# Questions tagged [vector-bundles]

A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.

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### Learning roadmap for holonomy theory

During my Master's thesis I encountered the theory of holonomy for the first time. Unluckily it was only tangentially related to the topic of my thesis, so I couldn't dive into it.
The book I was ...

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### Line bundles with meromorphic transition functions

I have the following situation: let $X$ be a projective complex manifold and let $f \in H^1(X,\mathcal{M}^{\times})$. So $f$ defines something like a line bundle with meromorphic transition functions.
...

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### Homogenization of Morse-Bott functions

Let $M$ be a compact manifold of dimension $n$.
A smooth function $f:M \to \mathbb{R}$ is called Morse-Bott if the set critical points of $f$ is a disjoint union of compact submanifolds $C_1,\ldots,...

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### Chern number of positive spinor bundle

What is the second chern number $c_2(V_+)$ of the positive spinor bundle on a 4-manifold, in particular $S^4$? Why is it that $V_+$ is the same as the quaternion line-bundle?
Thanks,

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### Orientation bundle and its flat connection

Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any ...

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### Bundles equivariant with respect to a transitive Lie algebra action

Suppose an algebraic group $G$ transitively acts on a variety $X$. Then it is well known that $G$-equivariant vector bundles on $X$ are in correspondence with representations of the stabilizer of a ...

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### Fundamental group of the moduli space of parabolic bundles with fixed determinant

I am looking for the fundamental group of the moduli space of parabolic bundles with fixed determinant over a smooth projective curve.
I know that the fundamental group of the moduli space of vector ...

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### Base locus of cotangent bundle of a surface

Is it easy to construct examples of smooth complex projective surfaces $S$ of general type such that $h^0(\Omega_S)>1$, $alb_S:S\rightarrow Alb(S)$ is generically finite (unto its image) and
$${\rm ...

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### local sections in general position

The following seems obvious-and-easy-to-prove, and yet I cannot find an argument beyond the trivial case $n=1$. Maybe someone knows about it?
Consider a real vector bundle of any rank $n$ over a base ...

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### exterior differentiation of foliations

Let $M$ be a differentiable manifold.
Let $T^*M$ be the cotangent bundle of $M$.
Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\Lambda^...

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### Local systems arising from higher rational homotopy groups

I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition.
I am aware that for a topological space $X$ and a point $x ...

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### On classifying space of normalizer of maximal torus

I am reading Marc Levine's paper 'Motivic Euler characteristics and Witt-valued characteristic classes'. In that paper he considers the $BN_T(SL_n)$, namely the classifying space of the group of ...

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### Fiberwise exponential map for vector bundle automorphisms

Let $p:E \to B$ be a smooth vector bundle of rank $n$ over a manifold $B$ and we identify $B$ with the image of the corresponding zero section.
For $b\in B$ denote by $E_b = p^{-1}(b)$ the fiber over $...

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### Confusion on a term related to connection and holonomy

This question is simply some of my confusions about a specific term.
Let $E\to X$ be a trivial complex vector bundle. When one says let $\nabla^E$ be a connection on $E\to X$ with trivial holonomy (...

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### Higher Schwarzenberger conditions for vector bundles

Has anyone computed higher Schwarzenberger conditions for vector bundles on projective space, even for rank 2 bundles? Schneider lists them for 2-bundles up to $\mathbb{P}^5$, and for 3-bundles on $\...

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### Computing the equivariant Chern character

Suppose I know the Chern character of an object $F \in D^b(X)$, where $X$ is some smooth complex projective variety with a finite group $G = \mathbb{Z}/m$ acting on it. In $D^b([X/G]) \simeq D^b(X)^G$ ...

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### Are projective bundles corresponding to non-isomorphic vector bundles always non-isomorphic?

Suppose we are given a scheme $S$ and two vector bundles $V$ and $W$ over $S$. Is it always true that $\mathbb{P}(V)\cong \mathbb{P}(W)$ implies that $V\cong W$ as $S$-schemes?
If the statement is ...

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### Reconstructing base manifold from its category of smooth vector bundles

It's probably a really naive question, but I didn't find any references. Given a category $V$ and we know that it is equivalent to the category $\mathbf{Vect}(X)$ of smooth vector bundles over a ...

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### Integral flow that can commute to Laplacian operator

Firstly, considering the vector field in $ \mathbb{R}^3 $, $ X=x_2e_1-x_1e_2 $, we can see that
$$
\phi(t,x)=\phi(t,x_1,x_2,x_3)=(t,x_1\cos t+x_2\sin t,-x_1\sin t+x_2\cos t,x_3)
$$
is the ...

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### Non-homogeneous line bundles over a homogeneous space

Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A homogeneous vector bundle over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form
$$
G \times_{\...

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### Linear span of tangential variety

Let $X \subset \mathbb{P}^N$ be a projective variety of dimension $n$. Let us denote with $TX=\bigcup_{x \in X}\mathbb{T}_xX$ the tangential variety, where $\mathbb{T}_x X$ is the projective tangent ...

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### Linear system giving the projective embedding of the tangential variety

I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this ...

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### On the positivity of the second Segre class of ample vector bundles

Let $E$ be an ample rank $r\geq2$ vector bundle over a smooth projective surface $X$ defined on an algebraically closed field $\mathbb{K}$ of characteristic $0$.
In Kleiman S. L. - Ample Vector ...

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### Do we have an equivariant version of integrability theorem of flat connections?

I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1:
Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a ...

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### Bundles vs. line bundles

Let $K$ be an algebraically closed field and consider the category $\text{Bun}$ of (finite dimensional) vector bundles over a $K$-variety $X$. Consider also the category of $\mathbb{G}^\times$-...

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### Normal bundle of a linear subspace

Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...

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### Confusion in notation of representation of Bastiani derivative

In the paper "Properties of field functionals and characterization of local functionals" at page 5 the Authors give the following definitions
Definition II.2. Let $U$ be an open subset of a ...

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### Reference for Schwartz kernel theorem on vector bundles

In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has
Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...

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### Torsion points of abelian variety as zeros of a section of a vector bundle?

Let $A$ be an abelian variety over $\mathbb{C}$ and let $X_m$ the subset of nontrivial $m$-torsion points on $A$. Can we realize $X_m$ as the zero locus of a global section of a suitable vector bundle ...

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### Classfication of vector bundles on projective line over a local ring

Let $R$ be a local ring. Let $\mathbb{P}^1_R=\rm{Proj}~R[x_0, x_1]$ be the Projective line over $R$.
Is there a classification of vector bundles of rank $n$ on $\mathbb{P}^1_R$ in terms of splitting ...

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### Problem In understanding distribuitional section

In this post Observables By Urs Schreiber he denotes the space of distributional sections in defenition 7.9 by $ \Gamma_{\Sigma}^{\prime}\left(E^*\right) $
That is if $u \in \Gamma_{\Sigma}^{\...

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### Understanding coherent sheaf obtained via sheaf injections of holomorphic vector bundles on TCP^1

My problem involves holomorphic vector bundles $E,F$ of the same rank on $T\mathbb{C}P^1$. I have a short exact sequence of sheaves $$0\rightarrow E\rightarrow F\rightarrow Q\rightarrow 0.$$
I want to ...

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### How to show the inclusion of the exceptional divisor is the zero section of the line bundle

Let $k$ be a field and $R$ be the ring $k[x,xy,xy^2,xy^3]$. Let $I$ be the ideal of $R$ generated by $x,xy,xy^2,xy^3$.Let $X$=Spec$(R)$ and $\tilde{X}$ be the blow-up of $X$ along $I$.I managed to ...

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### Why Gateaux derivative is a distribution?

Thanks to Jan Bohr answer and comment I edited this question.
Let $E$ be a vector bundle , $E^*$ the dual bundle and $D$ a density bundle. Denote by $\Gamma(E)$ the space of section of the bundle $E$....

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### How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?

I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-...

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### Model structure for dga of (endormorphism) vector bundle valued differential forms

I was browsing and came across this discussion on the model structure for a dga. They mostly explain the commutative case but then say some things about the non-commutative case.
Context Consider a ...

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### Polystable vector bundle contains a prescribed line bundle as a line subbundle

Let $X$ be a compact Riemann surface of genus $g \geq 1$, and let L be a line bundle over $X$ with $-g < \deg L \leq -\frac{1}{2}g$. Can we always find a flat line bundle $J \in \operatorname{Pic}^...

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### Vector subbundles of a given one in $\mathbb{CP}^1$

I apologize if this question is not suited for MathOverflow. This has been crossposted in MathStackExchange here and it is related to some open questions on that site that remain unsolved.
I would ...

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### Short exact sequence of equivariant line bundles on $\mathbb P^1$

I have a two-dimensional vector space ${\mathbb C}^2$ with basis $e_m, f_1$ and action of ${\mathbb C}^*$ by $t \cdot e_m = t^m e_m$ and $t \cdot f_1 = f_1$ and I have the projective line ${\mathbb P}^...

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### Semistable pure dimension one sheaves of rank 1 and degree 0 on a singular curve

We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $...

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### Does it make sense to define a holomorphic structure on $\mathbb{C}P^\infty$ and vector bundles over it?

Let $E \to \mathbb{C}P^\infty$ be any topological complex vector bundle over the infinite complex projective space. I'm wondering if it makes sense to possibly define a "holomorphic structure&...

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### Strategy to distinguish a rank 2 vector bundle from an extension class group

If you think this question is very basic, then I apologies my ignorance at the first stance.
Suppose $V$ is a rank 2 vector bundle on a rational ruled surface $\pi:\mathrm{X}=:\mathbb{P}(\cal{O}_{\...

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### Non vanishing of a cohomology class associated to a nef vector bundle

Lemma. Let $E$ be a rank $r$ nef vector bundle over a polarized smooth complex projective variety $(X,H)$ of dimension $n\leq r$. Then for any $t\in\mathbb{R}_{\geq0}$:
$$
\sum_{k=0}^nt^{n-k}\int_Xc_k(...

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### Ampleness of the normal bundle to the Albanese image

Let $X$ be a projective surface of general type over $\mathbb{C}$, and assume that $\Omega_X$ is globally generated. Then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is a local embedding ...

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

Given an $m$-dimensional smooth manifold $M$, and an $n$-dimensional submanifold $N$, with $n<m$, the tangent bundle of $N$ is a smooth $n$-dimensional subbundle of $TM|_N$.
I'm interested in ...

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### Is this blow-up a line bundle over the projective line

Let $R$ be the ring $\mathbb{C}[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $I$ be the ideal of $R$ generated by $a,d$. Let $X=\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of the affine scheme $X$ ...

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### Intersection form of $2n$-manifold for odd $n$

Let $M$ be closed orientable $2n$-manifold, where $n$ is odd. It is well known that the $\mathbb Z$-module $H^\bullet(M;\mathbb Z)$ has graded-commutative multiplication and $H^{2n}(M;\mathbb Z)\simeq\...

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### The Todd class and Weyl's character formula

Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...

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### Pullback of complex vector bundles along a retraction of compact Hausdorff spaces: a direct proof instead?

Consider a pointed compact Hausdorff space $(X,x_0)$ and a closed pointed subspace $i:A\subset X$ such that there exists a continuous map $r:X\rightarrow A$ such that $r|_A=\text{Id}_A$. Set
$$q:(X,...

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### Sections of the sum of two copies of the tangent bundle of the $2$-sphere

If $T$ is the tangent bundle of $S^2$, then $T\oplus T$ is trivial of rank $4$. Indeed, $T\oplus\mathbb R=\mathbb R^3$, so $T\oplus T\oplus\mathbb R^2=\mathbb R^6$ and $T\oplus T$ is stably trivial: ...