For questions on vector bundles.
1
vote
0answers
22 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 ...
3
votes
1answer
32 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
18 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
15 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
20 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
31 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
28 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
51 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
25 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 ...
5
votes
2answers
27 views
elementary 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 learn ...
2
votes
1answer
31 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
37 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
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0answers
49 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
65 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 ...
6
votes
1answer
119 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 ...
4
votes
4answers
112 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
70 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
24 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
88 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
1answer
80 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
118 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
38 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
54 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
82 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
1answer
62 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
38 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 ...
2
votes
0answers
73 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
30 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
78 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
87 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
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0answers
24 views
Nontriviality of pullback of tangent bundle on the Mobius bundle
Consider the Mobius 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 ...
1
vote
1answer
24 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
votes
0answers
36 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
54 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
81 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 ...
7
votes
1answer
117 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
2answers
88 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, ...
3
votes
1answer
89 views
Trivial tangent bundle and orientability
Let $M$ a (real) $n$-dimensional connected differentiable manifold.
(a) The tangent bundle $TM$ is trivial, $TM \simeq M \times \mathbb R^n$;
(b) $M$ is orientable.
Consider the ...
3
votes
0answers
43 views
Questions on a sub-bundle of $\mathbb R^3\setminus \{0\}$
Let us consider spherical coordinates $(r,\theta,\phi)$ on $\mathbb R^3$ and the manifold $M:=\mathbb R^3 \setminus \{0\}$.
Let us consider the 1-form on $M$
$$
\omega = zdz ...
4
votes
2answers
86 views
Describe tangent and normal bundle to a manifold
Consider the set
$X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$
I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
5
votes
1answer
60 views
Sub line bundle of a vector bundle
I am trying to read Friedman's "Algebraic surfaces and holomorphic vector bundles". I am unable to follow a claim (on pg 32) that any globally generated rank 2 vector bundle (say) $E$ on a complex ...
1
vote
0answers
44 views
Complex projective manifolds and holomorphic mappings
Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant.
...
3
votes
1answer
76 views
vector bundles on the affine line over a PID
Let $R$ be a PID. Is every finitely generated projective $R[T]$-module free? In other words, is every vector bundle on $\mathbb{A}^1_R$ trivial?
For $R=k[X]$ this is true by the Theorem of ...
6
votes
0answers
106 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
1answer
132 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 ...
4
votes
3answers
117 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 ...
5
votes
0answers
69 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 ...
4
votes
1answer
96 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 ...
2
votes
1answer
61 views
Open sets of the tangent bundle in a Riemannian manifold
Let $M$ be a Riemannian manifold with a metric $g$ and $(U,\varphi)$ a
chart around a point $p\in M$.
By a Remark page 63 of Riemannian Geometry by M. Do Carmo, it
seems that any open set ...
1
vote
1answer
82 views
Triviality/non-triviality of line/circle bundle over $S^3$
I am interested in whether I can find non-trivial bundles in the case of a fiber bundle with base space $S^3$ and fiber either $\mathbb{R}$ or $S^1$. I know that in the case of a principal bundle ...
