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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4
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467 views

Jets of Equivariant Vector Bundles

Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. ...
3
votes
1answer
253 views

Euler characteristics and the difference bundle construction

I am reading on K theory in Lawson and Michelson (Spin Geometry). One has the "exact sequence spaces" $L(X,Y)$ and there is the theorem that there is a unique equivalence of functors $\chi$ between ...
0
votes
1answer
83 views

Isotropic splitting for exact Courant algebroids

An exact Courant algebroid $E$ is one such that the sequence $0\to T^\star M\xrightarrow{\rho^\star} E^\star\simeq E\xrightarrow{\rho} TM\to 0$ is exact. Here $\rho$ is the anchor of the algebroid. ...
6
votes
2answers
645 views

Examples of excess intersection theory?

Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ ...
14
votes
3answers
979 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 ...
3
votes
1answer
493 views

1-jet bundle on vector bundle with metric connection

Background I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to ...
13
votes
2answers
909 views

Non trivial vector bundle over non-paracompact contractible space

The proof that the set of classes of vector bundles is homotopy invariant relies on the paracompactness and the Hausdorff property of the base space. Are there any known examples of: Non trivial ...
2
votes
2answers
452 views

Moduli spaces of vector bundles and stability conditions

Let $C$ be an algebraic curve. One of the easiest examples of stabilty functions is $$Z:Coh(C)/ \{ 0 \} \rightarrow \overline{\mathbb{H}};\ \ \ \ Z(E):=-deg(E)+i\cdot rk(E).$$ This induces the ...
9
votes
1answer
554 views

Rank two vector bundles on a curve of genus two

I recently learned of an interesting result of Narasimhan and Ramanan from 1969, which says that moduli space of rank two vector bundles with trivial determinant on a curve $X$ of genus two is ...
1
vote
2answers
465 views

Holomorphic bundles and maps to the Grassmannian ?

Hello, In the differentiable case it is quite easy to prove that vector bundles are equivalent to smooth maps to the Grassmannian $G_{k}(\mathbb{R}^N)$ for some integer $N>>0$. The proofs I ...
2
votes
2answers
463 views

Global Definition of the Dolbeault Complex of a Vector Bundle

For an $2n$-dimensional complex manifold $M$, and a smooth vector bundle $E$ over $M$, it is well-known (see Voisin, Huybrechts) that there exists an operator $\overline{\partial}$, built locally from ...
6
votes
2answers
397 views

Ample vector bundles on complex tori

Let $X$ be a $n$-dimensional complex torus and $\omega$ a Kähler form on $X$. Then, it is well known that a real $(1,1)$-class $[\alpha]\in H^{1,1}(X,\mathbb R)$ is a Kähler class if and only if for ...
4
votes
2answers
821 views

Riemannian metrics as sections of a vector bundle

Let $\pi : E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle ...
0
votes
1answer
201 views

eigen-bundles of a trivial vector bundle

Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on ...
6
votes
2answers
402 views

Homotopy invariance of vector bundles by parallel transport: reference needed for my students.

Let $M$ be a smooth manifold and $V \to [0,1] \times M$ be a smooth vector bundle. The homotopy invariance states that the restrictions $V_0$ and $V_1$ to the bottom and top of the cylinder are ...
1
vote
0answers
106 views

Topological index and Dirac operator with a non compact group

A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...
0
votes
1answer
309 views

Cotetrad, spin connection and Dirac operator

Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) ...
2
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0answers
70 views

Properties of special Cayley-Bacharach bundles on a K3-surface

Assume we have a $K3$-surface $X$ over $\mathbb{C}$ and two rational curves $C_1$ and $C_2$ on $X$ with $C_1.C_2=1$ and $C_i^2=-2$. Let $x$ be a closed point on the reducible curve $C_1\cup C_2$. We ...
4
votes
1answer
386 views

Representations of infinite dimensional Lie algebras as vector fields on manifolds

Suppose I have e.g. the Witt algebra, $\left[l_n,l_m \right] = -(n-m)l_{n+m}$. I want to realize the $l_n$ as vector fields on some manifold. The classical example is when the $l_n$ span the Lie ...
1
vote
1answer
365 views

Schemes associated to vector spaces

Let $k$ be a field. Let $F$ be a covariant functor on the category of $k$-algebras to the category of sets. Assume that the opposite functor $F^{op}$ on the category of affine $k$-schemes is a sheaf. ...
7
votes
2answers
366 views

References/surveys concerning characteristic classes of flat vector bundles

I'm looking for good surveys about characteristic classes of flat real vector bundles. Letting $G$ be $\text{SL}_n(\mathbb{R})$ with the discrete topology, orientable flat $n$-dimensional real vector ...
0
votes
1answer
406 views

Zero locus of a generic smooth section

Let $V$ be a smooth manifold, $E \rightarrow V$ a vector bundle over $V$ and $\Gamma$ be a finite group acting nontrivially on $V$ and $E$. Let $s \in C^\infty(E)$ be a generic $\Gamma$-equivariant ...
4
votes
1answer
484 views

etale homotopy and Adams conjecture

I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, ...
1
vote
1answer
282 views

dual of locally free sheaf

For simplicity, assume everythings occur on a smooth projective variaty $X$. Dual bundle of the given line bundle $\mathcal L$ is determined by $\mathcal L$ and $c_1(\mathcal L)$. $\mathcal L^*= ...
12
votes
6answers
887 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 ...
4
votes
3answers
2k views

Vector bundles vs principal $G$-bundles

It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres $$F=\pi^{-1}(x), \ \ \ x\in B $$ over any $x\in B$, are ...
9
votes
2answers
582 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. ...
37
votes
4answers
2k views

Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic

Hello, I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
8
votes
3answers
842 views

Relationship between monodromy representations and isomorphism of flat vector bundles

This question is somehow related to this one. Let $M$ be a smooth (compact, if you wish) connected manifold. Then, it is well known that there is an equivalence between the isomorphism classes of ...
7
votes
1answer
258 views

Does direct limit commute with functor of smooth sections?

Consider a countable family of finite-rank vector bundles $V_k$ over a finite-dimensional smooth manifold $M$. The direct limit of such a family is still a topological vector bundle even though it may ...
1
vote
1answer
205 views

Criteria for acyclicity

Let $X$ be a smooth projective variety. Let $E$ be a line bundle (or, more generally, a vetor bundle) on $X$. Are there any nice criteria for acyclicity of $E$ (that is, for the property $H^i(X,E)=0$ ...
3
votes
1answer
400 views

non-trivial locus of a holomorphic vector bundle

Let $X$ be a holomorphic vector bundle over $Y$ (where $Y$ is an arbitrary complex manifold, not necessary projective). Does there exist an analytic subset $Z$ of $Y$ such that the restriction of $X$ ...
0
votes
0answers
197 views

Variation of the Chern connection according to the variation of hermitian metric

Whats is the relation between the Chern connections of tow Hermitian metrics in a holomorphic vector bundle?
1
vote
2answers
420 views

complex vector bundles and curvature

Let us suppose that $X$, with a 2-form $\omega$. Suppose $J$ is an element of $su(2)$ such that $J^2=-e$ for $e$ the identity. Is there a necessary and sufficient condition on $\omega$ which will give ...
7
votes
1answer
229 views

Triviality of direct multiples of flat complex vector bundles

Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if $E$ is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold $X$ (more ...
3
votes
2answers
537 views

Stiefel-Whitney classes of a projective space bundle

Hi! Let $\gamma_1$ denote the twisted line bundle over $S^1$ and add a trivial $(2k-1)$-bundle $\mathbb{R}^{2k-1}$. Consider the projective bundle $P(\gamma_1 \oplus \mathbb{R}^{2k-1})$ over $S^1$. ...
4
votes
1answer
385 views

Associated vector bundles of infinite rank and induced connections

Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can associate to this ...
5
votes
1answer
327 views

A^1-invariant for Vector Bundles?

We know that if $X$ is a smooth connected variety over a field $k$, then any line bundle on $X\times_k\mathbb{A}^1$ is from a line bundle on $X$. This is simply because they have the same Picard ...
12
votes
1answer
634 views

Splitting principle for holomorphic vector bundles

Let $E \to X$ be a vector bundle over a decent space $X$. Then there is a space $Z$ together with a map $p: Z \to X$ which induces a split injection on cohomology and such that $p^* E$ splits as a ...
2
votes
1answer
242 views

Do cohomologically trivial line bundles affect morphisms?

Assume we have a locally free sheaf $R$ of associative $O_S$-algebras of rank $r^2$, possibly noncommutative, where $S=\mathbb{P}^2$ over $\mathbb{C}$. Furthermore given a locally free $O_S$-module ...
2
votes
1answer
359 views

fano moduli varieties of vector bundles

Let $M$ be a fine moduli space of vector bundles on curve which is an algebraic variety as well. The first example of such an object that I have in mind is rank 2, deg 1 VB on a genus 2 curve. This is ...
3
votes
1answer
307 views

A simple question about the degree of some vector bundle over rational curve.

Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, ...
8
votes
1answer
459 views

Weak Vector Bundles

The following notion has arisen in a paper I'm writing. Definition. A map $p: E\to B$ of spaces is said to be weak vector bundle if for all compact subspaces $K \subset B$ the restriction of $p$ to ...
2
votes
1answer
235 views

family of torsors and family of vector bundles

Suppose $X$ and $Y$ are smooth connected schemes over a field $k=\bar{k}$, $f: X\times_kY\to X$ is the first projection. You may assume $Y$ is proper if you like, my question is if $P\to X\times_kY$ ...
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vote
1answer
235 views

Intersection of subvector bundles

Suppose we have a smooth vector bundle $\pi: E \rightarrow B$ and two sub vector bundles $\pi_1: E_1 \rightarrow B_1$ and $\pi_2: E_2 \rightarrow B_2$ such that the bases $B_1$ and $B_2$ are ...
0
votes
2answers
367 views

Normal bundle of $CP^1$ in $CP^2$ [closed]

I'm studying the book "Differential forms of algebraic geometry" of Bott, Tu. At page 75 there is an exercise about the normal bundle of $CP^1$ in $CP^2$, and there is written that the transition ...
3
votes
1answer
681 views

Terminology of “covariant derivative” and various “connections”

I apologize for the long question. My ultimate aim is to understand the existing terminology and find appropriate terminology for covariant derivatives on vector bundles which generalizes to ...
2
votes
2answers
403 views

degree 0 vector bundles

If $X$ is a smooth projective curve over a field $k$, $V$ is a subbundle of some finite direct sum of $O_X$, then is it true that $V$ is of degree 0?
2
votes
1answer
260 views

When is restriction an equivalence of categories of equivariant vector bundles?

Suppose a (linear algebraic) group $G$ acts on a variety $X$ and that $U$ is a $G$-invariant open subvariety. My question is: under what conditions is the restriction functor $i^*: Vect^G(X) ...
0
votes
1answer
203 views

Restriction of the tautological class to a subbundle

Let $E$ a rank $r\geq 3$ vector bundle over a curve $C$ and let $E'$ a rank $r-1$ subbundle of $E$. Thus we have $\mathbb{P} (E') \subset \mathbb{P} (E)$; what can be said about $ ...