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0
votes
1answer
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
2answers
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
4answers
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
5answers
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
5answers
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
3answers
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
1answer
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
3answers
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
1answer
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
2answers
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
0answers
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
1answer
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
2answers
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
0answers
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
4answers
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
8answers
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
2answers
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
0answers
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
1answer
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
2answers
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
2answers
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
2answers
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
5answers
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
3answers
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
2answers
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
1answer
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
2answers
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
5answers
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
2answers
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
3answers
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
1answer
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
5answers
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
3answers
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
1answer
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
1answer
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
2answers
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
4answers
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
2answers
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
3answers
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
2answers
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
2answers
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
3answers
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
6answers
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
3answers
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 ...