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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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Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
Arrow's user avatar
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13 votes
1 answer
386 views

Does every vector bundle have a 'tensor inverse'?

For any vector bundle $E$ over a finite-dimensional CW complex, there is a vector bundle $E'$ such that $E\oplus E'$ is trivial. For a compact Hausdorff base, this is Proposition 1.4 of Hatcher's ...
Michael Albanese's user avatar
12 votes
0 answers
989 views

Every vector bundle has a metric connection?

Let $(E,g)$ be a vector bundle with a metric over a manifold $M$. Does $(E,g)$ always admit a compatible (metric) connection? If so, are there examples where there exists only one such metric ...
Asaf Shachar's user avatar
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12 votes
0 answers
178 views

What's Griffiths' ampleness in modern language?

When reading Griffiths' paper "Hermitian differential geometry, Chern classes and positive vector bundles", I found his definition of ampleness: Let $X$ be a compact complex manifold, $E\to ...
Golbez's user avatar
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11 votes
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When is a connection on the adjoint bundle induced by a principal connection?

Let $P\rightarrow M$ be a principal $G$-bundle with a connection 1-form $\omega$. In a local trivialisation $\tau_U \colon U\rightarrow P_U$ ($U \subset M$) we can pull the connection back to the ...
balintm's user avatar
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11 votes
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669 views

Null-correlation and Tango bundles on $\mathbb{P}^3$

Let $V$ be a four-dimensional complex vector space and $\mathbb{P}^3=\mathbb{P}(V)$. There are two interesting bundles $N$ and $T$ on $\mathbb{P}^3$, both of rank 2, called respectively a null-...
mahavishnu's user avatar
11 votes
0 answers
510 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 ...
unknownymous's user avatar
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10 votes
0 answers
262 views

Extending the tangent bundle of a submanifold to a subbundle of the manifold

Given an $m$-dimensional manifold $M$, and an $n$-dimensional submanifold $N$, with $n<m$, the tangent bundle of $N$ is a smooth $n$-dimensional subbundle of $TM|_N$. When can $TN$ be extended to ...
Rei Henigman's user avatar
  • 1,379
9 votes
2 answers
1k views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
PeterM's user avatar
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8 votes
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Show that exists a canonical symplectomorphism $T^{*}E^{*}\cong T^{*}E$

I need to prove that for $\pi: E → M$ a vector bundle then exists a canonical symplectomorphism $T^{*}E^{*}\cong T^{*}E$. I star with definitions: Let be $\pi : E → M$ a vector bundle, $\pi$ is a ...
weymar andres's user avatar
8 votes
0 answers
356 views

Axioms and uniqueness for the Euler class

In this question it was asked if the 4 properties listed on the wikipedia page uniquely characterise the Euler class. I answered no and claimed: For every oriented vector bundle $E\to X$ of rank $n$ ...
Jonas's user avatar
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533 views

Direct image of a line bundle is a vector bundle

Let $$f:X\to Y$$ be a holomorphic map between compact connected Riemann surfaces and let $$L\to X$$ be a holomorphic line bundle on $X$. I'm having some trouble understanding the fact that the direct ...
Simon Parker's user avatar
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8 votes
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Show that the Mobius bundle is a smooth real line bundle and it is non-trivial.

I'm reading John Lee's Introduction to Smooth manifolds 2nd edition. Consider the problem 10-1: Define an equivalence relation on $\mathbb{R}^2$ by $(x,y) \sim (x',y')$ if and only if $(x',y')=(x+n,...
bbw's user avatar
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1k views

Understanding the derivation of this form of the coderivative

Let $M$ be a smooth Riemannian manifold; Let $E$ be a vector bundle over $M$, equipped with a metric and a compatible connection $\nabla$. Denote by $d:\Omega^k(M,E) \to \Omega^{k+1}(M,E)$ the ...
Asaf Shachar's user avatar
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8 votes
0 answers
304 views

Leray–Hirsch theorem for K-theory

In Hatcher's book, Vector bundles and K-theory. He states the following version of Leray-Hirsch's theorem: Let $p:E \longrightarrow B$ be a fiber bundle with $E$ and $B$ compact Hausdorff and with ...
Chris Kuo's user avatar
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8 votes
0 answers
170 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 = \...
A.P.'s user avatar
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8 votes
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420 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
Tom's user avatar
  • 141
8 votes
0 answers
347 views

Do homotopic transition functions define isomorphic bundles (on smooth manifolds)?

It is fairly known that given a cover $\{U_i\}_{i \in \mathbb{N}}$ of some smooth manifold $M$ together with smooth transition functions $g_{\alpha \beta} \colon U_{\alpha} \cap U_{\beta} \rightarrow ...
Louis's user avatar
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8 votes
0 answers
765 views

de Rham Cohomology of Non-Flat Bundle

Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$. If $E$ ...
Michael Albanese's user avatar
8 votes
0 answers
142 views

How to understand the coordinate transition map in $E\otimes E'$?

This question seems embarrassingly basic. Let $X$ be some manifold and $E,E'$ two vector bundles over $X$ with rank $n$ and $n'$ respectively. Now assume to possible refinement we have $E,E'$ defined ...
Bombyx mori's user avatar
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8 votes
0 answers
1k views

understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
harlekin's user avatar
  • 8,770
7 votes
0 answers
108 views

Triviality of holomorphic vector bundles over $\mathbb{C}$

Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle. I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and ...
Alessio Di Prisa's user avatar
7 votes
1 answer
531 views

Bott Tu Exercise 6.14, integration along the fiber

Suppose $\pi:E\to M$ is an oriented $C^\infty$ vector bundle of rank $n$. We denote by $\Omega_{cv}^k(E)$ the set of all differential $k$-forms $\omega$ on $E$, such that for each compact $K\subset M$,...
blancket's user avatar
  • 1,850
7 votes
0 answers
152 views

What vector bundles are tangent bundles of smooth manifolds?

Given a smooth manifold $M$, we can naturally associate to it a vector bundle ${\rm T}M\rightarrow M$ called the tangent bundle of $M$. This operation induces a functor $\rm T:\rm Diff\rightarrow\rm ...
Dry Bones's user avatar
  • 697
7 votes
0 answers
490 views

A sheaf of differential forms is the sheafification of presheaf of skew-symmetric forms on vector fields?

Let $X$ be a real manifold, and $\mathcal{O}_X$ be the sheaf of smooth functions. We define a sheaf of $1$-forms as \begin{align*} \Omega_X\colon=\operatorname{\mathcal{Hom}}(\mathcal{O}_X(TM),\...
p_i's user avatar
  • 211
7 votes
0 answers
2k views

Proof details of the fact that the unit tangent bundle is compact in $TM$ if $M$ is a compact manifold

Let $M$ to be a manifold $m$-dimensional with a smooth hermitian metric $g$. The tangent bundle of $M$ is given by $TM= \bigcup_{p\in M} T_{p}M$, and the unit tangent bundle is given by $S=\{x \in ...
Pedro do Norte's user avatar
7 votes
0 answers
321 views

Relative Hitchin-Kobayashi correspondence and relative Hermitian Yang-Mills connections

Let $\mathcal E\to X$ be a stable vector bundle over a polarized projective manifold $(X,\omega)$.We know, that in this case $\mathcal E$ admits Hermitian-Einstein metric, i.e., a metric $h$, such ...
user avatar
7 votes
0 answers
1k views

Relationship between Principal Unitary Bundles and Complex Vector Bundles

I'm trying to prove that principal $U(N)$-bundles are in bijective correspondence to complex vector bundles. I understand the proof that principal $\text{GL}(N, \mathbb{R})$-bundles are equivalent to ...
Benighted's user avatar
  • 2,563
7 votes
0 answers
249 views

On the theorem of the cube and the semicontinuity theorem: intuition for proof of proposition?

Consider the following result from p. 89 of Mumford's Abelian Varieties. Proposition. Let $X$ be a complete variety, $Y$ any scheme and $L$ a line bundle on $X \times Y$. Then there exists a unique ...
KeD's user avatar
  • 2,137
7 votes
0 answers
722 views

Condition for a complex vector bundle to be holomorphic?

Suppose that $(E,\pi,M)$ is a complex vector bundle. I've seen it suggested in a few places that if $E$ and $M$ are complex manifolds, and $\pi$ is a holomorphic map, then $(E,\pi,M)$ is in fact a ...
user avatar
7 votes
1 answer
777 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let $...
beedge89's user avatar
  • 1,964
7 votes
0 answers
876 views

Orientability of the total space of a vector bundle over an oriented manifold

Let $M$ be a (smooth) manifold of dimension $n$, and let $\pi : E \to M$ be a (smooth) vector bundle of rank $r$. If I choose a connection on $E$, then I obtain a decomposition of $T E$ as $V E \...
Zhen Lin's user avatar
  • 90.5k
6 votes
0 answers
473 views

Well-defineness of Chern roots?

Let $\mathcal{E} \to X$ be a rank $n$ (complex) vector bundle on a space $X$ (possibly with some other mild conditions to make the splitting principle holds). According to the splitting principle, ...
Ray's user avatar
  • 1,280
6 votes
0 answers
183 views

Equivariant line bundle-divisor correspondence?

In reading about equivariant bundles, I've become a bit confused about how the usual line bundle-divisor correspondence $c_1:\text{Pic}(X)\xrightarrow{\cong}A^1(X)$ works in the equivariant setting. ...
Michael Mueller's user avatar
6 votes
0 answers
194 views

Is there a proof via Chern-Weil theory that the first Chern numbers of two $\mathbb R$-isomorphic complex vector bundles are equal mod 2?

Let $M$ be a compact, oriented surface. If we have a complex vector bundle $E$ over $M$, then we can define the first Chern number $c_1(E)$ via Chern-Weil theory. More precisely, if $\nabla$ is a ...
Hugo's user avatar
  • 3,805
6 votes
0 answers
1k views

Pushforward of a line bundle along a finite morphism of curves

Let $f:X\rightarrow Y$ be a finite morphism (a branched covering) of degree $n$ of smooth complex algebraic curves. It is a known result that for any line bundle $L$ on $X$, the pushforward $f_* L$ ...
G. Gallego's user avatar
6 votes
0 answers
477 views

Two definitions of holomorphic vector bundle

I am stuck in the task of understanding of the following. I am trying to learn about holomorphic vector bundle. So far, I have found two definitions. Definition 1: One starts with the complex (smooth ...
Jan Vysoky's user avatar
6 votes
0 answers
242 views

How can I compute simple examples of the associated vector bundle to a coherent sheaf?

Given a coherent sheaf $\mathcal{E} \in \text{Coh}(X)$ over a scheme there is a way to associate a relatively affine scheme over $X$. This is done by constructing the $\mathcal{O}_X$-algebra $$ \text{...
54321user's user avatar
  • 3,243
6 votes
0 answers
631 views

Basic question regarding complexification of a vector bundle

Let $E\rightarrow M$ be a real vector bundle over a real manifold $M$. By complexification of this bundle we mean the complex vector bundle $E_{\mathbb{C}}$ whose fibers are given by complexifying the ...
Hajime_Saito's user avatar
  • 1,823
6 votes
0 answers
225 views

Formal adjoint of curvature (Yang Mills)

Currently reading a paper on finding solutions to the Yang Mills equation $D^*\Omega=0$, where $\Omega$ is the curvature and $D^*$ is the formal adjoint of the exterior covariant derivative $D$. ...
beedge89's user avatar
  • 1,964
6 votes
0 answers
502 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
GFR's user avatar
  • 5,411
6 votes
0 answers
2k views

Existence of a holomorphic connection implies existence of a flat connection

Let $E$ be a holomorphic vector bundle on a complex manifold $X$, and let $D: E \to E \otimes \Omega_X$ be a holomorphic connection on $E$ i.e. $$ D(fs) = s\otimes \partial(f)+ fD(s), $$ for any local ...
Alex's user avatar
  • 6,284
6 votes
1 answer
177 views

Is there a natural ring structure on $\operatorname{Pic}(\mathbb{CP}^1)$?

The set of isomorphism classes of holomorphic line bundles on a complex manifold $X$ is a group under tensor product. This group is called the Picard group and is denoted $\operatorname{Pic}(X)$. We ...
Michael Albanese's user avatar
6 votes
0 answers
779 views

Sections of endomorphisms of a vector bundle

The following could be rather silly question but I haven'd found it stated explicitly; from the other side, it seems to me, that this fact is used often without comments. The problem is the ...
truebaran's user avatar
  • 4,650
6 votes
0 answers
149 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 ...
Steve's user avatar
  • 317
6 votes
0 answers
562 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 ...
Tony's user avatar
  • 6,758
6 votes
0 answers
215 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 $E_1$, ...
student's user avatar
  • 3,897
6 votes
0 answers
239 views

Complex vector bundles with real transition functions

After doing a bit of playing around (I think) I was able to show that the map $\operatorname{id}\otimes\ \psi : \Omega^{p,q}(X, E) \to \Omega^{p,q}(X, \bar{E})$, where $\psi $ is the conjugation map $...
Michael Albanese's user avatar
6 votes
0 answers
270 views

Alternate pullback bundle construction

If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is $$\phi^* F := \{(m,f) \in M \...
Victor Dods's user avatar
5 votes
0 answers
133 views

$K$-theory of $S^2$: spinor bundle vs tautological bundle over $\mathbb{C}P^1$

I'm trying to understand the relationship between different generators of the $K$-theory group of $S^2$. Part of my curiosity comes from reading this discussion about characteristic classes. The $K$-...
geometricK's user avatar
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