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Questions tagged [vector-bundles]

For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.

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Complex bundles and (quasi)-complex structures of manifolds

Could you help me with some hint or reference for the following questions? I'm reviewing the Milnor-Stasheff for references. Is there some $(2k)$-manifold with stably quasi-complex structure, such ...
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Visualizing the degree of a line bundle?

The current picture I have in my head is that a line bundle in general is some section "rotating" around the $0$-section and the degree counts how many times it crosses through the $0$-section, up to ...
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Relationship between locally trivial and locally non vanishing section.

What I want to show is that If a line bundle (not necessarily locally trivial) has locally nonvanishing sections for each open set in the open cover of the base space, then it is locally trivial. ...
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Conditions for a Connection to be a Metric(or Chern) Connection

Given a Hermitian metric on a holomorphic vector bundle we can easily define its Chern connection. But if we are given a connection $\mathcal{A}$, $$[De=\mathcal{A}e,]$$ where $e$ is a holomorphic ...
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Is the normal bundle construction idempotent?

Let $X$ be a submanifold of $M$. Inductively, let $N_0$ be the normal bundle of $X$ in $M$, and $N_{k+1}$ the normal bundle of $X$ in $N_k$. (Identify $X$ with the zero section of $N_k$, of course.) ...
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Are real linear maps of smooth sections locally determined?

Let $ \pi_{ 1 } \colon E_{ 1 } \to M $ and $ \pi_{ 2 } \colon E_{ 2 } \to M $ be smooth vector bundles (of finite rank) over a smooth manifold $ M $, and consider a map $ T \colon \Gamma ( E_{ 1 } ) \...
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Use clutching to show $TS^2=\mathbb{L}_{+1}^{\otimes 2}$

I'm looking for a way to show $TS^2=\mathbb{L}_{+1}^{\otimes 2}$. I'm told I should use a clutching construction, which I have very little understanding of (I've just looked at the part Karoubi's book ...
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Chern class of a vector bundle and the associated projective space bundle

I have a very basic question regarding Chern classes. Let $X$ be a smooth projective variety and $\mathcal{E}$ a vector bundle on it. Let $\pi:\mathbb{P}(\mathcal{E})\to X$ denote the projective space ...
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Proof Check for The Construction of The Quotient Bundle

I would like to see if my proof for the construction of a quotient bundle is correct. Proof: Given a subbundle $E'$ of rank $k'$ of the vector bundle $E$ of rank $k$ we may form the quotient bundle $...
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Sufficient conditions for triviality of pullback vector bundle to imply triviality of original vector bundle

Let $E$ be a vector bundle over a smooth manifold $N$ and $f\colon M \to N$ a smooth surjective map. Is it possible that $f^*E$ is trivial while $E$ is non-trivial? If the previous question has a ...
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Decompose vector fields on product manifolds

So, I know that tangent bundle of a product manifold $M \times N$ splits in a sum $$ T_{(x,y)}(M \times N) = T_xM \oplus T_yN, $$ so that is obvious that the sum $X \oplus Y$ of smooth vector fields $...
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Divisors and higher cohomologies

Given a divisor $D$ on a smooth, complex manifold there is an associated holomorphic line bundle $\mathcal{O}(D)$. The space of holomorphic sections $H^0(\mathcal{O}(D))$ is directly related to the ...
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Question about equivalent definitions of (holomorphic) line bundles

Definition: A complex line bundle is a smooth manifold $L$ with a projection map $\pi:L\to M$, so that each fiber $\pi^{-1}(p)$ has the structure of a complex vector space of dimension 1. And for each ...
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Equivalent Definitions of Compactly Supported Forms in The Vertical Direction

I've come across two definitions of compactly supported forms in the vertical direction and I'm trying to show they are equivalent. For the setup, let $\pi:E \to M$ be a vector bundle of smooth ...
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Pullback of tautological bundle on Veronese curve

Let's work in $\mathbb{RP}^n$. Let $N$ be $\frac{n!}{d!(n-d)!}$ We define the Veronese map $\mu_d : \mathbb{RP}^n \to \mathbb{RP}^N$ This map $[x_0:x_1:x_{n-1}] \mapsto [x_0..x_d, x_0..x_{d-1}x_{d+...
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Relation between the tangent bundle and the holomorphic tangent bundle of a complex manifold.

Let $M$ be a complex manifold. We define the tangent bundle $TM$ as we do in the differential case. The complex structure on $M$ naturally induces a complex structure on $TM$ which makes it a complex ...
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Existence of $S^1$-action on a vector bundle and computing its characteristic classes

The existence of an $S^1$ action sometimes helps us in computing topological invariants. For example we can compute the Euler characteristic looking at the fixed point set (see Euler characteristic ...
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Non-flat connection on trivial bundle?

From what I have read it seems like there exists non-flat connections on trivial vector/principal bundles. However I can't find any notes on it or examples. Can anyone confirm that such connections ...
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Subbundles of a rank-two trivial bundle on $\mathbb{P}^1$.

Can every line bundle $\mathcal{O}(a)$ on $ \mathbb{P}^1$ be realized as a subbundle of the trivial bundle $\mathcal{O}\oplus \mathcal{O}$ on $\mathbb{P}^1$? I know that this phenomena can happen in ...
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Sections of a quasi-coherent sheaf along the non-vanishing set of a section of a line bundle.

Let $(X,\mathcal O)$ be a quasicompact, quasiseperated scheme, $\mathcal L$ a line bundle on $X$ and $\mathcal F$ any quasi-coherent sheaf on $X$. Let $s \in \Gamma(X,\mathcal L)$ be any global ...
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Cobordant Map from May's Book

I have a question about an argument that occurred in the discussion about consequences of Bott periodicity in A Concise Course in Algebraic Topology by P. May on page 221. Here is the excerpt: ...
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Nontrivial Twists of a Vector Bundle

Let $k$ be a number field, and let $X$ be a projective $k$-variety. Let $\mathcal{V}$ be a vector bundle on $X$ that is defined over $k$. A vector bundle $\mathcal{V}'$ on $X$ that is defined over $k$ ...
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Cobordism Groups of Smooth Closed Manifolds

I have a question about an argument used in the proof the red tagged theorem below from A Concise Course in Algebraic Topology by P. May at page 220. Here the excerpt: We write $\mathcal{N}_n$ for ...
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Vector Subbundle and bundle morphism

I need a proof for the following statement: Suppose $ M \subset N, E \subset E_1 $ are two embedded submanifolds of $ N$ and $ E_1$. $ E_1 \rightarrow N $ and $E \rightarrow M $ are two vector bundles....
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Isomorphism of Vector Bundles

Let $f:X \to Y$ be continuous map between topological spaces and $E$ a $n$-vector bundle over $Y$. Then we know that the pullback $f^*E$ is also a $n$-bundle over $B$ and a pullback in sense of ...
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Classifying Map $p_{n,m}:BO(n) \times BO(m) \to BO(n+m)$

I have a question about a construction & its properties described in May's "A Concise Course in Algebraic Topology"(see page 190): Let $VB_n(-): (pcHoTop) \to (Set)$ the functor which assigns to ...
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1answer
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Triviality of complexified tangent bundle of a closed surface

Does anybody know how to prove the following statement: The complexified tangent bundle $TS\otimes\mathbb{C}$ of a closed surface $S$ is topologically trivial iff the Euler characteristic $\chi(S)$ ...
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Sections of a locally free sheaf

Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on a scheme $X$. We know that we can associate to it a vector bundle $F$ on $X$ such that $F_x \simeq \mathcal{F}(x)$, where with $\mathcal{F}(x)$ ...
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Inverse Image Sheaf and pullback of a Vector Bundle.

Let $M,N$ be smooth manifolds and $f:N\to M$ a smooth map. Denote by $\mathcal{O}_M,\mathcal{O}_N$ the corresponding sheaves of smooth functions. If we regard $\mathcal{O}_M$ as the sheaf of smooth ...
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Vector Bundle Transition Functions as Cech Cocycles

I am trying to understand the fact that vector bundles of rank $r$ over a space $X$ are classified by the Cech cohomology group $\check{H}^{1}\big(X, GL_{r}(\mathcal{O}_{X})\big)$. I believe this ...
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A question about Gieseker compactification

Let $\ell_{\infty} \subset \Bbb P^2$ be a fixed line, and $G = GL_r$. Let $\mathcal U^a_G$ be the Gieseker partial compactification of the moduli space $Bun^a_G(\Bbb A^2)$. The space $Bun^a_G(\Bbb A^2)...
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Restrict differential forms that vanish along fibres to the base

I am reading Jean-Luc Brylinski's book on loop space. At the end of section 1.6 Leray Spectral Sequence, he claims without proof that Let $f: Y\to X$ be a smooth bundle of paracompact manifolds. A $...
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Question about Chern Character in Hatcher's book

I have a question about an argument from Allen Hatcher's script Vector Bundles and K-Theory in Cor. 4.4 (see page 110). Here the excerpt: We consider a vector bundle $E \to S^{2n}$, Then for Chern ...
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parallel transport is independent of the bases chosen in each of the two tangent spaces

Connections on principal fibre bundles In the above set of notes on page-3 section 2.2 under the heading Parallel transport there is a statement that equivariance ensures that the parallel transport ...
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Loop Space of Classifying Space [duplicate]

Let $\operatorname{U}(n)$ be the group of unitary $n$-matrices. My question is why we have following isomorphism $$\Omega BU(n) \cong U(n)$$ where $BU(n)= E/U(n)$ is the classifying space and $\...
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Chern Classes Identification

I have a question about a remark/observation in A Concise Course in Algebraic Topology by P. May at page 199. Here the excerpt: Having introduced the Chern classes May claims that $c_1 \in H^2(BU(...
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What is the Künneth formula for complete varieties?

I'm reading a part of Mumford's Abelian Varieties, and in the Chapter The theorem of the cube: II he claims that some "Künneth formula" tells us that if $L_1$ is a line bundle on a product $X \times ...
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Thom Isomorphism Theorem

I have a question about an argument in a proof from J.P. May's "A Concise Course in Algebraic Topology" (page 196): We use following notations: Let $\zeta: E \to B$ be a $n$-vector bundle and $Sph(E)$...
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Exercise verification in John Lee's Introduction to Smooth Manifolds, Part 2

Let $\pi: E\to M$ be a smooth vector bundle, and $V$ an open subset of $E$ with the property that $V \cap E_p$ is non empty and convex for all $p\in M$. By "a section of $V$" we mean a (local or ...
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Exercise verification in John Lee's Introduction to Smooth Manifolds, Part 1

Let $\pi: E\to M$ be a smooth vector bundle, and $V$ an open subset of $E$ with the property that $V \cap E_p$ is non empty and convex for all $p\in M$. By "a section of $V$" we mean a (local or ...
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What is transition function of scheme theoretic vector bundle ??

Let X be a scheme X and $\mathscr{F}$ be a locally free sheaf of rank $n$ over $X$. I want to consider a vector bundle over an algebraic variety $X$ , that is , the relative spec over $X$ of ...
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Checking My Understanding About Operations on Vector Bundles

I'm reading Bott and Tu's Differential Forms in Algebraic Topology and the discussion they give for constructing direct sums, tensor products, and duals of vector bundles is too quick for me to get a ...
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Connection from a covariant derivative

It is a basic question on connections of vector bundles, more preciously how to obtain a connection of a vector bundle starting from a covariant derivative. Suppose $B \to M$ is a smooth vector ...
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Pullback bundle of the tangent bundle through a diffeomorphism and an application to parallel global sections

It is probably a stupid question and it is probably poorly written. Hopefully, it is not already answered somewhere else. (1) Let $f:M \to N$ be a diffeomorphism. Is it true that the pullback ...
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Criteria of a real vector bundle to be Stably trivial

I am interested to know the general condition for a real vector bundle $V$ over an unorientable manifold $X$ ($X$ can be either 4d or 5d) to be stably trivial. The case I've heard of is when $X$ is ...
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Why should one think of orientation as a homology class?

Let $\pi:E\rightarrow B$ be a smooth vector bundle. I call this vector bundle to be an oriented vector bundle, if I can choose an orientation on $\pi^{-1}(b)$ for each $b\in B$ and a trivialization ...
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A coherent sheaf is a vector bundle over subvariety?

Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset? Thanks in advance.
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Is the dual of a subsheaf of a stable sheaf itself stable?

Assume $(X,H)$ is a polarized $K3$-surface. We have a torsion free sheaf $F_0$ of rank $r$ on $X$ which we can represent by 2 exact sequences: 1) $0 \rightarrow F_0 \rightarrow F \rightarrow F'' \...
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What is the support of a vector bundle complex?

I'm doing a class on K-Theory, and I'm confused about what the support of a complex of vector bundles is. Consider the following complex: $$0\to V_1\to V_2\to \dots\to V_n\to 0$$ Assume that all ...
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Examples of non trivial vector bundles

Once you see the notion of vector bundle, next thing you want to see are examples of non trivial vector bundles. Here, I want to collect such examples with justification of one or two lines saying ...