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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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17 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 ...
3 votes
1 answer
247 views

What does it mean when a chern class of a vector bundle is postive(resp. negative)?

Recently i was studying line bundles on $\mathbb{C}P^1$. Here is my confusion: for any holomorphic map $f:\mathbb{C}P^1 \to M$, where $(M,E,\nabla)$ is a $r$-rank holomorphic vector bundle with a ...
1 vote
0 answers
41 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 ...
0 votes
1 answer
329 views

Curvature as infinitesimal holonomy via the path-ordered exponential formula

Exercise 95 (page 247) in Baez & Muniain's Gauge Fields, Knots and Gravity asks us to prove the well-known formula: $$v - v' = \epsilon^2 F_{\mu \nu} v + o(\epsilon^2)$$ where $v \in E_p$ is a ...
2 votes
1 answer
116 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 ...
4 votes
1 answer
175 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 ...
2 votes
0 answers
31 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 $\mathbb{C}^2/\Gamma$, ...
1 vote
0 answers
49 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 ...
1 vote
1 answer
34 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 ...
2 votes
0 answers
95 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) = ...
2 votes
1 answer
64 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 $T(E) = D(E) / S(E) \to T(E')$, ...
0 votes
0 answers
29 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 ...
4 votes
1 answer
37 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 $\newcommand{\Th}{\operatorname{Th}} \Th(\xi) := C(V \setminus X \hookrightarrow V)$, ...
0 votes
1 answer
40 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 \...
1 vote
1 answer
54 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 ...
1 vote
0 answers
14 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 ...
2 votes
2 answers
45 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. ...
0 votes
0 answers
12 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 $\operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0$ ...
0 votes
0 answers
42 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 ...
1 vote
0 answers
48 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 ...
0 votes
0 answers
42 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 ...
0 votes
0 answers
42 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 ...
6 votes
2 answers
839 views

Holomorphic line bundles over $\mathbb{CP}^n$ and the Hirzebruch surfaces

In Huybrechts' text Complex Geometry, I am told that any holomorphic line bundle over $\mathbb{CP}^n$ is of the form $\mathcal{O}(k)$ for some $k\in\mathbb{Z}$, where \begin{equation} \mathcal{O}(k)=\...
0 votes
1 answer
144 views

Exact sequence of smooth vector bundles over a smooth manifold induces a s.e.s. on sheaves of sections

Let $M$ be a smooth manifold. Denote $C_M^\infty$ to the sheaf of real-valued functions on $M$. For a smooth vector bundle $\pi:E\to M$, denote $\Sigma_E$ to the sheaf of smooth sections of $\pi$ (it ...
6 votes
1 answer
103 views

Three topologies on the space of sections of a vector bundle

Let $E\to M$ be a Riemannian vector bundle over an oriented Riemannian manifold $(M,g)$ with a connection $\nabla$. Let $\Gamma(E)$ denote the vector space of sections of $E\to M$. For $\sigma \in \...
2 votes
1 answer
73 views

Table of Clifford Algebras

Thanks to Hans Lundmark, I just found the table of Clifford algebras $C_k=C_{\Bbb R}(0,k)$ and $C'_k=C_{\Bbb R}(k,0)$, $0\leq k\leq 7$. But, I couldn't understand some of the slots in the table. $\...
0 votes
1 answer
31 views

The total space of the direct sum of Mobius band and trivial vector bundle

We can consider the open Mobius band $M$ as a real line bundle over $S^1$, and it is nonorientable as a vector bundle. Let $E_n$ denote the trivial real rank $n$ vector bundle over $S^1$. Then is the ...
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 ...
7 votes
2 answers
2k views

Sum of normal bundle and tangent bundle.

Would somebody be able to prove that the whitney sum of the normal and tangent bundles of a submanifold of $\mathbb{R}^n$ is trivial? Would apreciate a detailed proof...I'm struggling a little. Tina
0 votes
1 answer
75 views

How to define twisted connection on vector bundle

I got motivation from twisted Cohomology where we twisted the derivative $d_\psi=d+\psi\wedge$ and find cohomology class $H_{\psi}^k(M)$ where $\psi$ is closed one form. I try to define twisted ...
1 vote
2 answers
139 views

Existence of complex line subbundle of complex bundles

Suppose $M$ be a real smooth manifold, and let $E$ be a rank $r>1$-complex vector bundle over $M$. I would like to know that if there is a complex line subbundle of $E$. By usual obstruction theory,...
3 votes
1 answer
508 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 ...
0 votes
0 answers
46 views

Clarification on Defining Hermitian Structure on Vector Bundles

In Huybrechts's text on complex geometry, he defines a Hermitian structure on a vector bundle $\pi: E \to M$ as Hermitian scalar products on each fiber $E_x$. Specifically, given a trivialization $\...
4 votes
3 answers
426 views

Is the square of a vector bundle trivial?

Let $E$ be a smooth vector bundle with a metric. Then $E^*$ is isomorphic to $E$ by mapping $v \in E$ to $\langle v, \circ\rangle \in E^*$. Because $E \otimes E^*$ is a trivial bundle, can we say ...
1 vote
1 answer
65 views

An oriented rank 3 vector bundle over a 4-manifold with $p_1=0$ has finite structure group

Let $E\to M$ be an oriented rank 3 real vector bundle over a smooth 4-manifold $M$ and suppose $p_1(E)=0$. Why does this imply that the structure group of $E\to M$ can be reduced to a finite group $G$?...
1 vote
1 answer
66 views

Connections on Vector Bundles

I am reading the book on Geometric Analysis by Jost and I am stuck on the derivation of $F = dA + A\wedge A.$ In particular I am not sure if I understand why $$A \circ d \mu + A \circ A \mu = A\wedge ...
0 votes
1 answer
29 views

Transition function of the tautological bundle on $\mathbb{P}^n_k$

This problem comes from Vakil 17.1.H (Jul3123 version). Problem 17.1.H. Suppose $k$ is a field. Define the subset $X\subseteq \mathbb{A}^{n+1}_k\times \mathbb{P}^n_k$ corresponding to points on the ...
0 votes
0 answers
24 views

Clifford Module Bundle of Exterior Algebra of Differential Forms

The definition of Clifford Module Bundle is: A Clifford bundle over $(M, g)$ is a vector bundle $S \rightarrow M$ with connection $\nabla^S$ with the following properties: i) for each $x \in M, S_x$ ...
2 votes
0 answers
49 views

Curvature of a connection in a local frame

Let $M$ be a smooth manifold and $E$ a vector bundle over $M$ with a connection $\nabla$. Denote by $F_{\nabla }\in \Omega ^{2}(\mathrm {End} (E))=\Gamma (\Lambda ^{2}T^{*}M\otimes \mathrm {End} (E))$ ...
7 votes
1 answer
2k views

Formula for the Stiefel-Whitney classes of a 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 ...
0 votes
0 answers
38 views

Why $K_\nabla$ is the same as $K_{\omega_S}$?

Let $G\subset GL_n(\mathbb{R})$ be a Lie subgroup and let $M$ be a manifold. Consider $S$ a $G-$structure on $M$ and $\nabla$ a connection compatible with $S$. We know that if $E=E(P,V)$ is the vector ...
0 votes
0 answers
25 views

Restriction of a vector bundle to a nodal curve

Let $S\to B$ be an elliptic surface with one nodal singular fiber C (a nodal projective curve $C$ of genus $g=1$). Let $\mathcal F$ be a slope-semistable rank-$2$ vector bundle on $S$. What can we say ...
5 votes
2 answers
335 views

Flat Bundle vs Trivial bundle

In R.W. Sharpe's Differential Geometry, a flat fibre bundle is defined as a bundle whose transition functions are constant. I don't understand the difference between this and a trivial bundle. Because,...
0 votes
0 answers
48 views

Computing the tangent space of the orbit of a gauge group action at a connection

Let $E\to M$ be a smooth real vector bundle, and let $\mathfrak{G}$ be the group of smooth bundle automorphisms. (The Lie algebra of $\mathfrak{G}$ is the space $\Omega^0(\text{End}(E))$.) For a ...
3 votes
1 answer
554 views

Fiber-preserving diffeomorphism

What does it mean in the definition 7.1(ii) that the diffeomorphism $$\phi_U$$ is fiber-preserving? In what sense/how technically it preserves fibers? They have never defined this in the book before.
0 votes
0 answers
16 views

Induced morphism of smooth variety to the Grassmannian on global sections?

In Huybrechts & Lehn, The Geometry of Moduli Spaces of Sheaves, page 143, it reads Let $X$ be a smooth variety. Suppose $E$ is a locally free sheaf of rank $r$ which is generated by its space of ...
0 votes
1 answer
47 views

$H^i\left(BO_m ; \mathbb{Z}_2\right) \cong H^i\left(B O_n ; \mathbb{Z}_2\right)$ for $i \leq m$ and $i \leq n.$

The notes I am reading say that groups in the title are isomorphic. Could someone explain to me why it is the case? Here by $BO_n$ I mean the infinite Grassmann manifold of $n$-dimensional subspaces. ...
1 vote
1 answer
38 views

Two $SO(2)$ connections on a smooth $SO(2)$-vector bundle

Let $L\to M$ be a smooth $SO(2)$-vector bundle. We can identify the Lie algebra $\mathfrak{so}(2)$ with $i\Bbb R$. Suppose $\nabla, \nabla'$ are two $SO(2)$-connections on $L$ such that $\nabla=\nabla'...
1 vote
1 answer
48 views

When does the map from Fundamental Group to Holonomy Group Injective?

We know that over a rank $n$ vector bundle $E$ (with base space being $M$, a manifold), if the connection $\nabla$ is flat, then the parallel transport along loop $\gamma:I\to M$ for $E$ will only ...
0 votes
0 answers
42 views

Family of vector spaces over a scheme

In Example 2.12 of the notes by Victoria Hoskins, the concept of a naive moduli problem is introduced, focusing on vector bundles (locally free sheaves) on a fixed scheme ( X ) up to isomorphism. The ...

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