For questions on vector bundles.
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0answers
19 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
52 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
2answers
87 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 ...
1
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0answers
30 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 $V$, ...
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0answers
31 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 ...
3
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0answers
61 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 ...
3
votes
1answer
51 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 ...
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.
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0answers
40 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} ...
2
votes
0answers
28 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
39 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 ...
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 ...
4
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0answers
44 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 ...
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
34 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 ...
0
votes
1answer
63 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 ...
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vote
0answers
18 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 ...
8
votes
1answer
130 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 ...
1
vote
1answer
33 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 $$\displaystyle\nu(F)=\frac{T_pM}{F_p}.$$
2
votes
1answer
37 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
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1answer
23 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. ...
4
votes
1answer
55 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
39 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 ...
2
votes
1answer
38 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
39 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
52 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
83 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
4answers
126 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
73 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 ...
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 ...
3
votes
0answers
31 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 ...
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 ...
3
votes
1answer
90 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
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 ...
3
votes
0answers
41 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 ...
1
vote
1answer
187 views
If bundle map is a local isomorphism, then bundle map is diffeomorphic?
Suppose $\phi$ is a bundle map from $E$ to $F$, where $E$ and $F$ are two vector bundles over a manifold $M$. Now for each $x\in M$, $\phi$ restricted to $E_x:E_x\to F_x$ is a isomorphism, how to show ...
3
votes
0answers
75 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 ...
2
votes
1answer
60 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
90 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 ...
3
votes
0answers
39 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 ...
4
votes
1answer
138 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 ...
1
vote
1answer
35 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
79 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 ...
1
vote
0answers
47 views
Complex projective manifolds and holomorphic mappings
Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant.
...
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 ...
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 $E_1$, ...
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 ...
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1answer
33 views
Vector bundles and principal $G$-bundles
I am trying to understand the notion of a principal $G$-bundle versus a vector bundle. Here $G$ is a Lie group.
Supposedly, principal $G$-bundles are a generalization of vector bundles. My problem ...
3
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0answers
37 views
Any vector bundle on $\mathbb R$ is a trivial bundle
How to prove that any vector bundle on a Euclidean space is a trivial bundle? It is enough to prove it for the case of dimension $1$ and I hope it will be a nice exercise for me to generalize to the ...
5
votes
1answer
87 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 ...
