The vector-bundles tag has no wiki summary.

**0**

votes

**1**answer

456 views

### Is a fibre bundle over a vector bundle trivializable on each fibre?

Let $\pi:E\to M$ be a vector bundle over a closed smooth manifold and supose $\Pi:F\to E$ is a fibre bundle over the total space of $\pi$. I'd like to know if, restricted to $E_p$, the second bundle ...

**6**

votes

**2**answers

1k views

### Reference request: moduli space of vector bundles

I am trying to study the moduli of holomorphic vector bundles fast and I'm primarily interested to understand:
1) Why and were the stability is important.
2) How are the construction methods.
3) some ...

**10**

votes

**4**answers

565 views

### Algebraic analogue of the Moebius bundle over the circle

Let $R$ be the ring $R[X,Y]/(X^2+Y^2−1)$. The space of $\mathbb{R}$-rational points of the affine scheme associated to $R$ is the topological circle $S^1$.
An algebraic vector bundle over $R$ is an ...

**1**

vote

**5**answers

745 views

### indecomposable vector bundles having proper sub-bundles.

Over rational curve we know that any vector bundle is decomposable to direct sum of line bundles.
In higher dimensions there are examples of indecomposable bundles.
some indecomposable vector ...

**7**

votes

**5**answers

2k views

### What is a square root of a line bundle?

If ${L}$ is a line bundle over a complex manifold, what does the square root line bundle $L^{\frac{1}{2}}$ mean?
After some google, I got to know that there are certain conditions for the existence of ...

**4**

votes

**3**answers

1k views

### The correspondence between affine vector bundles and f.g. projective modules

The definition of a (geometric) vector bundle over a scheme $X$ can be rewritten as follows in terms of 'not-so-geometrical algebra' if $X=Spec R$ is affine and if I am not missing something.
A ...

**5**

votes

**1**answer

497 views

### Rank 2 vector bundle on a product of elliptic curves

Let $E$, $F$ be two complex elliptic curves, and $A=E \times F$. Let us denote by
$\pi_E \colon A \to E, \quad \pi_F \colon A \to F$
the natural projections. For all $p \in F$ let us write $E_p$ ...

**3**

votes

**3**answers

567 views

### Geometry and Integrability in Other Bundles

Background: Suppose $E=TM$ is the tangent bundle to some differentiable manifold $M^n$. If we specify some subbundle $D\subset TM$ (distribution of $k$-planes) then there are two natural situations ...

**9**

votes

**1**answer

1k views

### How do you describe vector bundles on elliptic curves?

Throughout "curve" means smooth projective curve over an algebraically closed field.
Motivation and Background
I read somewhere that Atiyah has classified vector bundles on elliptic curves. My ...

**7**

votes

**2**answers

341 views

### Swan like theorem and covering spaces

Let $X$ be a finite CW complex. Swan's theorem provide an equivalence
\[
Vec(X)~\xrightarrow{\sim} ~ProjMod(hom_{Top}(X,\mathbb{R}))
\]
between the category of ...

**2**

votes

**0**answers

653 views

### Vector bundles on some non-projective surfaces

Let $X$ be a smooth projective curve over a field $k$ and let $L$ be a line bundle on $X$.
I will denote by $S$ the total space of $L$ -- this is a smooth surface over $k$ containing
$X$ (as the zero ...

**4**

votes

**1**answer

751 views

### How to resolve a wedge product of vector bundles

Let $X$ be an algebraic variety. Consider an exact sequence
$$0\to A\to B\to C\to 0$$
of vector bundles on $X$. I have seen in different papers the following type resolution of wedge product of $C$ ...

**2**

votes

**2**answers

253 views

### Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated?

Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth ...

**1**

vote

**0**answers

139 views

### Partial ordering of vector bundles on projective spaces

I would like to know if there are some interesting partial orders defined on the isomorphism classes of vector bundles on $\mathbb P^n_k$ (you can assume $k$ is $\mathbb C$ if that helps).
...

**8**

votes

**4**answers

608 views

### Which sets of Stiefel-Whitney characteristic numbers can be realized as coming from a manifold?

EDIT: The original question was answered very quickly (and very nicely!) but the answer leads to a pretty obvious subsequent question, which I will now ask. The original question is maintained for ...

**17**

votes

**8**answers

6k views

### What are some open problems in algebraic geometry?

What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...

**4**

votes

**2**answers

776 views

### Holomorphic vector bundles and Swan's theorem

Is every holomorphic vector bundle a direct summand of a trivial vector bundle on submanifolds of C^n? What about projective varities? I believe Swan's theorem says something about the first question. ...

**3**

votes

**0**answers

388 views

### Topological obstructions to extending algebraic vector bundles

Ariyan and Kevin Lin have asked about the problem of extending vector bundles defined on an open subvariety across the rest of the variety. There can be subtle commutative algebra obstructions, as in ...

**4**

votes

**1**answer

271 views

### Is there a “simple commutation” relation between $D^{''}$ and $\delta^{'}$, with $D^{''}$ the (0,1) part of the chern connection of a vector bundle and $\delta^{'}$ the adjoint of the (1,0) part?

Hi,
as the title says i'm wondering if there's a "simple" and known commutation relation between the following two differential operators. Let $E$ be a holomorphic vector bundle over a compact kahler ...

**3**

votes

**2**answers

381 views

### Vanishing of Self-Ext groups of vector bundles

Let $E$ be a rank two vector bundle on $\mathbb{P}^n$. Assume that $\text{Ext}^1(E, E)=0$. Will $\text{Ext}^2(E, E)$ be zero? Why? Any geometric explanation (in terms of deformation theory?)?
Edit: ...

**4**

votes

**2**answers

258 views

### Endomorphisms of bundles associated to codimension 2 subvarieties

Preamble
I initially decided to post this question on math.stackexchange a few days ago, as I consider it to be much less of a research question and much more of "I'm learning" question. But there ...

**28**

votes

**2**answers

1k views

### Symmetric powers and duals of vector bundles in char p

Suppose that $X$ is a smooth projective variety (eg $P^n$) and $E$ is a vector bundle (eg the tangent bundle). If the characteristic is zero, then taking symmetric powers "commutes" with taking duals:
...

**6**

votes

**5**answers

1k views

### Vanishing of Euler class

Given a real oriented vector bundle E over the base space B of rank n, such that the Euler characteristic class in the n-th cohomology group of B vanishes, is it true that there exists a global ...

**10**

votes

**3**answers

1k views

### Is it true that if the pushforward of a coherent sheaf is locally free, then the original sheaf is locally free?

I think the title says it all. If I have a finite map $p:X\to Y$ between schemes, and $F$ is a coherent sheaf on $X$ such that $p_*F$ is locally free, can I conclude that $F$ is locally free?
...

**1**

vote

**2**answers

466 views

### Nakano semipositivity

Let $X$ be a compact Kaehler manifold.
What is a good, possibly algebraic-geometric, way to think to Nakano semipositivity of holomorphic vector bundles on $X$?
Is the trivial line bundle ...

**0**

votes

**1**answer

218 views

### Algebraic Correspondences 'Expressible' as Vector Bundles

For algebraic curves $C$ over a closed field, a correspondence on $C$ is a the same thing as a divisor, and so, a line bundle on $C \times C$. Can I assume that this simplification does not extend to ...

**8**

votes

**2**answers

654 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 ...

**16**

votes

**5**answers

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 ...

**4**

votes

**2**answers

508 views

### Deformations of sheaves via automorphisms. How to express $Ext^1$?

Let $X$ be a complex manifold (for example $\mathbb CP^n$), let $v$ be a holomorphic vector field on $X$, and let $F$ be a coherent sheaf (for example a vector bundle or a structure sheaf of a point). ...

**10**

votes

**3**answers

954 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 ...

**3**

votes

**1**answer

774 views

### Quick ways to calculate cohomology of vector bundle/local system from transition functions?

Suppose I have a vector bundle (or local system, or something else given by transition functions) on a Riemann surface (or generally a (complex) manifold), and I want to compute its cohomology. The ...

**4**

votes

**5**answers

2k views

### When is the push-forward of the structure sheaf locally free

Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module?
Example 1. Suppose that $f$ is affine. Then ...

**46**

votes

**3**answers

4k views

### Intuitive explanation for the Atiyah-Singer index theorem

My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem.
I'm trying to learn the ...

**5**

votes

**1**answer

654 views

### Maslov index of a pullback bundle

This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology.
Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex ...

**2**

votes

**1**answer

729 views

### How far is the tangent bundle from projective space?

Is there a general theory of embeddings of the (total variety of) the tangent bundle on a (nonsingular) projective variety into projective space? I suppose what I really mean is (and to be more ...

**2**

votes

**2**answers

772 views

### Connection on line bundle on projective curve

Let $C$ be a smooth projective curve. It is known that a line bundle on $C$ is of degree 0, if we can impose a connection structure on it.
Now my question is:
Given a line bundle $L$ of degree 0, if ...

**11**

votes

**4**answers

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

**2**answers

914 views

### K-Theory and the Stack of Vector Bundles

I have some understanding that vector bundles provide a basic, familiar example of what I should call a stack. Namely, consider the functor $Vect$ that assigns to a space $X$ the set of isomorphism ...

**10**

votes

**3**answers

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 ...

**3**

votes

**2**answers

715 views

### How to prove that w_1(E)=w_1(detE) ?

How to prove that the first Stiefel-Whitney class $w_1 (E)$ of a real rank $n$ vector bundle over a manifold M is equal to $w_1(\operatorname{det} E)$, where $\operatorname{det} E$ is the $n$-th wedge ...

**1**

vote

**2**answers

359 views

### Relative minimality for conic bundles

Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces.
The definition is ok: the fiber over ...

**21**

votes

**3**answers

1k views

### When can we cancel vector bundles from tensor products?

Let $E,F,G$ be algebraic vector bundles over $\mathbb P_{\mathbb C}^n$. My general question is:
Assume $E\otimes G \cong F\otimes G$, under what conditions can one conclude that $E\cong F$?
Some ...

**3**

votes

**6**answers

1k views

### Reference request: Moduli spaces of bundles over singular curves

I would like to know some reference (articles, books...) about any kind of moduli spaces of any of the following objects:
vector bundles
torsion-free sheaves
principal bundles
parabolic bundles
...

**3**

votes

**1**answer

412 views

### Proof of a Theorem in the paper “Construction of bundles on P^n” by Horrocks

I am trying to understand Horrocks's construction of vector bundles. However I have been stuck on the proof the first theorem in the paper.
In the paper, a trivial bundle is a direct sum of Hopf ...

**10**

votes

**1**answer

547 views

### Semiring of algebraic vector bundles on projective space

Let $K$ be a field and $n \geq 1$. Then the set of isomorphism classes of vector bundles over $\mathbb{P}^n_K$ is a semiring (i.e. almost a ring, but no additive inverses are possible). By introducing ...

**5**

votes

**2**answers

455 views

### About the Gauss map of a surface in euclidean 3 space

Regarding the sphere as complex projective line (take $(0,0,1)$ as the infinite point), the Gauss map of a smooth surface in the 3 dimensional space pulls a complex line bundle back on the surface.
...

**1**

vote

**1**answer

370 views

### Computing the dimension of the module of global holomorphic vector fields for complex projective n-space

Here by $P^n$ I mean $CP^n$, and what I want to do is to calculate the number of global sections of the holomorphic tangent bundle of $CP^n$.
If $n=1$, it is well known that $h^0(P^1, ...

**2**

votes

**1**answer

482 views

### On finite endomorphisms of $\mathbf{P}^r$

This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot ...

**5**

votes

**1**answer

371 views

### Uniqueness of Chern/Stiefel-Whitney Classes

This question is closely related to this previous question.
Chern and Stiefel-Whitney classes can be defined on bundles over arbitrary base spaces. (In Hatcher's Vector Bundles notes, he uses the ...

**15**

votes

**3**answers

1k views

### How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?

The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually always the best ...