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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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Projective bundle formula for Connective $K$-theory

I know how to prove that for every vector bundle $E \rightarrow X$ of degree $d$ we have: $$K^{*}\mathbb{P}(E) = K^{*}X[h]/(\lambda_{-1}[E](h)),$$ where $\lambda_{-1}[E](h) = \sum_{i = 0}^{d}(-1^{i})[\...
Eduardo4313's user avatar
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1 answer
69 views

Vector bundle construction theorem

Let $M$ be a k-dimensional smooth manifold, and let $\{U_\alpha\}_{\alpha\in I}$ be an open cover of $M$. Further, for any $\alpha, \beta \in I$, let there be given smooth maps $\varphi_{\alpha\beta}:...
Gao Minghao's user avatar
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34 views

Example of semi-stable non sufficiently smooth vector bundle

Let $X$ be a smooth projective complex variety and $H = c_1(\mathcal{O}_X(1))$ an ample class (if you prefer, you can take more generally $X$ to be a compact Kähler manifold with $H = [\omega]$ its ...
Cactus's user avatar
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Lee's Smooth Manifolds Problem 10-18

Lee's Introduction to Smooth Manifolds Problem 10-18 asks us to prove the following theorem: Theorem. Let $S$ be a properly embedded codimension-$k$ submanifold of $\mathbb R^n$. Then the following ...
Joseph Kwong's user avatar
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37 views

How to extend an ordinary vector bundle to a $\langle k \rangle$-vector bundle?

I have this question while I am studying Steimle's paper "An additivity theorem for cobordism categories". In this paper, a $\langle k \rangle$-vector bundle is defined to be a $P(\underline{...
Yuxun Sun's user avatar
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35 views

Global sections of $E \to M$ with measurable coefficients

I'm reading Demailly's Complex Analytic and Differential Geometry and I stumbled upon the following paragraph. Now, assume that $M$ is oriented and is equipped with a smooth volume form $dV(x)=\gamma(...
donovan's user avatar
  • 250
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1 answer
44 views

Extension of a vector bundle over the smooth compactification of an algebraic variety

This questions arises while reading an article by Cornalba and Griffiths. The following is the link to the paper (actually the whole volume of the journal). https://eudml.org/doc/142315 On page 23, ...
Yongmin Park's user avatar
3 votes
1 answer
49 views

Can two different spin structures on a manifold induce the same spin$^c$ structure?

Let $(M,g)$ be an oriented Riemannian $n$-manifold with transition functions $g_{\alpha\beta}:U_{\alpha \beta}\to SO(n)$. A spin structure on $M$ is a lift of $g_{\alpha\beta}$'s to functions $\tilde{...
user302934's user avatar
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1 answer
42 views

Rank $1$ locally free quotient sheaves of the cotangent bundle restricted to a smooth generic curve

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Let $X$ be a smooth projective variety over $\mathbb{K}$ such that $K_X$ (the canonical bundle) is ample. For $m\gg0$ there ...
Armando j18eos's user avatar
1 vote
1 answer
61 views

Total space of the normal bundle

I'm studying the total space of the normal vector bundle of a manifold $M$ differentiable embedded in $\mathbb{R}^n$, defined by: $N = \{(q,v) | q \in M, \text{ vperpendicular to M at q}\} \subset M \...
Claudia Mazzella's user avatar
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Error in "Principles of Algebraic Geometry" by Griffiths and Harris

At page $148$ of "Introduction to Algebraic Geometry", Griffiths and Harris define a positive line bundle as a line bundle $L\to M$ with a metric such that $(i/2\pi)\Theta$ is a positive $(1,...
Temoi's user avatar
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5 votes
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55 views

Curvature form and Curvature tensor in complex vector bundle

Let $X$ be a complex manifold and $\pi:E\rightarrow X$ be a complex vector bundle. A connection on $E$ is a $\mathbb C$-linear operator: $$\nabla:\mathcal C^\infty(X,E)\rightarrow\mathcal C^\infty(X,\...
N00BMaster's user avatar
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Induced action on vector valued differential forms

Let $F(X)$ be the frame bundle of a smooth $n$-dimensional manifold $X$. Let $G$ be a finite subgroup of $GL_n(\mathbb C)$ acting by automorphisms on $X$ and let $V$ be a $GL_n(\mathbb C)$-...
Flavius Aetius's user avatar
2 votes
1 answer
46 views

How to show that a sheaf is itself a sheaf of modules?

I am currently writing the proof for the following proposition: **Show that the sheaf of sections on a vector bundle $V$ over $X$ is a sheaf of modules over a sheaf of continuous function on $X$. ** I ...
SourBiscuit's user avatar
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1 answer
62 views

Fundamental equations of a Riemannian immersion

The Gauss, Codazzi-Mainardi and Ricci equations are the three fundamental equations of a Riemannian immersion. The Gauss equation inputs 4 tangent vectors, the Codazzi-Mainardi inputs 3 tangent ...
AlexInorbit's user avatar
2 votes
1 answer
68 views

Scheme-theoretic construction of tensor product of vector bundles on a scheme.

Let $X$ be a scheme and let $\mathcal{E}$ and $\mathcal{F}$ be locally free sheaves of finite rank (maybe works with coherent sheaves?). Let $E = \mathrm{Spec}(\mathrm{Sym}(\mathcal{E}^\vee))$ denote ...
Calculus101's user avatar
1 vote
1 answer
55 views

Sign discrepancy in covariant derivative

Suppose that $E\rightarrow M$ is a vector bundle over a differentiable manifold, equiped with a connection $\nabla$. The connection $\nabla$ induces connections on the various vector bundles ...
Richard Muniz's user avatar
1 vote
1 answer
91 views

Equivalent forms of second Bianchi identity on $TM$

$\DeclareMathOperator{End}{\mathrm{End}}$ This question is already asked here Second Bianchi identity on tangent bundle but with no answer. Let $M$ be a smooth manifold, and $E \to M$ a smooth vector ...
Alex Pawelko's user avatar
1 vote
1 answer
49 views

How to find out metric for different line bundles over complex projective space

(Tautological line bundle on $\mathbb{C P}^n$) The point on $\gamma_n$ is $(\ell, z) \in \mathbb{C P}^n \times \mathbb{C}^{n+1}$. It is natural to define a Hermitian metric $h$ on $\gamma_n$ by $h(\...
falamiw's user avatar
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Do orthogonal trivializations induce orthonormal frames?

Let $E$ be a real vector bundle with a bundle metric $g$ over $M$, and $\{U_i,\phi_i\}$ an orthogonal trivialization; that is for all $i$ and $j$ we have that the map $\phi_{ij}:U_i\cap U_j\rightarrow ...
Chris's user avatar
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1 vote
0 answers
39 views

Equivalence between K-theory for C*-algebras and K-theory for rings.

My question is motivated for the Swan’s theorem that give us an isomorphism between the $K(X)$ and $K(C(X))$. When you think as $K(C(X))$ as the algebraic K theory everything works perfectly. Because ...
Gomífero's user avatar
2 votes
1 answer
58 views

An equivalent condition of parallel mean curvature vector in normal bundle (in an orthonormal frame)

Question: Let $f:M\rightarrow (\bar{M},\bar{g},\bar{\nabla})$ be an isometric immersion of Riemannian manifolds, let $\{e_1,\dots,e_m,e_{m+1},\dots, e_{m+p}\}$ be a (local) orthonormal frame, with its ...
Zoudelong's user avatar
  • 347
1 vote
0 answers
57 views

K-theory of inverse limit

Is there any textbook reference for the topological K-theory to be a continuous functor? It is well known for C*-algebras, but I don't see a direct proof for (compact Hausdorff) spaces. More precisely,...
mathable's user avatar
  • 444
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0 answers
50 views

How can we check the amplness of a normal bundle?

Let $P^{n^2-1}$ be a projective space of n by n matrices. And let $Y\subset P^{n^2-1}$ be a projective variety defined by the $(n-1)\times (n-1)$ minors. Hence $Y$ parametrizes the rank $(n-2)$ locus. ...
Romanism's user avatar
3 votes
1 answer
101 views

Stable bundles over $\Bbb P^1$.

Is there a direct way to see that the only stable vector bundles over $\Bbb P^1$ are line bundles? I think that this can be shown using the Birkhoff–Grothendieck theorem, but it seems like an overkill ...
Iman's user avatar
  • 171
2 votes
1 answer
160 views

Constructing a $1$-form that vanishes iff the connection is compatible with the $G$-structure

I'm studying $G$-structures from Crainic's lecture notes. I'm stuck on the proof of Proposition $4.18$ at page $122$. Let $\mathscr{N}(\mathfrak{g}_S)$ be the normal vector bundle, i.e.: $$\mathscr{N}(...
Armando Patrizio's user avatar
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1 answer
52 views

Uniqueness of Harder-Narasimhan filtration for coherent sheafs

I am reading the following lecture notes https://people.math.harvard.edu/%7Elurie/205notes/Lecture20-HarderNarasimhan.pdf on the Harder-Narasimhan filtration. I am having trouble understanding the ...
ASHoa BSGa's user avatar
2 votes
1 answer
78 views

The natural map from compact vertical cohomology to de Rham cohomology is not injective

Let $\pi:E\to M$ be an oriented vector bundle. In Bott-Tu's book Differential Forms in Algebraic Topology, the compact vertical cohomology $H^*_{cv}(E)$ is defined by using differential forms $\omega$ ...
user302934's user avatar
  • 1,630
0 votes
0 answers
17 views

Equivalent definition for bundle orientability.

We know there are several equivalent definition for orientability of manifolds(see Lee's introduction to smooth manifolds): Existence of a choice of orientation at each point $p\in M$, such that ...
Eric Ley's user avatar
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1 vote
0 answers
71 views

Twisting global sections of sheaf with line bundle $H^0(E,\mathcal{O}_E)\otimes L_x^{\otimes k}\cong L_x^{\otimes k}$ [closed]

This is from the proof of the Kodaira embedding theorem (p.249 Huybrechts's Complex Geometry). By $E$ is denoted the exceptional divisor of the blow up of $X$, $L$ is an ample line bundle and $L_x$ ...
領域展開's user avatar
  • 2,407
1 vote
0 answers
74 views

Chern classes over projective space

My goal is to calculate the first Chern class of line bundles over the projective space, i.e. $c_1(\mathcal{O}(n)), n\in \mathbb{Z}$ Wikipedia states that $c_1(\mathcal{O}(n))=n$. I was really ...
領域展開's user avatar
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1 vote
0 answers
50 views

Global sections and meromorphic functions $(H^0(X,L)\setminus \{0\})\to \text{Div}(X))$

In Huybrechts's Complex Geometry, page 82 deals with the map $H^0(X,L) \setminus \{0\} \to \text{Div}(X)$. If we fix a trivialization on $L$ with the cocycle $(U_i, \psi_{ij})$ and take a non-trivial ...
領域展開's user avatar
  • 2,407
1 vote
0 answers
54 views

When does the Kodaira dimension equal the algebraic dimension?

I am using the notation from Huybrechts's Complex Geometry. The canonical ring of a complex manifold is defined as: $$R(X) := \bigoplus_{m \geq 0} H^0(X, \mathcal{K}_X^{\otimes m}).$$ The Kodaira ...
領域展開's user avatar
  • 2,407
4 votes
1 answer
193 views

Definition of Reduction of a Structure Group Implies Triviality?

Let $\pi:E\rightarrow B$ be a (smooth) principle bundle with structure group $G$. By definition, there exists a reduction of the structure group to a subgroup $H<G$, if there exists a global ...
LarsB's user avatar
  • 362
3 votes
1 answer
81 views

Canonical divisor of Kummer Surface

I want to prove that a Kummer surface is K3 so, first of all, I want to focus on the canonical divisor $K_X$. Just to fix some notations: $A$ is a complex torus of the form $\mathbb{C}^2/\Gamma$, ...
WindUpBird's user avatar
0 votes
0 answers
43 views

Question about the Vector Bundle Chart Lemma

In Lee's Introduction to smooth manifolds, one may find the following lemma: I was trying to construct a similar criterion for continuous vector bundles. Would we need condition (iii) for that? As ...
Julius Maximus's user avatar
4 votes
1 answer
61 views

Functoriality of Thom Space as Mapping Cone

One of the most general definitions of the Thom space associated to a real vector bundle $\xi\colon V \to X$ is as $\newcommand{\Th}{\operatorname{Th}} \Th(\xi) := C(V \setminus X \hookrightarrow V)$, ...
Ben Steffan's user avatar
  • 4,883
0 votes
1 answer
45 views

Vector bundle associated to the universal cover $\mathbb{R}\to S^1$

It's a well known fact that, given a principal $G$-bundle (where $G$ is a Lie subgroup of $\text{GL}(r,\mathbb{R})$) $$\pi_P:P\to X$$ there is an associated vector bundle $$\pi_E:E(P):=(P\times \...
Kandinskij's user avatar
  • 3,536
1 vote
1 answer
42 views

Equivariant Multiplicative Formula for Sphere Bundles in the Index Theorem

I am reading through the proof of the Atiyah-Singer index theorem, out of Lawson-Michelson, and I'm a bit confused about the proof of multiplicative property for indices of sphere bundles, where I ...
Elie Belkin's user avatar
1 vote
1 answer
55 views

What is the topology of the zero set of this quadratic form?

Consider a quadratic form in $\mathbb{R}^4$ defined as $q(x) = x_1^2 + x_2^2 - x_3^2 - x_4^2$. I am trying to gain as much intuition as possible about the set of vectors such that $q(x) = 0$. One ...
Leo's user avatar
  • 861
2 votes
0 answers
98 views

Understanding Chern Classes [closed]

Let $E$ be a vector bundle over a complex manifold $X$ with curvature form $F_{\nabla}$ of the Chern connection. The Chern classes are defined by: $$ \det\left(\frac{i}{2\pi}tF_{\nabla} + I_n\right) = ...
領域展開's user avatar
  • 2,407
2 votes
1 answer
150 views

Contraction by a form on the definition of Hermitian-Einstein metric-Local expression of the curvature form

In Huybrechts Complex Geometry, on page 217, the definition of a Hermitian-Einstein structure is given as: $$i\Lambda_{\omega}F_{\nabla} = \lambda \, \mathrm{Id}_E,$$ where $\Lambda_{\omega}$ denotes ...
領域展開's user avatar
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2 votes
1 answer
84 views

Why do maps $E \to E'$ descend to maps between the Thom space $T(E) \to T(E')$?

Let $f: E \to E'$ be a vector bundle homomorphism between $E$ and $E'$, both equipped with metrics. Why does it follow that there exists a map between the Thom spaces $T(E) = D(E) / S(E) \to T(E')$, ...
Chris's user avatar
  • 3,431
1 vote
0 answers
15 views

Why is the lift of a foliate vector field to the transverse bundle a foliate vector field w.r.t. the lifted foliation?

Let $M$ be a manifold with a foliation $F$ of codimension $q$, and let $p: B \rightarrow M$ be the transverse frame bundle (which over each point of $M$ consists of frames in the quotient of the ...
rosecabbage's user avatar
  • 1,697
2 votes
2 answers
60 views

Circle bundle over $\Bbb CP^n$ whose total space is $S^{2n+1}$

Viewing $S^{2n+1}$ as the unit sphere in $\Bbb C^{n+1}$, the natural projection $S^{2n+1}\to \Bbb CP^n$, the generalized Hopf fibration, is a circle bundle. What I am curious about is its converse. ...
user302934's user avatar
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0 votes
0 answers
13 views

$\operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0$ defines a complex curve $C$ with kernel as holomorphic line bundle

This is Exercise 12.4. Riemann Surfaces (Simon Donaldson). Let $T_0, T_1, T_2$ be generic $n \times n$ complex matrices. Show that the equation $\operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0$ ...
HelloMaths's user avatar
0 votes
0 answers
54 views

Is the torus as a bundle over the circle a vector bundle?

I was looking for an example of a smooth fiber bundle over a manifold that is not a vector bundle. My idea was that $S^1 \times S^1 \overset{\pi}{\rightarrow} S^1$ has $S^1$ as Fibre, which doesn't ...
Pastudent's user avatar
  • 870
1 vote
0 answers
53 views

Sheaves on a smooth manifolds and bump functions

Let $M$ be a smooth manifold and let $\mathscr{F}$ be a sheaf of abelian groups on $M$. Clearly if $\mathscr{F}$ is the sheaf of sections of a vector bundle, then every section defined on a compact ...
Kandinskij's user avatar
  • 3,536
0 votes
0 answers
50 views

Metric of Normal Bundle

If I had a circle bundle over some base manifold $\mathcal M$, I could write the line element in local coordinates as $$ds^2=g_{ij} dx^i dx^j+\phi(x)(dz^2+A_i dx^i)^2,$$ where $g_{ij}$ is a Riemannian ...
arow257's user avatar
  • 334
0 votes
0 answers
46 views

Parallel transport of a hermitian form on the fiber $E_x$

I have the following problem: Let $E\rightarrow M$ be a complex vector bundle over a complex manifold $M$, and let $v_x$ a hermitian form defined on the fiber $E_x$ such that it is invariant under the ...
kahlerian's user avatar

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