Questions tagged [vector-bundles]
For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.
2,133
questions
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Explicitly exhibit that vector fields over the 2-sphere is a projective module
We know by the Serre Swan theorem that smooth vector fields over a smooth manifold form a projective module over the ring of smooth functions. We also know that the hairy ball theorem implies that the ...
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60 views
Does every vector bundle have a 'tensor inverse'?
For any vector bundle $E$ over a finite-dimensional CW complex, there is a vector bundle $E'$ such that $E\oplus E'$ is trivial.
For a compact Hausdorff base, this is Proposition 1.4 of Hatcher's ...
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19 views
Open subset moduli space of parabolic vector bundles
Let $X$ be a connected, compact Riemann surface and let $p_1\dots p_k$ distinct points on it. We denote by $D$ the associated divisor. After the choice of so-called weight vectors $\alpha_i$ for each ...
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1answer
79 views
Kernels and Cokernels of differentials on a bounded long exact sequence.
Given a bounded long exact sequence on a smooth projective variety where objects are direct sums of $\mathcal{O}(n)$ for different integer $n$'s i.e. $\bigoplus_i\mathcal{O}(n_i)$. Is there anything ...
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27 views
Induced representations built on sections of an associated vector bundle. Questions on notations
Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions $\varphi$ on this group:
$$
{\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\...
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28 views
Chern character of the restriction of a canonical bundle over product space
I am new to StackExchange, and self-taught in the field of characteristic classes, vector bundles, etc... so apologies in advance if my question is somewhat trivial or ill-posed.
I am trying to do a ...
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21 views
Show that the kernel of a bundle morphism with constant rank is a sub-bundle.
I'm trying to solve the following problem:
Let $M$ be a smooth manifold and $\pi: E \to X$ and $\pi ':F \to X$ be smooth vector bundles of $X$. Suppose we have a (smooth) bundle morphism $f:E\to F$ of ...
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1answer
54 views
Local expression of differential forms of a vector bundle
Let $ p: M \to B $ be a smooth vector bundle over B.
I'm wondering how to express a differential form on M in local coordinates. I'm aware of the local expression of a differential form on a ...
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52 views
Computing cohomology of tautological bundles on Fano threefold
Consider the Fano threefold $V=V_5$ of index $2$ and degree $5$. Note that $V = \mathrm{Gr}(2,5) \cap \mathbb{P}^6 \subset \mathbb{P}^9$. Let $U$ and $Q$ be the tautological and tautological quotient ...
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1answer
31 views
Why is the set of all smooth sections of a smooth vector bundle closed under addition?
Let $\pi:E\to M$ be a smooth vector bundle, where $E$ and $M$ can have nonempty boundaries. Let $\sigma_1$ and $\sigma_2$ be smooth global sections of $E$. I want to show that $\sigma=\sigma_1+\...
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1answer
33 views
Quasi-coherent sheaf which is a vector bundle on curves.
This question is inspired by this question. Given a quasi-coherent sheaf on a smooth variety $X$ such that its restrictions to curves are finite dimensional vector bundles. Does it follow that the ...
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21 views
Simply connected and rationally connected varieties in char $p$.
Consider a smooth projective variety in char $p$ with trivial etale fundamental group. You can assume it is also rationally connected. I believe there are a lot of examples for these types of ...
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30 views
understanding of structure equation
For a vector bundle $E$, we have the structure equation relating a curvature form $\omega$ and its connection form $\Omega$, which is
$$\Omega=d\omega +\omega \wedge \omega$$
Under local ...
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39 views
Let $X$ be a smooth manifold and $\pi:E \to X$ be a smooth vector bundle. Show that the set of sections $\Gamma(X,E)$ is an $C^{\infty}(X)$ module.
I'm trying to show the following:
Let $X$ be a smooth manifold and $\pi:E \to X$ be a smooth vector bundle. Show that the set of sections $\Gamma(X,E)$ is an $C^{\infty}(X)$ module.
I am stuck showing ...
2
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1answer
28 views
Any complex curve in a complex surface is the zero set of some holomorphic section of a holomorphic line bundle
Let $Y$ be a closed complex surface, $L\to Y$ be a holomorphic line bundle, $\sigma:Y\to L$ a holomorphic section, and $B\subset Y$ the zero set of $\sigma$. If the first Chern class $c_1(L)$ of $L$ ...
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83 views
Conjugating the curvature with a parallel transport operator does not make sense
I'm trying to understand the proof of the Ambrose-Singer theorem, following, for instance, Werner Ballmann's notes. I understand that the gist of the theorem is that, when you have a vector bundle $(E,...
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1answer
49 views
Hint for proof of proposition 10.15, lines a) and c) in John M. Lee's book “An Introduction to Smooth Manifolds”
I have been thinking about how to prove proposition 10.15, lines a) and c) in John Lee's book "An Introduction to Smooth Manifolds" (I leave a screenshot of the proposition below), but I can'...
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73 views
Tangent bundle of the connected sum of two smooth manifolds
Let $M,N$ be two smooth connected real $n$-dimensional manifolds. Suppose they are both orientable for simplicity. Let $i:D_1\hookrightarrow M$ and $j:D_2\hookrightarrow N$ two embedded disks so that $...
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103 views
On the complexification of a holomorphic bundle $E$
Let $E\to M$ be a holomorphic vector bundle over a complex manifold. Considering it as a real vector bundle one can complexify $E$ to $E^{\mathbb{C}}$. The bundle $E^{\mathbb{C}}$ is again holomorphic ...
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29 views
Equivalence between Ehresmann and Koszul definition of linear connection
I was taught, in different courses, the following definitions of connection.
An Ehresmann connection on a fibre bundle $\pi:B\rightarrow M$ is a distribution $H\hookrightarrow TB$ such that:
$$\...
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42 views
Eheresmann connection and covariant derivative
I was introduced the following definition of connection on a fibre bundle (actually my professor gave the same definition in the context of fibred manifolds).
Which I understood is due to Ehresmann
A ...
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63 views
Algebraically closedness on a theorem of Hartshorne.
The following theorem is from Hartshorne's book on ample subvarieties:
Some definitions:
The theorem:
This is Corollary 1.2:
This is prop 1.3: (The proof of this one is very long not sure if I can ...
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1answer
42 views
Function defined on the tangent bundle
Suppose $F$ is a smooth real-valued function defined on the tangent bundle $TM$ of a Riemannian manifold $M$, i.e. $F:TM\to\mathbb{R}$ given by $F(p, v)\in\mathbb{R}$. Consider the function $f:M\to\...
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Orientability of vector bundles and the map $BSO(n) \to BO(n)$
Edit: (1) below is now clear, but I'm still looking for an answer for (2).
There are two issues I have when trying to understand the classifying spaces of (oriented) vector bundles:
(1) Let $E \to X$ ...
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0answers
29 views
Few questions regarding formal vector bundles on formal completions.
This question is regarding formal vector bundles on formal schemes. The formal schemes that I'm interested are formal completion of a scheme along a closed sub-scheme. The category of formal coherent ...
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42 views
Lemma 1.45 in the book of heat kernels and dirac operators
I'm trying to understand the proposition which says that if $p: \mathcal{V} \rightarrow B$ is a real vector bundle and $j : B \rightarrow \mathcal{V}$ is the inclusion from B to $ \mathcal{V}$ as ...
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32 views
Existence of fibre metric compatible connections on vector bundle
Given a fibre metric $h$ over a vector bundle $E$, how can I show there always exist connections on $E$ such that it is compatible with $h$ in the sense that given any two smooth section $\sigma,\psi$,...
2
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0answers
52 views
Unimportance of the metric for spin structures
I have some pedantic confusions when it comes to spin structures. Let $B$ be a nice space (like a CW complex) and let $\xi$ be an oriented vector bundle on $\xi$. If I choose a metric on $\xi$, then ...
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26 views
Decision problem for the splitting type of rank 2 bundle from the given exact sequence
Let $\mathcal{F}$ be a rank 2 vector bundle over some algebraic variety.
After restrict to the given line $L\cong\mathbb{P}^1$, it fits into an exact sequecne
$$
0 \rightarrow \mathcal{O}_L(-2)\...
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31 views
High rank vector bundle over surfaces (reference request)
I would like to find a reasonably concise reference for the following fact.
Let $E$ be an orientable vector bundle over a closed orientable surface $\Sigma$ with rank at least 3. Suppose the second ...
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55 views
Orientability vs $n$-frames via Stiefel-Whitney classes
I am trying to understand Stiefel-Whitney classes as obstructions for the existence of linearly independent sections of a vector bundle $E \to B$, that is a section of the Stiefel bundle $V_k (E) \to ...
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1answer
44 views
Is it possible for each vector bundle to find another one that their direct sum is trivial on an affine?
Given a smooth projective variety $X$ and a hyperplane section complement $U$. For any vector bundle $E$ on $X$, is it possible to find another vector bundle $F$ such that $E\oplus F$ is a trivial ...
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19 views
Canonical dilation vector field on a vector bundle
I am reading a paper by Langerock, called "A connection theoretic approach to sub-Riemannian geometry".
He writes:
[Let $\pi:E\rightarrow M$ be a vector bundle] ... Let $\{\phi_t\}$ ...
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0answers
21 views
Coproduct in the category of differential bundles
Just for clarity I briefly state the definition I am using. A differential bundle is a differentiable submersion $\pi: E\to B$, where $E$ and $B$ are differential manifolds. Observe that I am not ...
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59 views
Proving that $\frac{\nabla^2 u}{ds \, dt} - \frac{\nabla^2 u}{ds \, dt} = k_{\nabla}(\frac{d\gamma}{ds}, \frac{d\gamma}{dt})(u)$
I am currently trying to make Exercise 20 of these lecture notes.
It should be a very simple exercise, but I am swamped due to the amount of bookkeeping. I fear I am missing a very simple nuance that ...
4
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0answers
58 views
sheaf inclusion of line bundle
Suppose $(X,\mathcal O _X)$ is a complex manifold, $L$ a line budle with nontrivial global section $s\in H^0(X,L)$. Then in sheaf category, one has a inclusion $\mathcal O_X\rightarrow L$. Notice that ...
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20 views
Effect of square integrable Chern curvature on underlying bundle
Let $\left(E,h\right)$ be a hermitian holomorphic vector bundle over a complex manifold $M$. Then there exists a unique hermitian connection $\nabla^h$, the $\left(0,1\right)$-part of which is the ...
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1answer
49 views
Why is a vector bundle called E?
Vector bundles are often denoted as $p:E \to B$, where $p$ is a projection map, $B$ is the base space and $E$ is the total space. Here the choice of the letters $p$ and $B$ is clear, but is there also ...
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2answers
104 views
Rank of coherent sheaf on complex manifolds
Let $X$ be a complex smooth connected manifold of dimension $n$ and let $\mathcal{O}_{X,x}$ be the ring of germs of holomorphic functions at $x$. Since the stalk of $\mathcal{O}_X$ at $x$ does not ...
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1answer
32 views
Is it possible for the pullback of an ample line bundle under projection to be big?
Given an ample line bundle on a curve is there any chance for its pullback to the projective bundle of some vector bundle to be big? It is known that first cohomology of inverse of nef and big line ...
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1answer
70 views
Bianchi Identity in Chern's book
I'm reading Chern's book "complex manifolds without potential theory".
In chapter 5 he claims the following: Let $E\to M$ be a vector bundle, $D$ a connection, $\omega$ the corresponding ...
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1answer
37 views
Direct summands of successive extensions of line bundles
On a projective variety any direct summand of a direct sum of line bundles is going to be direct sum of a subset of initial line bundles. This follows from the Krull-Schmidt property of vector bundles ...
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63 views
Christoffel symbols $\Gamma$ as an $\text{End}(E)$-valued one-form implies it is a tensor field?!
I'm currently reading John Baez's Gauge Fields, Knots and Gravity for fun, and I stumbled across the following when talking about connections in a vector bundle. So, let $\pi:E\to M$ be a smooth ...
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33 views
Generalizing the clutching construction to more contractible open sets
In computing the topological $K$-theory of $\mathbb{R}P^2$ I had an idea to mimic the clutching construction: Recall that this says that for the open cover $S^k = D^k_+\cup D^k_-$ a rank $n$ vector ...
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0answers
24 views
Symplectic potential determines local trivialization?
Perhaps this is pretty basic, but somehow I'm not getting it:
Given a symplectic manifold $(M,\omega)$, a Hermitian line bundle $B \rightarrow M$, pick a symplectic frame and set
$$ \theta = \frac{1}{...
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1answer
52 views
A question about the complex conjugate bundle
If a complex vector bundle is constructed by the complexification of a real vector bundle, say $E=F\otimes \mathbb{C}$, then there's a conclusion that $E$ is isomorphic to its conjugate bundle by ...
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1answer
41 views
Degree of vector bundle under extension of scalars.
I have a somewhat of a trivial question, just want to confirm it (or maybe its wrong). Degree of a vector bundle doesn't change when we extend the scalars is that right? So if a vector bundle is semi-...
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47 views
Kernels of the surjection from trivial vector bundle to $\mathcal{O}(k)$.
Is there a way to understand the Harder-Narasimhan filtration (the slopes appearing) for the kernel bundle of the surjection $\mathcal{O}^n \rightarrow \mathcal{O}(k)\rightarrow 0$ on an algebraic ...
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1answer
54 views
Pullback of locally free sheaf over the associated vector bundle
Let $X$ be a scheme over $k$, $\mathcal{F}$ be a locally free sheaf on $X$ of rank $r$, $F$ be the vector bundle associated to $\mathcal{F}$ i.e. $F=\operatorname{Spec}\operatorname{Sym}(\mathcal{F}^{*...
2
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1answer
34 views
About trivial restrictions of a vector bundle
Let $M$ be a smooth manifold of dimension $n$. Let $U \subset M$ be an open subset such that the restriction $TM|_U$ is trivial. Let $N$ be a closed submanifold of $M$ contained inside $U$. Does it ...
