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 $X$.

Filter by
Sorted by
Tagged with
1
vote
1answer
22 views

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 ...
7
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
2
votes
0answers
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\...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
1
vote
0answers
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 ...
2
votes
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+\...
2
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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
votes
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 $L$ ...
2
votes
0answers
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,...
0
votes
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'...
4
votes
0answers
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 $...
2
votes
0answers
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 ...
0
votes
0answers
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: $$\...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
1
vote
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\...
3
votes
0answers
91 views

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 $E \to X$ ...
1
vote
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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
votes
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 ...
0
votes
0answers
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)\...
1
vote
0answers
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 ...
2
votes
0answers
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 ...
0
votes
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 ...
1
vote
0answers
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 $\{\phi_t\}$ ...
1
vote
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 ...
4
votes
0answers
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
votes
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 ...
2
votes
0answers
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 ...
1
vote
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 ...
4
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
4
votes
0answers
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 ...
2
votes
0answers
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 ...
3
votes
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}{...
0
votes
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 ...
1
vote
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-...
1
vote
0answers
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 ...
0
votes
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
votes
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 ...

1
2 3 4 5
43