For questions on vector bundles.

learn more… | top users | synonyms

1
vote
0answers
11 views

Obstruction to the splitness of an exact sequence of holomorphic vector bundles

In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma (3.A.3) on the page 153, section Royden's extension lemma, whose statement is Every exact sequence of holomorphic vector ...
2
votes
0answers
27 views

Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
3
votes
1answer
33 views

Is the total space of a vector bundle over an irreducible scheme irreducible?

Let $X$ be an irreducible scheme over $\mathbb{C}$ and let $F$ be a locally free sheaf of rank $r$ on $X$. Is the total space $Y$ of the associated vector bundle to $F$, $Y=Spec(Sym(F^{\vee}))$, ...
1
vote
1answer
228 views

Is a sub-bundle of a vector bundle a vector bundle?

Could anyone please help me with this question? (1) Let $(E, p, B)$ be a vector bundle where $E$ is the total space, $B$ is the base, and $p$ is the structure map, that is, $p:E\to B$. Now suppose ...
3
votes
1answer
26 views

Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don't really ...
3
votes
1answer
30 views

Vanishing of the first Chern class of a complex vector bundle

Suppose that $E\to M$ is a $\mathbb{C}^n$-bundle with a metric. This is equivalent to saying that there exists a chart $\{U_\alpha\}$ of $M$ and $\phi_{\alpha,\beta} \colon U_\alpha\cap U_\beta\to ...
3
votes
1answer
46 views

Metric on Homogeneous Space $G/H$

For simplicity, assume $G$ is compact and semi-simple Lie group, and $H$ is a closed subgroup of $G$. Therefore the homogeneous space is reductvie, say $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ where ...
1
vote
0answers
30 views

spin structures definition

One way to define a spin structure on a $SO(n)$-bundle $p:E\to B$ is to require that there is an element $\sigma\in H^1(E;Z_2)$ such that restricted to each fiber gives a generator of ...
1
vote
0answers
15 views

Tangent Bundle of Product Manifold

Suppose $M,N$ are manifolds, and consider the product $M\times N$. From this answer, I know that: $T_{(m,n)}(M \times N) \cong T_m M \oplus T_n N $ Can we conclude that $T(M\times N) \cong T(M) ...
2
votes
0answers
25 views

Reduction to the special orthogonal group

It is well known that an $SL_n$-bundle $E$ on an algebraic curve $X$ is self dual (i.e $E\cong E^*$) iff it is an $SO_n$-bundle However, I can't see why, because the isomorphism $E\cong E^*$ means ...
2
votes
1answer
30 views

Applying $\bar \partial$ to a tensor product of holomorphic vector bundles bundles

Reading Griffiths and Harris, it is stated as an "obvious" fact that for a tensor product $E \otimes E'$ of holomorphic vector bundles, $(D_E \otimes 1 + 1 \otimes D_{E'})'' = \bar \partial$ where ...
3
votes
1answer
147 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
4
votes
1answer
45 views

To Reconcile Two Different Descriptions of the Dual Bundle

$\newcommand{\mc}{\mathcal}$ Let $\pi:E\to M$ be a smooth vector bundle with typical fibre a $k$-dimensional vector space $\mc V$. There are (at least) two ways to construct the dual bundle of $E$. ...
3
votes
2answers
79 views

triviality of tensor product of vector bundles

Let $\xi$ be a $O(n)$-bundle with fibre $\mathbb{R}^n$. Let $\xi\otimes \mathbb{C}$, $\xi\otimes \mathbb{H}$ be complex vector bundles and quaternionic vector bundles. If $\xi$ is not a trivial ...
3
votes
1answer
20 views

Vector fields spanning the tangent bundle at each point

Let $M$ be a smooth manifold of dimension $n.$ It is well known that it is not allways possible to find such global vector fields $X_1,\ldots, X_n$ which would be a basis for the tangent space at each ...
2
votes
0answers
20 views

Quotient space of equivariant vector bundle.

Let $p: P \to B$ be a principal $G$-bundle, and $\pi : E \to P$ a vector bundle with action of $G$ on $E$ such that $G$ acts by vector bundle isomorphisms and $\pi$ is equivariant. Is it always the ...
1
vote
1answer
27 views

Trivialization of vector bundles

Is it true that any $R^n$-bundle over a space (say a simplicial space) of dimension $k<n$ is trivial? It seems to me any $R^n$-bundle ,for $n>1$, over $S^1$ is trivial. But I cannot figure out ...
2
votes
0answers
19 views

The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\mc}{\mathcal} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\pr}{pr} \newcommand{\at}{\big|} \DeclareMathOperator{\GL}{GL}$ Let $\pi:E\to N$ be a smooth vector bundle over a smooth ...
0
votes
0answers
29 views

Rank 2 vector bundle V and its k$_V$

This question is from the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle ,Ex4.6 Suppose o$\le$a$\le$b$\le$c are three integers, show that there exist vector bundle V which fit ...
1
vote
1answer
21 views

Relating holomorphic sections of a line bundle to holomorphic functions on the line bundle

I am in the following situation: $X$ is a complex manifold and $L$ a holomorphically trivial line bundle over $X$. My objective is to understand the space of holomorphic functions $\mathcal{O}(L)$ on ...
0
votes
0answers
16 views
1
vote
1answer
47 views

Compute a chern class $c(K^n)$ for a non-singular algebraic hypersurface $K^n$ of degree $d$ in $P^{n+1}(C)$.

This is Problem 16-D in Characteristic classes by John W. Milnor and James D. Stasheff. Problem 16-D) If the complex manifold $K^n$ is complex analytically embedded in $K^{n+1}$ with dual ...
5
votes
1answer
48 views

Rank of a coherent sheaf in terms of coefficients of the Hilbert polynomial

Let $X$ be a projective scheme over a field $k$. Let $\mathcal{O}(1)$ be an ample line bundle on $X$, then the Hilbert polynomial $P(E)$ is given by $m\mapsto\chi(E ⊗ O(m))$. The explicit polynomial ...
3
votes
5answers
270 views

Why can we always take the zero section of a vector bundle?

$\require{AMScd}$ As I understand it, a rank $k$ vector bundle is a pair of topological spaces with a map between them $$ E\xrightarrow{p}B $$ such that there exists an open cover $(U_\alpha)$ of $B$ ...
1
vote
1answer
23 views

Lie Derivative of Connection 1 form

On Page 106 of Kobayashi & Nomizu's 'Foundations of Differential Geometry', the authors write \begin{align*} (L_X \omega)(Y)&=X(\omega(Y))-\omega([X,Y]). \end{align*} Here, $\omega$ is the ...
4
votes
1answer
44 views

Prove that $w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$

Let $\gamma^n$ be the canonical $n$-plane bundle over the infinite Grassmann manifold $G_n(\mathbb{R}^{\infty})$. I'm asked to prove that $$w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$$ (exercise 9-A ...
2
votes
1answer
22 views

Orientability of $\gamma^n\oplus \gamma^n$ WITHOUT characteristic classes

I was curious to find an argument to show orientability of the $2n$-bundle $$\gamma^n\oplus \gamma^n$$ where $\gamma^n$ is the canonical $n$-bundle over the infinite grassmannians ...
2
votes
0answers
20 views

Exact sequence of vector bundles

I'am working on a shorter proof of a theorem but to manage it I need to know if a lemma is true. Conjecture: Given a manifold $M$ and an short exact sequence of vector bundles $$ 0 \rightarrow E' ...
2
votes
1answer
38 views

Definition of Hölder Spaces of maps between manifolds (also Sobolev Spaces of these maps)

Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$ Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are ...
0
votes
0answers
24 views

(Co)Tangent bundle of Cone manifold

Given a Riemannian manifold $(M,\bar{g})$, we can construct the Riemannian cone manifold $(C(M), g )$ as follows. Topologically, $C(M)$ is $M \times \mathbb{R}_{>0}$. We equip this with the ...
1
vote
1answer
79 views

Products of Vector Bundles

Suppose that $E$ is a vector bundle over a compact, Hausdorff space $X$. Then $E^n$ is a vector bundle over $X^n$. If $D(E)$ is the disk bundle, there is a map on fibers $D(E^n)_x \rightarrow ...
0
votes
0answers
25 views

question on a vector bundle

let be $E=S^{1}\times \mathbb{R}$ be a trivial vector bundle on $M=S^{1}$ and $\nabla$ be a connection on it defined by $$\nabla:\chi(M)\times \Gamma(E) \to \Gamma(E)$$ and ...
2
votes
1answer
43 views

underlying real vector bundle of a complex vector bundle

Let $\eta^\mathbb{C}$ be a complex line bundle. If the underlying $2$-dimensional vector bundle $\eta$ is not trivial as a real vector bundle, can we obtain that $\eta^\mathbb{C}$ is not trivial as a ...
38
votes
2answers
2k views

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have ...
1
vote
0answers
25 views

Classifying all $E$-torsors up to isomorphism?

If $0\to E\to F\xrightarrow{\phi}\mathbb{A}^1_X\to 0$ is an exact sequence of bundles over the scheme $X$, then it is known that if $s\colon X\to \mathbb{A}^1_X$ is the constant section sending ...
0
votes
0answers
18 views

2-form as a curvature form

Let $M$ be a manifold and let $F$ be a global-defined $2$-form over $M$. Are there any conditions of $M,F$ such that $F$ is a curvature form of a connection on some line bundle ?
2
votes
0answers
56 views

Formula for Stiefel-Whitney of tensor product

I need help for solving Ex. 7C from 'Characteristic Classes' by Milnor/Stasheff: The exercise asks to find a formula for the (total) Stiefel-Whitney class of $\xi^m\otimes\eta^n$ over a paracompact ...
4
votes
2answers
155 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 ...
1
vote
2answers
43 views

commuting property of connections and bundle homomorphisms

I have the following situation: $E,F$ are (smooth) vector bundles over a smooth manifold $M$. Assume we are given connections $\nabla^E,\nabla^F$ on $E,F$ and a homomorphism of $M$-bundles ...
0
votes
1answer
18 views

Trivial bundle on projective space?

On $P^n(\mathbb R)$ I consider the open sets $U_j$ (with $x_j \neq 0$) and the transiction functions of a linear vector bundle $E_d$, $f_{hk}:p\to(x_k /x_h)^d$. I have to demonstrate that, if $d$ is ...
3
votes
0answers
34 views

Vector Bundle Structure on $\sqcup_{p\in M}\mathcal L(T_pM, T_{f(p)}N)$.

Let $f:M\to N$ be a smooth map between smooth manifolds. Is there a natural way to give a smooth vector bundle structure to $\bigsqcup_{p\in M} \mathcal L(T_pM, T_{f(p)}N)$. where $\mathcal L(V, ...
2
votes
1answer
50 views

Projection map of a vector bundle induce isomorphism on top cohomology.

I'm reading a passage in Milnor-Stasheff about Euler class, and I noticed that he states that the projection map $$\pi \colon E \to B $$ where $(E,\pi,B)$ is a n-dim vector bundle, induces a canonical ...
1
vote
1answer
9 views

vector bundle associated to the representation of a lattice

Suppose that $G$ is a connected semisimple Lie group and that $\Gamma$ is a cocompact lattice in $G$. Given a complex-linear representation $\chi$ of $\Gamma$ on a finite-dimensional complex vector ...
2
votes
0answers
31 views

Connection and reduction of the structure group

I am writing a memoir about gauge theory. I have trouble with a small proof which should be simple and have the feeling that I am missing something obvious. I want to show that the set of connections ...
0
votes
1answer
49 views

Endomorphisms in an exact sequence of vector bundles

Let X be a smooth projective variety over $\mathbb{C}$. And suppose we have an exact sequence of vector bundles over $X$. $\qquad\qquad\qquad\qquad\qquad 0\longrightarrow A\longrightarrow ...
2
votes
0answers
61 views

Examples of vector bundles

I know three examples of interesting vector bundles (not taking into consideration what one gets from them using algebraic operations): Tangent bundle of a smooth manifold Tautological bundle over ...
0
votes
0answers
30 views

$H^1$ of some vector bundle on a cubic 3-fold

This question is a sequel to the following one Dimension of moduli space of some stable vector bundles on a cubic 3-fold. Let $E$ be a stable rank 2 vector bundle on a cubic 3-fold, say $X$, with ...
4
votes
0answers
23 views

Why is it called *adjunction* formula?

Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy $$ \omega_X \big|_Y = ...
1
vote
2answers
40 views

Vector bundle and its definition

Take this definition of vector bundle: its a triple $(V,\pi,M)$ where $V$ is a set, $\pi:V \rightarrow M$ a sutjective map such that for every $m \in M$ the set $\pi^{-1}(m)$ has the structure of real ...
1
vote
1answer
66 views

Dimension of moduli space of some stable vector bundles on a cubic 3-fold.

I'm trying to understand the claim that the moduli space of stable rank 2 vector bundles on a (general?) cubic 3-fold, say $X$, with $deg c_2=6$ and $det=O_X(2)$ is of dimension 9. I belive what I ...