The vector-bundles tag has no wiki summary.

**18**

votes

**2**answers

902 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

**1**answer

571 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

**3**answers

581 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

**0**answers

293 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**

votes

**5**answers

828 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**

votes

**1**answer

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

**3**answers

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**

votes

**1**answer

995 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

**0**answers

431 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

**2**answers

664 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

**0**answers

300 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**

votes

**0**answers

374 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 ...

**54**

votes

**4**answers

2k views

### Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?

**14**

votes

**2**answers

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

**1**answer

686 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 ...

**20**

votes

**3**answers

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

**1**answer

462 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 ...

**10**

votes

**3**answers

791 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 ...

**1**

vote

**3**answers

590 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 ...

**1**

vote

**2**answers

729 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, ...

**5**

votes

**2**answers

766 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**

votes

**2**answers

897 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 ...

**1**

vote

**1**answer

433 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 ...

**0**

votes

**1**answer

471 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 ...

**7**

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 ...

**11**

votes

**4**answers

579 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

782 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 ...

**8**

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 ...

**5**

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 ...

**6**

votes

**1**answer

561 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

590 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 ...

**10**

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

342 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

655 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

814 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

258 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

141 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

617 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

815 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

397 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

275 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

382 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

2k 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

476 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

221 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

667 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 ...