For questions on vector bundles.

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Simple exact sequences of vector bundles

I've come across some simple exact sequences of vector bundles that make manifest some basic confusions I have. These questions may be quite intertwined, in ways that my limited understanding obscures,...
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50 views

Vector bundle on connected component of the base space

Let $B$ denote a fixed topological space, which will be called the base space. A real vector bundle $\xi$ over $B$ consists of a total space $E=E(\xi)$, a projective map $\pi: E\to B$ and for each $b\...
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55 views

On the definition of projective vector bundle.

Definition 1 : Let $B$ be a topological space. A complex vector bundle of rank $n$ over $B$ consists of the data $(E,B,\pi,\{U_i\}_i,\{\phi_i\}_i)$ where $E$ is a topological space, $\pi:E\to B$ is a ...
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1answer
37 views

How do these sections arise in a Bott tower?

A Bott tower of height $n$ is a sequence of $\mathbb CP^1$ bundles $\require{AMScd}$ \begin{CD} B_n @>{\pi_n}>> B_{n-1} @>{\pi_{n-1}}>> \cdots @>{\pi_2}>>B_1@>{\pi_1}...
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1answer
34 views

Sections of tensor bundle are tensor product of sections

Given $E,F$ vector bundles over a manifold $M$, I would like to know a proof of $\Gamma(E\otimes F) = \Gamma(E) \otimes_{C^\infty(M)} \Gamma(F)$. Where $\Gamma$ denotes the smooth sections over $M$. ...
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1answer
39 views

Construction of connections given in “From Calculus to Cohomology”

I'm struggling to understand part of the descripition of a connection on page 167 of Madsen, Tornehave "From Calculus to Cohomology". Immediately after defining a connection $\nabla$ on a bundle $\xi$ ...
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1answer
29 views

The dimension of a projectivised, complexified tangent bundle

I am looking at page 15 here, starting from the line 'Let us suppose we are given a...' The Zoll projective structure is not relevant to this question, so don't worry about that. We take, in this ...
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1answer
41 views

Constructing symplectic structure on $T^*M$

I read the picture below, but I don't know how to get the equation above red line. Whether by using $T_{\xi_x}(T^*M)\cong T^*_xM$ ?But which isomorphism should be choice ? Then , how to check the ...
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1answer
48 views

How can I equip the universal vector bundle over $\mathbf{G}(k,n)$ with a connection?

When constructing the grassmannians $\mathbf{G}_\mathbb{F}(k,n)$ for $\mathbb{F} = \{\mathbb{R},\mathbb{C} \}$ there is a natural vector bundle $$ \mathbf{G}_{k,n} \to \mathbf{G}_\mathbb{F}(k,n) $$ ...
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1answer
80 views

Topology of the tangent bundle

Let $M$ be a smooth manifold, and let $\pi:TM\to M$ be its tangent bundle. We define the topology of $TM$ by declaring a subset $V$ of $TM$ to be open if and only if $\psi_\phi(V\cap\pi^{-1}(U))$ is ...
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1answer
26 views

Naive question on characteristic classes of $Gr(k,n)$

Let $Gr(k,n)$ be the Grassmann manifold of $k$-planes in $\mathbb{R}^n$ and $\gamma_k$ its tautological $k$-plane bundle. Is it obvious to see (or even true) that the Stiefel-Whitney classes $w_i(\...
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49 views

Normal bundle of a point

Let $X$ be a projective variety over a field $k$. I am trying to understand the notion of the normal bundle of a closed immersion. Let $x$ be a closed point of $X$. What is the normal bundle of $x$ ...
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1answer
26 views

Relating the normal bundle and trivial bundles of $S^n$ to the tautological and trivial line bundles of $\mathbb{R}P^n$

On page $10$ of Hatcher's Vector Bundles and K Theory, he gives a proof that the Whitney sum of the trivial line bundle over $\mathbb{R}P^n$ and the tangent bundle is equal to the Whitney sum of ...
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33 views

What does it mean for a tangent bundle to be parallel?

I am looking at page 8 of the paper here, just below definition 2.13. I have never come across a phrase such as '$T\mathfrak{C}\subset{TM}$ is parallel' - what does it mean for a tangent bundle to be ...
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parallelizable sphere product closed disk

From Wall's Surgery on Compact Manifolds, P9: Observe that $S^r \times D^{m−r}$ is parallelisable. If $m > r$, this is true, because spheres can be embedded in Euclidean space of one ...
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1answer
35 views

Why doesn't this argument show the Möbius bundle is trivial?

I wrote the following argument to prove that $S^1$ is parallelizable, that is, to show that the tangent bundle is trivial. It looks fairly reasonable to me. Let $\tau=2\pi$. We define a map $\...
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26 views

Vector bundle over an open set of $\mathbb{R}^n$

I can't see or understand if it is true or not if all vector bundles on over an open set of $\mathbb{R}^n$ are trivial or not. Is there an easy way to see it? The problem comes from the fact that we ...
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1answer
43 views

Normal bundle of an embedding of a parallelizable manifold into a parallelizable manifold

This question is motivated by an observation of Milnor. Theorem: Let $M^m$, $N^n$ be parallelizable smooth manifolds, and $i:M\to N$ an embedding. If $n>2m$, the normal bundle is trivial. The ...
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1answer
19 views

$r$-jet of a smooth function and its fiber bundle.

Let $M$ be a smooth manifold of dimension $n$. Let $E$ denote the bundle of germs of smooth functions on $M$. For every stalk $E_x$ we can define the ideal $$I_x^k=\{ f \in\mathcal{C}^{\infty}(M) \...
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2answers
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Under what type of transformations are characteristic classes and characteristic numbers of a manfold invariant?

When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they ...
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1answer
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What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers?

I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that ...
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1answer
71 views

Is $T\mathbb{C}\mathbb{P}^n$ globally generated?

A vector bundle $E\to X$ is globally generated if there exists global holomorphic sections $s_1,\dots,s_n$ such that $E_x$ is spanned by $s_1(x),\dots,s_n(x)$ for all $x\in X$. Consider the ...
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Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
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Motivation for equivalence of Tautological Line Bundle and ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$

I am currently reading a book on Algebraic Geometry and both the Tautological Line Bundle $L \to \mathbb{P}_k^n$ and the blow up ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$ are defined set ...
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Definition of cotangent and conormal bundle

I have read the following definition of cotangent bundle: Let $X$ be a $n$-dimensional smooth algebraic variety. For any $p\in X$ there exist a neighbourhood $U_{p}\subseteq X$ and functions (...
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1answer
55 views

Pullback of global sections?

Let $X$, $Y$ be abelian varieties and let $f:X\to Y$ be a morphism. They told me that we can define the pullback $f^*s$ of a global section $s\in\Gamma (L)$ where $L$ is an ample line bundle on $Y$, ...
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25 views

Global holomorphic vector field on a two-sphere

I'm sure this question has been asked before... Till today, I thought that one cannot define a global holomorphic vector field on a two-sphere due to the hairy ball theorem. However, here's an ...
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1answer
34 views

Degree of filtered vector bundle

Suppose I have the sheaf $\mathscr{M}$ defined by $$0\to \mathscr{M}\to \mathscr{O}_{\mathbf{P}^r}^{r+1}\to \mathscr{O}_{\mathbf{P}^r}(1)\to 0 $$ that is, $\mathscr{M}\simeq \Omega_{\mathbf{P}^r}^1(1)...
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1answer
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Cohomology class of zero set of a section

Say we have a rank $r$ smooth vector bundle $E\to X$ and a smooth section $s:X\to E$ of it. Shortly after the 30 min mark in this video Joe Harris defines the $r$th Chern class of this bundle to be ...
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1answer
37 views

Vector subbundle and frame field relation

Question: Let $E \to M $ be a vector bundle of rank $k$. Suppose that for each $p \in M $ we are given a subspace $E'_p$ of $E_p$ and consider the set $\displaystyle E' = \bigcup_{p \in M} E'_p $....
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1answer
88 views

Cohomologies with line bundle vs. coherent coefficients

I recently learned in a lecture that the derived category of a smooth variety is generated/spanned by (complexes of) locally free sheaves. (Unfortunately I haven't been able to find a more precise ...
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1answer
56 views

Relation between projective equivalence and linear equivalence of divisors

For the whole question I'll be working in $\mathbb{P}^n_{\mathbb{C}}$ and assume that everything is smooth. We say that two sets $U,V\subseteq \mathbb{P}^n$ are projectively equivalent if there ...
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How to show $Hom(V,V)\rightarrow Hom(V_x,V_x)$ is injective, V being semi-stable

Let $V$ be a semi-stable vector bundle over a smooth irreducible projective curve of genus $g\geq 2$. Let $x\in X$. How do we show that the canonical map $Hom(V,V)\rightarrow Hom(V_x,V_x)$ which ...
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0answers
61 views

Chern classes of a double cover

Let $X$ be a compact complex surface and let $D$ be a double cover of $X$. Let $\pi:D\to X$ be the double cover map (a 2:1) map. If $E$ is a vector bundle (rank at least 2) on $X$ with $c_1(E) = A$ ...
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1answer
65 views

The non-existense of the fine moduli scheme of vector bundles. Why?

The reference I am using is this enter link description here. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a ...
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1answer
90 views

Show that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$

Let $M, N$ be compact manifolds and $\Omega^*$ its algebra exterior. How to prove that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$? I thought about the function $f(\omega,...
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1answer
53 views

Are there characteristic classes for symplectic vector bundles?

Given a real vector bundle, say the tangent bundle of a manifold, some obstructions to this being the underlying real bundle of a complex bundle come from characteristic classes. These can prove the ...
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What does “factoring out an (group) action $\tau$ of a group $G$ acting on some set $E$” mean?

I am reading a survey article where they define the following objects: $\Gamma:=\mathbb{Z}^{n}$ seen as a group of translations. $\mathbb{T}:=\mathbb{R}^{n}/\Gamma$ is the $n$-dimensional ...
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Bundle isomorphism $\Phi : TM \oplus T^*M \to T(T^*M)$.

Prove that there is a bundle isomorphism $\Phi : TM \oplus T^*M \to T(T^*M)$ which identifies the summand $T^*M$ with the vertical vectors. If $\omega_{can}$ is the canonical symplectic ...
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1answer
70 views

Whitney sum bundle vs. direct product bundle [closed]

The question is simple: are Whitney sum bundle and direct product bundle the same? When are they different? PS: I have realized my mistake: let $E_1 \to B$ and $E_2 \to B$ be two bundles, their ...
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1answer
21 views

On the zero in the fibre of a vector bundle.

Let $X$ be a differentiable manifold, let $\{U_i \mid i\in I\}$ be an open cover of $X$, let $\{g_{ij}:U_i\cap U_j\to\mathbb{R}^r\}$ be a set of differentiable maps satisfying the cocycle condition, ...
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60 views

Roots of canonical line bundles that are not necessarily square roots

I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root. But what about further roots? ...
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1answer
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Computing Curvature of a Connection (Dirac Monopole)

I'm trying to verify some computations in a paper I'm reading and am feeling a little lost. In particular I haven't been able to properly compute curvature of a connection acting on a line bundle. ...
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1answer
21 views

Extending a section to a basis of sections in a trivial bundle.

Suppose we have a manifold $M$ of dimension $m$ and let $E$ be the rank $n$ trivial vector bundle on $M$. Let $\Gamma$ be a section of the trivial bundle $E$. My question is, are there some conditions ...
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1answer
52 views

Exact sequences of bundles and orientations

If we have an exact sequence of finite-dimensional vector spaces $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow0$$ then an orientation of any two induces an orientation of the third. I have ...
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20 views

Linear connection in a vector bundle in terms of the vertical projection

Let $\pi:E \rightarrow M$ a vector bundle. Can we define a linear connection as a connection $\Phi \in \Omega^1(E,VE)$ such that the horizontal projection $p_H = \text{id}_{TE} - \Phi$ is a vector ...
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2answers
64 views

Why does a tangent bundle have dimension 2n instead of n?

Let $n=dim(T_pM)$ for every $p\in M$, where $M$ is a smooth manifold. I understand that specifying $p$ is not enough to determine an element of $TM$, but what if do we specify only $v\in T_pM\subset ...
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Proving that every tangent bundle is direct summand of a trivial bundle.

I am trying to prove that there exists a formal immersion from a manifold $M$ of dim $m$ into $\mathbb{R}^{2m}$. Formal immersion is just an injective bundle map from $TM$ into $T\mathbb{R}^{2m}$ that ...
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Definition of isomorphism classes of vector bundles with reduced structure group

I'm looking for a definition of the notion of isomorphism between vector bundles $E$ and $F$, over the same base $X$, whose structure groups have been reduced from $GL(n)$ to some subgroup $G\subseteq ...
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2answers
104 views

Kernel of $\omega^\#$ is $k$-dimensional

Let $M$ be a smooth manifold with coordinates $\{q^i\}_{i=1}^n$ .The variables $(q^i,p_i)$ are coordinates on the cotangent space $\Omega=T^*M$. Any cotangent space carries a natural one-form $\tilde{\...