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7
votes
2answers
659 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
0answers
293 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
0answers
373 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
4answers
2k views

Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
14
votes
2answers
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
1answer
681 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
3answers
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
1answer
460 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
3answers
784 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
3answers
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
2answers
725 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
2answers
765 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
2answers
887 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
1answer
422 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
1answer
468 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
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 ...
11
votes
4answers
578 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
772 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
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
505 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
583 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
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
0answers
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
1answer
798 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
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
0answers
140 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
616 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
810 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
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
1answer
274 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
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
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
472 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
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
2answers
665 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
523 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
985 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
791 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 ...
5
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
669 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
770 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
791 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 ...