# 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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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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
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### 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 ...
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### 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
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### 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 ...
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### 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. ...
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### $\operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0$ 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 $\operatorname{det}(Z_0T_0 + Z_1T_1 + Z_2T_2) = 0$ ...
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### 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
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 \mathcal{O}(k)=\...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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
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### 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,...
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### 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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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'...
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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 ...