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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178 views

Extending intersection bundles

Let $X$ be the product $Gr_i(V)\times Gr_j(V)$ of two Grassmannians where $V$ is a complex vector space of dimension $d$. There is an open $U\subset X$ formed by all those $(V',V'')\in X$ such that ...
2
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1answer
674 views

Linearization of a vector field

In a paper that I was reading, I stumbled across the following theorem: Let $X$ be a vector field with $$X= > a^ix^i\partial_{x^i} + > \mathcal{O}(|x|^2),$$ where $x$ is some chart and ...
11
votes
0answers
318 views

Atiyah-Bott from Beauville-Laszlo

This is a question about the cohomology groups of the stack of vector bundles (with fixed discrete invariants) on an algebraic curve. Explicit formulas for these cohomology groups are known, and they ...
9
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2answers
739 views

Good reference for globally formulated calculus of variations on Riemannian manifolds?

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...
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0answers
292 views

Splitting of vector bundles on a complex torus

Let $X$ be a complex torus (a finite dimensional complex vector space modulo a lattice) and let $E$ be a smooth (not necessarily holomorphic) complex vector bundle over $X$. Is it true $E$ is ...
1
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1answer
209 views

A specific degeneration of a rank 2 bundle

I wish to know if there is a rank 2 vector bundle $E$ on $\mathbb{P}^1 \times \mathbb{P}^1$ such that $\mathbb{P}(E)$ when restricted to $\mathbb{P}^1 \times [0:1]$ is the $n$th Hirzebruch surface and ...
15
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0answers
481 views

Characteristic Classes for $E_8$ Bundles

Given a principal $E_8$ bundle $P\rightarrow X$ one can take the adjoint representation $\rho :E_8\rightarrow SU(\mathbb C^{248})$ and form the associated vector bundle $V=P\times_{\rho}\mathbb ...
4
votes
1answer
358 views

A vector bundle with a given jumping line

I'd like to know if there exists a holomorphic rank 2 sub-bundle of $T\mathbb{P}^3$ which, when restricted to a given line is $\mathcal{O}(-a)\oplus \mathcal{O}(a)$, but is trivial when restricted to ...
2
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1answer
581 views

Spectral sequence of symmetric or exterior algebras?

This question is inspired by Hartshorne's exercise II.5.7 (c-d): the problem reads: Let $0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow0$ be a short exact sequence ...
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2answers
304 views

Top self-intersection of the tautological line bundle

Let $\mathcal E$ be a rank $n$ vector bundle over a curve $Y$ and let $X=\mathbb P(\mathcal E)$ and let $\pi: X \to Y$ be the projection. I would like to compute the value of the top self-intersection ...
28
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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 ...
4
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1answer
300 views

Reference for nef coherent sheaves?

The definition and basic properties of nef locally free sheaves appear for instance in the second volume of Lazarsfeld's book "Positivity in Algebraic Geometry" (beginning of chapter 6). However, I ...
2
votes
1answer
166 views

Closed immersion into (relative) projective bundle.

Hello, I have a question about certain immersions. If $X$ is a base scheme, and $E$ is a loccally free quasi coherent sheaf over X, then we can build $P(E)$ the projective bundle associated to E. ...
2
votes
0answers
108 views

Bundles of (co)chain complexes

A stupid question: does anybody know a good book or something in which invariants of a vector bundle of finite-dimensional acyclic (co)chain complexes (which, I believe is equivalent to a acyclic ...
2
votes
1answer
511 views

Extending a vector bundle to a torsion free sheaf

Let's say that $X$ is an integral scheme of finite type over a field and $Y \subset X$ is a closed subscheme. Given a vector bundle $E$ on $Y$, is $E$ the restriction to $Y$ of a vector bundle on a ...
5
votes
1answer
403 views

K-Theory as a special $\lambda$-ring

I wonder if there is a nice and short proof that the $K$-theory of a topological space is a special $\lambda$-ring without invoking the splitting principle and alike. Is it possible to show directly ...
3
votes
1answer
520 views

moduli of vector bundles on a surface

Let $S$ be a smooth projective surface with an ample divisor $X\subset S$. Consider the moduli stack of vector bundles $F$ on $S$ such that 1) $c_1(F)=0$ 2) $c_2(F)=n$ 3) The restriction of $F$ to ...
3
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0answers
211 views

Ricci flat metrics on holomorphic vector bundles over Riemann surfaces

I am interested in the local geometry of holomorphic curves in Calabi-Yau threefolds. The setup and question are then the following: Consider a $\mathbb{C}^2$ bundle over a compact Riemann surface ...
4
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2answers
2k views

Rank 2 flat bundles on an elliptic curve, via extensions

I have some hopefully elementary questions about rank 2 flat bundles on an elliptic curve $E$. Take $p\in E$, and consider the exact sequence $$0\to \mathcal{O}(-p) \to V \to \mathcal{O}(p)\to 0$$ ...
5
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5answers
2k views

Why is $\mathbb{R}^{\infty}$ defined the way it is?

I've been thinking about Grassmannians recently. Think of $\mathbb{R}^k$ as a $k$-dimensional vector space. Let $\text{Gr}_n(\mathbb{R}^k)$ denote the Grassmannian of all $n$-dimensional vector ...
3
votes
3answers
582 views

Polynomial Vector Fields on the 3-Sphere

EDIT(3): I am looking for a basis for the Lie algebra of polynomial vector fields on $S^3$. EDIT(2): I am fairly certain now that my question is more along the lines of, what does the Lie algebra of ...
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0answers
459 views

Ample vector bundles, $H^1=0$ and global generation in characteristic $p$

This is a follow up from this question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective curve $X$ is ample if and ...
2
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2answers
501 views

Elementary transformations of ruled surfaces as maps of vector bundles

This comes as a question in Beauville's Algebraic surfaces book (III.24 (2)). We work over $\mathbb{C}$. All geometrically ruled surfaces (grs) $p:S\longrightarrow C$ over a curve $C$ can be seen as ...
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2answers
933 views

Nice example of a topologically trivial bundle with nontrivial connection

So, I've been trying to understand what exactly an anomaly is, and how they arise in physics. Apparently an anomalous theory is some theory whose action is given by a section of some bundle (rather ...
5
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1answer
617 views

when is a section of vector bundle determined by its zero locus?

Let $V_X$ be a vector bundle of rank $r>1$ on a smooth (connected) projective variety of dimension $r$. Let s be the global holomorphic section, whose zero locus is a zero dimensional subscheme ...
9
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1answer
484 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 ...
18
votes
2answers
933 views

If the total Chern class of a vector bundle factors, does it have a sub-bundle?

Motivation: $T_{\mathbb P^2}$ isn't an extension of line bundles Here's a trick to show that the tangent bundle $T$ of $\mathbb P^2$ is not an extension of line bundles. If it were, we would have a ...
2
votes
1answer
595 views

Mehta-Seshadri and Parabolic bundles

In the original paper of Mehta-Seshadri, it seems like they treat the case of zero parabolic degree (i.e. they prove that zero parabolic degree stable parabolic bundles correspond to irreps of the ...
14
votes
3answers
585 views

Is analytic Quillen-Suslin simple?

This question is motivated by a sentence on the Wikipedia entry for Quillen-Suslin theorem. This theorem states that every algebraic vector bundle on affine space is trivial. The analogous result is ...
4
votes
0answers
296 views

Vector bundles of schemes and their topological realizations

Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$. Does $R_\mathbb{R}$ send an ...
10
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5answers
853 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 $A(t)$ ...
8
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1answer
376 views

Can curves differentiate vector bundles on P^2?

Not much is known about vector bundles on $\mathbb{P}^2$ but I wonder if the following is a tractable question: If $E,E'$ are non-isomorphic vector bundles on $\mathbb{P}^2$, then is there always a ...
10
votes
3answers
2k views

Why is the integral of the second chern class an integer?

I'm currently reading the paper "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase" by Barry Simon. Imagine a vector bundle with a connection $\nabla$. For simplicity, we assume that this is ...
12
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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
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0answers
438 views

Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles. I will ...
7
votes
2answers
684 views

Which torsion classes in integral cohomology are Chern classes of flat bundles?

Chern-Weil theory tells us that the integral Chern classes of a flat bundle over a compact manifold (i.e. a bundle admitting a flat connection) are all torsion. Given a compact manifold $M$ whose ...
4
votes
0answers
309 views

Harder-Narasimhan filtration of rank 4 vector bundles on $P^2$

Given a non-semistable vector bundle on $P^2$ of rank 4, are explicit conditions known for when the ranks in its Harder-Narasimhan filtration are (3,1), (2,2) and (1,3) respectively? I would be very ...
10
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0answers
378 views

Exotic smoothness and Parallelizability

Regarding the parallelizability of the Milnor's seven dimensional exotic spheres: Parallelizability of the Milnor's exotic spheres in dimension 7 The following question naturally arises: Suppose ...
55
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4answers
3k views

Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
14
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2answers
2k views

First Chern class of a flat line bundle

A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one? Let $X$ be a nice space ...
12
votes
1answer
706 views

Is there an alternative characterisation of vector bundles with vanishing characteristic classes?

This question came up yesterday during our index theory seminar. Let $M$ be a 1-connected smooth manifold and let $E \to M$ be a finite-rank complex vector bundle over $M$. If all the Chern classes ...
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3answers
1k views

The third axiom in the definition of (infinite-dimensional) vector bundles: why?

Serge Lang's Differential and Riemannian Manifolds is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers ...
2
votes
1answer
470 views

Sections of Grassmannian bundles

Let $X$ be a smooth projective variety of dimension $n$. Take the bundle $TX \oplus Sym^2(TX)$ over $X$ where $Sym^2(TX)$ is the second symmetric product of the tangent space. The Grassmannian bundle ...
6
votes
1answer
848 views

Does a Trivial Tangent Bundle Induce a Multiplication?

Let $M$ be a connected smooth manifold, and assume that it is parallelisable; that is, its tangent bundle is trivial. Does $M$ admit an H space structure? That is, does there exist a smooth map ...
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3answers
849 views

Calculating the decomposition of a vector bundle over rational curve

Consider the rational curve (conic) given by image of the map $$ u([z,w])=[z^2,-z^2,w^2,-w^2,zw] \in \mathbb{P}^4 $$ which lies in quintic 3-fold $X: x_1^5+\cdots+x_5^5- x_1\cdots x_5=0$. By ...
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3answers
594 views

Topology of maps between fibers of vector bundles

First of all sorry for the (possible) incorrect english. I don't know english very well. I'm with a doubt about topology of maps between fibres of vector bundles. Consider $E$ and $F$ vector bundles ...
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2answers
744 views

Calculating normal bundle

I just realize that even though I know what normal bundes are, I dont know how to compute them. The main objective is to show that a ration curve C on a quintic threefold doesnt move. If C is a line, ...
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votes
2answers
772 views

Finite vector bundles over punctured affine spaces

Let $X$ be a connected scheme. Recall that a vector bundle $V$ on $X$ is called finite if there are two different polynomials $f,g \in \mathbb N[T]$ such that $f(V) = g(V)$ inside the semiring of ...
5
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2answers
923 views

trace of the atiyah class equals chern class

In several textbooks ("The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn, "Calcul differentiel et classes caracteristiques..." by Angeniol and Lejeune-Jalabert) it is mentioned that the ...
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1answer
441 views

Why is the Cotangent Space of Complex Projective Space Not $U(1)$-Equivariant?

I'm looking at the cotangent bundle of $CP^{N}$ at the moment in the context of equivariance. For many reasons, it seems to me that this bundle is not $U(1)$-equivariant, or, in other words, cannot be ...