a smooth projective, irreducible, curve, $\mathcal{L}$ be an invertible sheaf on $X$ and $\mathcal{L}' \subset \mathcal{L}$, an invertible subsheaf. Is $\deg(\mathcal{L}') \le ...
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0answers
14 views

Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
3
votes
1answer
47 views

Transformation behavior of connection on vector bundle.

Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be ...
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4answers
72 views

commuting property of connections and bundle homomorphisms

I have the following situation: $E,F$ are (smooth) vector bundles over a smooth manifold $M$. Assume we are given connections $\nabla^E,\nabla^F$ on $E,F$ and a homomorphism of $M$-bundles ...
4
votes
1answer
51 views

Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
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0answers
20 views

Classification of rank $\geq 2$vector bundles over Grassmanians

Are there classification results of higher rank (complex) vector bundles over (complex) Grassmanian manifolds? For example, we know that line bundles are in correspondence with the $H^2(G)$, the ...
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0answers
11 views

If $\xi$ is a vector bundle and $f^{*}(\xi)$ is the bundle induced by $f$ then $f^{*}(\xi)$ is orientable.

Let $\xi = \pi : E \rightarrow X$ be a vector bundle and $f: Y \rightarrow X$ a continuous map. Let $E' \subset Y \times E$ be the set of all $(y,e)$ with $f(y) = \pi(e)$. We can define a vector ...
2
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1answer
36 views

Is there a natural Dolbeault operator on a almost holomorphic vector bundle?

For vector bundles $(\pi: V \rightarrow M )$ over a complex manifold, there is a notion of holomorphicity that can be defined in two equivalent ways : $V$ is a complex manifolds and $\pi:V ...
0
votes
1answer
24 views

Prove that $s: B \rightarrow E$ is a section ( vector bundles)

I'm very very unfamiliar with vector bundles, so maybe this question is quite trivial. Let $\pi : E \rightarrow B$ be a vector bundle and $s: B \rightarrow E$ a map sending each $p \in B$ to the $0$ ...
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1answer
20 views

construction of linear independent local section in vector bundle [duplicate]

Let $X_1,\ldots,X_k$ be lineary independent local sections of a vector bunlde $V$, $rank(V)>k$, defined on some open set $U$. Can I construct a local section $X_{k+1}$ on a (maybe smaller) open ...
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0answers
57 views

Proof of the formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
3
votes
1answer
65 views

Explaining the definition of vector bundles

Recall the definition of a vector bundle: Let $M$ be a topological space. A $k$-dimensional vector bundle over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E \to ...
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1answer
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1answer
32 views

Why are compact complex manifolds Liouville?

I know this is true but strangely can't find references. Also, consider the trivial $n$-bundle over any connected compact manifold, does Liouville imply that all holomorphic sections are constant? ...
0
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1answer
44 views

$\bigwedge^k T^*M$ is a $\binom{n}{m}$-dimensional Subbundle of $\bigotimes^k T^*M$.

I am trying to prove the following: Let $M$ be a smooth manifold. Then $\bigwedge^k T^*M$ is a smooth subbundle of dimension $\binom{n}{k}$ of $\bigotimes^kT^*M$. To do this, I think the ...