For questions on vector bundles.
13
votes
7answers
536 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\ ...
10
votes
1answer
120 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 ...
8
votes
2answers
138 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 ...
8
votes
2answers
86 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 ...
8
votes
1answer
133 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
92 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 ...
7
votes
2answers
80 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
111 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 ...
7
votes
1answer
121 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 ...
6
votes
3answers
150 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
3answers
223 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 $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ ...
6
votes
2answers
97 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, ...
6
votes
1answer
120 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
1answer
333 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 ...
6
votes
3answers
59 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
91 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 ...
6
votes
1answer
133 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 ...
6
votes
1answer
150 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
137 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
0answers
111 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
151 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
4answers
128 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 ...
5
votes
2answers
117 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 ...
5
votes
1answer
88 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
1answer
169 views
Pasting Together Fibers of a Vector Bundle
Everyone:
Please forgive that I do not yet know LaTex, bro, and my English ( I am from UCV in Venezuela).
I think I understand concept of bundles almost well, and that, once a vector bundle with a ...
5
votes
1answer
121 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 ...
5
votes
1answer
200 views
On Frobenius reciprocity theorem
The classical Frobenius reciprocity theorem asserts the following:
If $W$ is a representation of $H$, and $U$ a representation of $G$, then $$(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res ...
5
votes
1answer
57 views
Tangent bundle : is a manifold
I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector ...
5
votes
1answer
64 views
Sub line bundle of a vector bundle
I am trying to read Friedman's "Algebraic surfaces and holomorphic vector bundles". I am unable to follow a claim (on pg 32) that any globally generated rank 2 vector bundle (say) $E$ on a complex ...
5
votes
2answers
436 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 ...
5
votes
0answers
93 views
Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both mobius 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 ...
5
votes
0answers
75 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 ...
5
votes
0answers
174 views
understanding this differential operator on a tensor product
I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
4
votes
1answer
557 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, ...
4
votes
2answers
98 views
Describe tangent and normal bundle to a manifold
Consider the set
$X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$
I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
4
votes
1answer
84 views
Seemingly contradictional facts on whether Chern classes determine a line bundle or not.
All varieties will be smooth when necessary.
Earlier i learned that the first Chern class of a line bundle on an algebraic variety does not determine the bundle up to algebraic isomorphism, i.e. the ...
4
votes
3answers
126 views
What is a tangent bundle? (Aubin)
Here's what I read in A Course in Differential Geometry by Thierry Aubin.
2.5. Definition. The tangent bundle $T(M)$ is $\bigcup_{P\in M} T_P(M).$
And then
2.6. Definition. Let $\Phi$ be a ...
4
votes
1answer
228 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.
...
4
votes
1answer
147 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 ...
4
votes
1answer
90 views
How to show $\det(E)\cong M\times \mathbb{R}$ when $M$ is orientable?
If we have an orientable bundle $E\rightarrow M$, then the transition maps can be adjusted by Gram-Schmidt process to be in $SO(n,\mathbb{R})$. So the determinant bundle $\det E$ is isomorphic to ...
4
votes
1answer
93 views
Question on the definition of ample vector bundles
Let $X$ be a compact complex manifold. According to Fulton and Lazarsfeld, a vector bundle $E$ on $X$ is called ample if the Serre line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ on the projectivized ...
4
votes
1answer
56 views
Vector Bundle Doubt..
Well I have a doubt about a rank $k$ vector bundle. My definition of vector bundle is: A rank $k$ vector bundle is a triple $(\pi, E, M)$ where $E$ and $M$ are smooth manifolds and $\pi:E\rightarrow ...
4
votes
1answer
75 views
Pull-back of sections of vector bundles
I'm sure this is a silly question but I'm stuck at the concept of pulling back sections of a vector bundle. Let $\pi:E\to X$ be a vector bundle on a variety $X$ and $f:Y\to X$ a morphism. We have a ...
4
votes
1answer
84 views
Specific homotopy between complex conjugation and the identity.
Consider the set $\mathcal{C} = C^{\infty}(\mathbb{C}^*, \mathbb{C}^*)$, where $\mathbb{C}^* = \mathbb{C}\backslash\{0\}$.
Both $f(z) = z$ and $g(z) = \bar{z}$ can be seen as elements in ...
4
votes
2answers
94 views
Bundle Automorphisms, Structure Groups and Gauge Groups
I am trying to get my head around the mathematical foundations of gauge theory and wanted to check that I am correct in thinking the following is true.
If $E$ is a $G$-principle bundle over $M$ then ...
4
votes
1answer
118 views
Existence of Complex Structures on Complex Vector Bundles
Let $E$ be a real vector bundle on a smooth manifold $X$. Let $J : E \to E$ be a vector bundle morphism (i.e. $\pi \circ J = \pi$, where $\pi : E \to X$ is the projection map) with $J^2 = ...
4
votes
1answer
116 views
Image of smooth vector bundle morphism
Let $\pi_1:V_1 \rightarrow B_1$ and $\pi_2:V_2 \rightarrow B_2$ be smooth vector bundles and write $\phi_V: V_1 \rightarrow V_2$ as well as $\phi_B:B_1 \rightarrow B_2$ respectively for the total and ...
4
votes
1answer
57 views
Computing Chern Classes of Tautological Line Bundles
I cannot find any references with how to handle this tautological line bundle $V \rightarrow P(\mathbb{H}^2)$ where $\mathbb{H}$ are the Hamilton quaternions.
I know it is a complex rank 2 vector ...
4
votes
1answer
139 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 ...
4
votes
0answers
45 views
How should we think of 'differences' of vector bundles?
Given a Hausdorff space $X$, the set of all vector bundles of finite dimensions $Vect(X)$ (up to isomorphism) with respect to the direct sum is a monoid. We can use the group completion construction ...
