The vector-bundles tag has no wiki summary.

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### Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...

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### Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...

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### Cohomology of Sym(E) [closed]

Let $U,V$ be two vector bundles on some scheme $X$. Let $E$ be some bundle in $0\to U\to E\to V\to 0$. If I can show that $H^i(X; Sym(U\oplus V))=0$ for $i>0$, can we conclude the same property for ...

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### a section invariant under Reeb flow

Let $M$ be a compact contact manifold with $R$ the Reeb vector field. Let $E$ be some vector bundle of gauge group $G$, say, $G = SU(2)$ and $E$ is the adjoint vector bundle.
So my question is:
If ...

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### analytic vector bundles

Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as a smooth vector bundle, is a trivial bundle.
Is $E$ a trivial analytic vector bundle?
I need to the ...

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### Possible homotopy-theoretical approach to Gauss-Bonnet

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...

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### Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...

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### Double vector bundle vs vector bundle over supermanifolds

Double vector bundle is roughly a vector bundle over (horizontal) vector bundle.
Vector bundle is a supermanifold in a nature way (non-nature on the other way around).
My question: is a double ...

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### Construction of a Hermitian metric on a locally free subsheaf which is not a subbundle

Assume $X$ is a smooth projective curve over $\mathbb{C}$ and let $\mathcal{E}$ be a locally free sheaf of rank 2 on $X$. We pick a closed point $x\in X$ and a surjection $f: \mathcal{E}\rightarrow ...

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### Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles?
Can you give me a text or an article?
For example:
Consider a vector bundle $E$ with fiber $V$ and base manifold $M$.
Consider ...

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### Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.
We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.
For ...

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### Do Hermitian metrics also split on the Riemann sphere?

Maybe this is well known, but i could not find a pointer to some literature:
Let us assume $E$ is a rank n vector bundle on the Riemann sphere $\mathbb{C}\mathbb{P}^1$. We know that ...

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223 views

### A generalized Thom Isomorphism

The ordinary Thom isomorphism says $H^{*+n}(E,E_{0}) \simeq H^{*}(X)$, where $E$ is a vector bundle over $X$ and $E_{0}$ is $E$ minus the zero section. Now assume that $S$ is a non vanishing section ...

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### Degeneracy divisor of the “trace” morphism

Let $f\colon X\to Y$ be a finite morphism of smooth curves over an alg. closed field of characteristic zero. I recently asked how methods reminiscent of basic algebraic number theory can be used to ...

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### Vector bundles on vector bundles

Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base?
In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...

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### Canonical n plane bundle over Lagrangian Grassmanian

Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets ...

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### Integration from vector bundles

Let $(E,M,p)$ be a smooth n dimensional vector bundle. Then $(TE,TM,Dp)$ is a 2n dimensional vector bundle. We restrict this bundle to $M\subset TM$. We denote this restricted bundle by $F$, as a ...

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### Are these two bundles, stably equivalent?

Let $(E,M,p)$ be a n dimensinal smooth vector bundle where $M$ is a k dimensional manifold. We assign to $M$, two different vector bundles $F_{1}$ and $F_{2}$ over $M$ as follows:
1)$TE$ is a ...

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### Extending the tautological bundle of $G(1,3)$?

It is well known that the Grassmanian of lines in $\mathbb P^3$ is isomorphic
to a quadric in $\mathbb P^5$. I would like to ask if the tautological rank two bundle on the grassmanian extends to a ...

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### Does $\mathbb P^1 \times \mathbb P^1$ admit an Ulrich bundle?

In an answer to a MathOverflow question on the following link
Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$, it is mentioned that $\mathbb P^1 \times \mathbb P^1$ has an Ulrich sheaf. However, ...

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### Connectedness of a section of an algebraic bundle

Let $X$ be a complex projective variety, $E$ be a rang $n$ bundle with $n<dim X$ and $s$ be a (holomorphic) section of $E$.
There is a relatively straightforward criterium to check if the space ...

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### Two ways to differentiate a section of vector bundle

Let $\pi:E\rightarrow M$ be a vector bundle, and $D$ a connection on it. Suppose $\sigma_1,\sigma_2\in\Gamma(E)$, $p\in M$, $V\in T_pM$ such that $\sigma_1(p)=\sigma_2(p)$. Are the following two ...

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### Subbundles of conformal blocks bundles over $\overline{\mathcal{M}}_{g,n}$

Let $\mathfrak{g}$ be a semisimple split Lie algebra over a field $k$ of characteristic zero. Let $l$ be a non-negative integer and let $P_{l}^{n}$ be the set of $n$-tuples of dominant integral ...

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### stable vector bundle and space surves

I am sure this is well known, but I am not an expert...so I appreciate any help
Let $C \subset \mathbb{P}^3$ be the complete intersection of two hypersurfaces of degree $d_1$ and $d_2$. Let ...

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### Definition of complex smooth vector bundle

Let $\xi:E \overset{\pi}{\rightarrow} B$ be a $2n$-dim. real vector bundle. Are the following two definitions for $\xi$ to be smooth and complex equivalent?
1) For every point $x \in B$ there is a ...

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### Non-equivariant vector bundles over complex projective $N$-space

From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant.
I wonder, do there exist ...

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### Restriction of a line bundle to a two-cycle

I am reading a paper on Chiral Differential Operators
http://arxiv.org/pdf/hep-th/0604179v3.pdf
and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...

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### Is this sphere bundle over SL3/SO3 trivial?

The space $SL_3(\mathbb{R})/SO_3(\mathbb{R})$ can be though of as some kind of 5-(real)dimensional generalized upper half-space.
Fix any copy of $SO_2(\mathbb{R})$ sitting inside $SO_3(\mathbb{R})$, ...

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### Formal adjoint of covariant derivative on endomorphism bundle

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$, equipped with an integrable, unitary connection $D=D'+D''$. Let $\beta\in\Lambda^{p,q}(\mathrm{End}\,E)$ be ...

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### Vector bundle connection over complex manifold vs. over underlying real manifold

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$ of rank $r$. Denote by $(X^{\mathbb{R}},g^{\mathbb{R}})$ the underlying Riemannian manifold of $(X,g)$.
...

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### A question on parallelizable manifolds [closed]

Let $M$ be a manifold with the property that $f^{*}(TM)$ is isomorphic to TM, for every diffeomorphism $f$ on $M$. Does this imply that $M$ is parallelizable?

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### A question on long line

Assume that $M$ is the long line. Is $TM$, the tangent bundle, isomorphic to $T^{*}(M)$, the cotangent bundle?

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### top chern class

Let $E\to M$ be a holomorphic vector bundle ($M$ is compact) and assume $s\colon M\to E$ is a non-trivial section transverse to the zero section.
Is it be possible that $s^{-1}(0)\neq \emptyset$, ...

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### Orthonormal frame bundle orthogonal to a curve

This is a duplicate of this question on math.stackexchange, since I got there not a single answer.
Let $M$ be a $n$-dimensional smooth riemannian manifold and ...

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### How can we show that a transverse section exist

Let $c_i,d_j <n$, be a set of integers and define
$$ M=\prod Gr(c_i,n),\quad N=\prod Gr(d_j,n),$$
where $Gr(k,n)$ is the grassmannian of k-planes in $\mathbb{C}^n$.
Let $E_M=\oplus E_i^*$, where ...

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### Totally non parallelizable manifold

Does there exist a manifold M which all iterated tangent bundles are non parallelizable manifolds? That is$ M, TM , T^2(M), \ldots ,T^n(M)\ldots$ are non parallelizable manifold?
What is ...

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### How to show the Euler Characteristic is equal to self-intersection number of zero-section [duplicate]

myThe definition of the Euler characteristic (given in Guillemin and Pollack's "Differential Topology") of a compact oriented manifold $X$ is the self-intersection number of the diagonal $\Delta$ in ...

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### Unicity of a vector field on $S^1$-bundle

Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( ...

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### Do there exist equivariant sheafs that are not equivariant vector bundles?

For $F \subset G$ two algebraic groups, consider a homogeneous space $H$ of the form $G/F$. Now every vector bundle over $H$ is a coherent sheaf, but the converse is not true. What happens in the ...

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### Spaces over which every vector bundle is a summand of the trivial bundle

Let X be a Hausdorff space such that every real vector bundle on X is summand of a trivial bundle. Does this imply that X is homotopy equivalent to a compact Hausdorf space? This question ...

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### Connections on the Hodge bundle?

Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of ...

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### Global sections of determinant bundle of symmetric powers

Let $E$ be a globally generated vector bundle of rank $r$ on a normal irreducible projective variety $X$. Suppose that $E$ induces a finite map
$$X \to \mathbb{G}r (H^0(E), r)$$
to the Grassmannian of ...

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### On the definition of convergence of a sequence of sections of a bundle

Convergence of a sequence of sections of a bundle is defined as follows:
Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on ...

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### chern classes of push-pulled vector bundles

Let $f:X\to Y$ be a finite cover of smooth algebraic varieties, branched along a divisor $R\subset Y$. Let $E$ be a vector bundle on $Y$. What is the relation between the chern classes of $E$ and the ...

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### Vector bundles on a weighted projective stack

Put $X := \mathbb A^{n+1}\!-\lbrace0\rbrace$. Let $G=\mathbb C^*$ act on $X$ with (positive) weights $w_0,\dots,w_n$. The quotient stack $[X/G]$ is called the weighted projective stack.
Each vector ...

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### Push-forward of a nef bundle

Let $f:X\rightarrow Y$ be a finite morphism between normal varieties. Let $E$ be a vector bundle on $X$ and let us consider its pushforwad $f_{*}E$.
Does anyone know an example where $E$ is nef but ...

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### How to compute the index of such operator?

Let $M$ be a compact Riemannian manifold, with $R$ nowhere-vanishing vector field on $M$(whose orbit may be closed/ not closed). $E$ and $F$ are two vector bundle (Edit: which are sub-bundles of ...

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### Is the moduli space of stable vector bundles over a smooth projective curve fano?

Let $K$ be a field of characteristic zero but not algebraically closed. Let $C$ be a smooth projective curve over $K$. Let $r, d$ be two positive integers that are coprime. Consider the moduli space ...

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### Stiefel classes and generic sections

I asked this question in math.stackexchange few days ago.
Unfortunately, I haven't seen any simple answer.
One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly ...

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### Existence of connections making a bundle endomorphism parallel

Let $M$ be a manifold, $TM$ its tangent bundle, and $N:TM\to TM$ a vector bundle morphism. It is possible to find a torsionless linear connection $\nabla$ on $TM$ such that $\nabla N=0$?