For questions on vector bundles.
6
votes
2answers
81 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 ...
3
votes
2answers
48 views
Showing that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is a smooth vector bundle over $S^1$
I want to show that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is the total space of a smooth vector bundle over $S^1$. Here $\sim$ is defined as follows: fix a linear isomorphism $L$ of ...
3
votes
1answer
49 views
Transition functions of trivial vector bundle
I have a question on trivial vector bundles. The question is as follows:
Can we characterize the transition functions of a trivial vector
bundle in some way?
To be very concrete: suppose we ...
4
votes
0answers
40 views
How many sections of a vector bundle send a point outside a divisor?
Let $\pi:E\to X$ be a holomorphic vector bundle on a complex algebraic variety $X$, and assume $E$ has nonzero global sections; fix a divisor $D\subset E$ and a point $P\in X$.
I have the vague ...
1
vote
1answer
45 views
Dropping the orientable condition from the Thom isomorphism theorem.
My first question is: what are some examples of un-oriented n-plane bundles over $ \mathbb{Z}$ where it is easy to see that there is no Thom class?
I would like to know some examples because the real ...
0
votes
0answers
30 views
Exactness of $\Gamma^\infty$ Functor
Does anybody know a reference for fact that the Functor $\Gamma^\infty$, assigning to every smooth vector bundle $\mathcal{E}\to M$ the $C^\infty(M)$-module $\Gamma^\infty(\mathcal{E})$ of smooth ...
2
votes
0answers
47 views
Chern class of tautological line bundle
I'm studying characteristic classes from the Chern-Weil construction (via connection and curvature). I'm trying to compute some simple examples. Let $E$ be the tautological line bundle over projective ...
2
votes
0answers
50 views
An orbit of a group action and the implicit function theorem
Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function ...
3
votes
0answers
29 views
Maps between total spaces of holomorphic vector bundles
I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles.
Let me outline a situation that is a bit more concrete, to help focus ...
1
vote
0answers
24 views
Non-(stable)-triviality of the tautological bundles
The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold $G_k(\mathbb{K}^N)$ of all $k$-planes in $\mathbb{K}^N$ (for $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or ...
4
votes
1answer
80 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 ...
1
vote
0answers
32 views
Representation of Homogeneous vectorbundle = Induced representation
Hello friends of mathematics :)
I have a question about the induced representation. Suppose $G$ is a group and $H$ a subgroup of $G$. Suppose $\rho$ is a representation of $H$ on the vectorspace $V$, ...
3
votes
1answer
57 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 ...
3
votes
0answers
70 views
How to calculate characteristic classes of tensor products?
I was given the following as an execrise:
Prove that the tensor product of complex line bundles over $X$ satisfies the following relationship:
$$c_{1}(\otimes E_{i})=\sum c_{1}(E_{i})$$
It is ...
2
votes
0answers
32 views
Curvature form projective spaces
Let $T\mathbb{C}P^n$ tangent bundle over $\mathbb{C}P^n$. We have an hermitian metric on $T
\mathbb{C}P^n$ defined as $h=\frac{dzd\bar{z}}{(1+|z|^2)^2}$. If we consider Levi-Civita connection we can ...
2
votes
1answer
45 views
Is Whitney sum of vector bundle a categorical colimit?
We known that the direct sum of two vector spaces is the categorical colimit of vector spaces. My question is whether Whitney sum of vector bundle is a categorical colimit (in the category of vector ...
4
votes
0answers
64 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 ...
1
vote
0answers
41 views
Canonical bundle and Möbius bundle
I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Möbius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
0
votes
0answers
17 views
Isomorphism canonical and Moebius bundle.
I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Moebius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
1
vote
1answer
37 views
Change of frame of a vector bundle
Let $(E, \pi, M)$ be a complex vector bundle of rank $k$. Let $U \subset M$ be an open set and let $f = (s_1, \dots, s_k)$ be a frame (i.e. s_i are linearly independent). Of course $U$ is supposed ...
6
votes
2answers
142 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.
...
0
votes
1answer
69 views
Universal bundles and classificant maps.
We have this theorem: Let $\xi=(E,p,X)$ be a vector fiber bundle with rank($\xi)=n$. We can find a map $f: X \rightarrow Gr_n(\mathbb{C}^{\infty})$ such that $\xi=f^*(\gamma_n)$. I denote with ...
1
vote
0answers
23 views
Bott connection
Can anyone help me showing the following: Let $E$ be a smooth vector bundle over $M$ and $F\subseteq TM$ a smooth distribution. A $F$-connection is a $\mathbb R$-bilinear aplication ...
1
vote
1answer
29 views
Positive curvature on holomorphic vector bundles
There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize:
Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. ...
2
votes
1answer
39 views
Prove that a tensor field is of type (1,2)
Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let
$$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$
Prove that $N$ is a tensor field of ...
1
vote
1answer
36 views
Distribution and Tangent Bundle
Let $F=\{F_p; p\in M\}\subseteq TM$ be a rank $k$ smooth distribution. Can anyone explain-me what is the set $$\displaystyle\nu(F)=\frac{T_pM}{F_p}.$$
4
votes
1answer
59 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
0answers
47 views
Which is the correct universal line bundle: the tautological bundle or its dual?
With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$
In ...
6
votes
3answers
65 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 ...
2
votes
1answer
47 views
Classification of flat complex line bundles
I'm having a contradiction with two different classifications of flat complex line bundles over a manifold $X$. Suppose for simplicity that $H^2(X;\mathbb Z) = 0 = H^1(X;\mathbb C)$. Then the only ...
3
votes
2answers
44 views
Vector Bundles and Distributions
How can I show that following: If $F\subseteq TM$ is a smooth distribution then $F$ is vector bundle and the inclusion $F\hookrightarrow TM$ is a morphism of vector bundles?
1
vote
0answers
59 views
A funny condition for ampleness on a curve
Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$.
Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample ...
4
votes
1answer
93 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 ...
8
votes
1answer
134 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 ...
5
votes
4answers
137 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 ...
2
votes
1answer
81 views
How do we define ample vector bundles
Let $X$ be a smooth projective variety over $\mathbf C$. How do we define an ample vector bundle $E$?
Do we just ask its determinant $\det $ to be ample?
Is it the same as saying that $f^\ast E$ is ...
3
votes
0answers
33 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 ...
5
votes
0answers
106 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 ...
4
votes
1answer
101 views
Geometric meaning of Line-bundle product
I was wondering,
What could be a geometric/intuitive meaning of taking the (tensor) product of two line bundles on a smooth variety. Could you depict this to me with some example?
...
6
votes
1answer
121 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 ...
3
votes
0answers
43 views
Two questions on jet bundles
I am working with jet bundles on compact Riemann surfaces. So if we have a line bundle $L$ on a compact Riemann surface $C$ we can associate to it the $r$-th jet bundle $J^rL$ on $C$, which is a ...
2
votes
1answer
61 views
Trivial Tangent and Cotangent Bundles
If we have a smooth manifold $M$, why is the tangent bundle $TM$ trivial (as a vector bundle) iff the cotangent bundle $T^*M$ is trivial as well?
6
votes
1answer
95 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 ...
4
votes
2answers
119 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 ...
3
votes
0answers
44 views
Triviality of the tangent space of an abelian variety
The tangent bundle of an abelian variety $A/K$ is trivial. Mumford's proof of this fact goes something like this: given the (non-zero) value of a section in a fiber, one can determine the value in all ...
3
votes
0answers
79 views
A question of extension of vector bundles.
Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong ...
1
vote
1answer
40 views
Vector space structure on $(-1,1) \subset \mathbb{R}$ (or: möbius strip as vector bundle)
I'm first putting the question into it's context, so probably you can see if i'm asking the wrong question to get what i want.
The Task is to show that the Möbius (Moebius) strip is a Vector bundle ...
7
votes
2answers
84 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 ...
8
votes
1answer
101 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 ...
1
vote
0answers
36 views
Nontriviality of pullback of tangent bundle on the Möbius bundle
Consider the Möbius strip $M$ as a vector bundle over $\mathbb R$, in which it is embedded as the zero section. We have the tangent bundle $TM$ over $M$. How to prove that the pullback of this to ...
