**3**

votes

**1**answer

258 views

### Are rational sections of a vector bundle useful?

Let $X$ be a complex manifold or variety and $L$ a line bundle on it. Given a rational section $s$ of $L$, we get a divisor $D=Div(s)$ and may recover $L$ as $\mathcal{O}(D)$. What about vector ...

**1**

vote

**1**answer

283 views

### Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bundles on a manifold?

Let's say we denote by $T^{(n,m)}M$ the vector-bundle of rank $(n,m)$ tensors on a manifold $M$ and by $\Lambda^pM$ the vector-bundle of $p$-forms on $M$. Is there a relationship (perhaps a ...

**4**

votes

**1**answer

157 views

### A question on $Isom(p_1^*E,p_2^*E) \rightrightarrows X$

I am learning the moduli stacks of vector bundles and have trouble understanding some definitions. Let $E$ be a rank $n$ vector bundle over the scheme $X$. We denote by $p_i$ the $i$th projection ...

**11**

votes

**0**answers

930 views

### conformal blocks for beginners

I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...

**5**

votes

**2**answers

535 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 ...

**6**

votes

**3**answers

515 views

### How does one go from Chern--Weil to cohomology classes on BGL(n,C)?

Let's assume we start with Chern--Weil theory in the following form:
Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfrak l_n(\mathbb C)$ ...

**3**

votes

**2**answers

241 views

### Triviality of Associated Bundles

Let $P\rightarrow M$ be a principal (right) $G$-bundle, where $G$ is a Lie group. Given a finite-dimensional representation of $G$, $V$ say, we can define the associated bundle ...

**1**

vote

**1**answer

245 views

### Direct sum of two stable bundles of same slope

How to prove that the direct sum of two stable vector bundles of the same slope over a smooth curve is a semistable bundle?

**0**

votes

**1**answer

190 views

### Sommese's theorem (generalized Weak Lefschetz) in arbitrary characteristic?

Sommese's theorem is a natural generalization of the Weak Lefschetz; for a smooth projective (connected) $X$, an ample vector bundle $E/X$ of rank $e$, and a section $s:X\to E$ it states that the ...

**4**

votes

**2**answers

489 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

**1**answer

274 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

**1**answer

94 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. ...

**7**

votes

**2**answers

776 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 $s$ ...

**17**

votes

**3**answers

1k 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

**1**answer

747 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 ...

**14**

votes

**2**answers

1k 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

**2**answers

485 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 ...

**10**

votes

**1**answer

673 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

**2**answers

521 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

**2**answers

511 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

**2**answers

440 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 ...

**6**

votes

**2**answers

972 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 ...

**1**

vote

**1**answer

211 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

**2**answers

433 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

**0**answers

112 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

**1**answer

329 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**

votes

**0**answers

77 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

**1**answer

404 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

**1**answer

394 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

**2**answers

410 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

**1**answer

427 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

**1**answer

512 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

**1**answer

322 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

**6**answers

971 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 ...

**7**

votes

**3**answers

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 ...

**10**

votes

**2**answers

627 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.
...

**41**

votes

**6**answers

3k 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

**3**answers

1k 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

**1**answer

269 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

**1**answer

215 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

**1**answer

408 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

**0**answers

206 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

**2**answers

444 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

**1**answer

266 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

**2**answers

551 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

**1**answer

435 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

**1**answer

340 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 ...

**13**

votes

**1**answer

730 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

**1**answer

248 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

**1**answer

387 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 ...