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3
votes
2answers
405 views

Vector Bundles on normal surfaces

Let $X$ be a projective normal surface over $\mathbb{C}$. In this related question it is stated as soon as $X$ is smooth any vector bundle defined on the compliment of a codimension 2 subset extends ...
21
votes
4answers
2k views

Using linear algebra to classify vector bundles over P^1

There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let ...
14
votes
4answers
1k views

What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
28
votes
3answers
2k views

When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
11
votes
4answers
2k views

Extending vector bundles on a given open subscheme

Many people seem to know the following. Personally, I don't quite understand it though and maybe I'm wrong. It's the fact that "a vector bundle on an open subscheme extends in only one way to a vector ...
10
votes
5answers
835 views

Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same. Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
15
votes
1answer
409 views

Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
14
votes
3answers
944 views

Is the category of vector bundles over a topological space abelian?

Consider the trivial bundle $V=\mathbb{R}\times\mathbb{R}$ and the map $f:V\rightarrow V$ given by $(t,x)\mapsto(t,tx)$. This has fibrewise kernels and cokernels, but the ranks jump at 0, so the ...
12
votes
1answer
1k views

A question on classification of almost complex structures on $4$-manifolds

I have a (basic?) question in topology. Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by ...
10
votes
3answers
1k views

Are schemes that “have enough locally frees” necessarily separated

Let me motivate my question a bit. Thm. Let $X$ be a locally noetherian finite-dimensional regular scheme. If $X$ has enough locally frees, then the natural homomorphism $K^0(X)\longrightarrow ...
10
votes
3answers
2k views

Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$

I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle ...
6
votes
1answer
555 views

“Vector bundle” with non-smoothly varying transition functions

I'm working my way through Lang's Fundamentals of Differential Geometry, and when he introduces vector bundles, he states that for finite dimensional bundles, the third axiom is redundant. I'm hoping ...
11
votes
2answers
611 views

Vector bundles on vector bundles

Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base? In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...
9
votes
2answers
572 views

Configuration spaces and non homeomorphic vector bundles

Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign. ...
12
votes
6answers
861 views

Does every vector bundle allow a finite trivialization cover?

Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$. (a) Is ...
9
votes
1answer
479 views

Is every closed embedded codimension-n submanifold cut out by a section of a rank-n vector bundle?

Let $M$ be a smooth manifold (over $\mathbb R$) and $N \hookrightarrow M$ a closed embedding. Locally near any point in $N$, I can find coordinates $x^1,\dots,x^{\dim M}$ on $M$ so that $N$ is the ...
5
votes
6answers
2k views

Differences between reflexives and projectives modules

Let R be a normal noetherian domain. What is the difference between a finitely generated reflexive module and a finitely generated projective module? Can anybody recommend any references or make ...
16
votes
5answers
1k views

A ring such that all projectives are stably free but not all projectives are free?

This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
8
votes
3answers
764 views

How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?

Edit: It seems I had two different constructions mixed up in my head, namely the frame torsor and the automorphism bundle of a vector bundle. This made the main question a bit confusing. The first ...
5
votes
1answer
347 views

Spaces over which every vector bundle is a summand of the trivial bundle

Let X be a Hausdorff space such that every real vector bundle on X is summand of a trivial bundle. Does this imply that X is homotopy equivalent to a compact Hausdorf space? This question ...
3
votes
2answers
191 views

Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle

I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up. Let $X$ be a ...
3
votes
1answer
661 views

Orthogonal complements in Hilbert bundles

It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle. What is known about the ...
3
votes
1answer
249 views

Totally non parallelizable manifold

Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold? What is ...
1
vote
0answers
108 views

The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
1
vote
1answer
254 views

Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$

I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...
1
vote
1answer
221 views

Possible homotopy-theoretical approach to Gauss-Bonnet

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
1
vote
1answer
82 views

Are these two bundles, stably equivalent?

Let $(E,M,p)$ be a n dimensinal smooth vector bundle where $M$ is a k dimensional manifold. We assign to $M$, two different vector bundles $F_{1}$ and $F_{2}$ over $M$ as follows: 1)$TE$ is a ...