The tag has no wiki summary.

learn more… | top users | synonyms

9
votes
1answer
397 views

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 ...
-3
votes
1answer
269 views

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 ...
1
vote
1answer
81 views

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 ...
1
vote
2answers
237 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 ...
1
vote
0answers
91 views

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 ...
11
votes
2answers
612 views

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 ...
3
votes
2answers
178 views

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 ...
1
vote
1answer
78 views

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 ...
1
vote
1answer
82 views

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 ...
1
vote
2answers
99 views

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 ...
1
vote
1answer
183 views

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, ...
3
votes
2answers
177 views

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 ...
3
votes
3answers
233 views

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 ...
2
votes
1answer
187 views

Frobenius rank of a manifold

The rank of an smooth manifold M is defined by Milnor, as follows: "The maximum number of independent commuting vector fields on M" For example it is well known that the rank of $S^{3}$ is 1 (Lima, ...
2
votes
1answer
126 views

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 ...
0
votes
0answers
59 views

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 ...
1
vote
1answer
93 views

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 ...
-1
votes
2answers
172 views

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 ...
2
votes
1answer
176 views

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 $SO_3(\mathbb{R})$, ...
2
votes
0answers
57 views

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 ...
3
votes
0answers
159 views

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 $(X,g)$. ...
-2
votes
1answer
117 views

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?
1
vote
1answer
93 views

A question on long line

Assume that $M$ is the long line. Is $TM$, the tangent bundle, isomorphic to $T^{*}(M)$, the cotangent bundle?
3
votes
1answer
315 views

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 $s^{-1}(0)\neq \emptyset$, ...
3
votes
0answers
120 views

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 ...
1
vote
1answer
129 views

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 ...
3
votes
1answer
249 views

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 ...
1
vote
0answers
96 views

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 ...
1
vote
1answer
91 views

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( ...
1
vote
1answer
153 views

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 ...
5
votes
1answer
348 views

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 ...
2
votes
2answers
245 views

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 ...
6
votes
1answer
248 views

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 ...
-1
votes
2answers
172 views

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 ...
3
votes
2answers
210 views

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 ...
2
votes
1answer
129 views

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 ...
0
votes
2answers
219 views

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 ...
2
votes
1answer
191 views

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 ...
1
vote
2answers
307 views

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 ...
2
votes
1answer
162 views

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 ...
4
votes
2answers
189 views

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 $\nabla N=0$?
2
votes
2answers
162 views

Preimage of $1 \in H^n(M^n)$ under Chern character

Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how ...
25
votes
2answers
923 views

vector bundle trivial over every compact subset, then it is globally trivial

Let $X$ be a non-compact metric space (though if the answer to the question is positive, then it probably also holds for more general spaces like, e.g., paracompact Hausdorff) and $E \to X$ a vector ...
0
votes
0answers
113 views

Mumford's vector bundle stability equivalent the notion orbit stability for a G-space?

Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class ...
8
votes
3answers
261 views

Duality relations for Lebesgue spaces of sections of vector bundles

Suppose $X$ is a topological space, and $\mu$ is a Borel measure on $X$. Also suppose we have an $n$-dimensional vector bundle $E \to X$, with an inner product $\langle \cdot,\cdot \rangle_x$ on the ...
2
votes
0answers
133 views

A natural tower of bundles over Grassmannian manifolds

There is a well-known exact sequence of vector bundles $$ 0\to U\to T\to N\to 0 $$ over a Grassmannian manifold $Gr(V,n)$: the fibers over $L\in Gr(V,n)$ are, respectively, $L$ itself, the whole $V$ ...
8
votes
1answer
356 views

Does there always exist a line bundle whose Chern class represents an integer symplectic form?

Let $(M, \omega, J)$ be a compact symplectic manifold with a compatible almost complex structure $J$, such that the symplectic form determines an integer cohomology class, ie $$ [\omega] \in H^2(M, ...
1
vote
0answers
150 views

extending a vector bundle

given a vector bundle $V \rightarrow N$ over a manifold $N$ and let's assume $N \hookrightarrow M$ is embedded into a manifold $M$ is there a way to extend $V$ to a bundle over $M$, i.e. is there a ...
5
votes
0answers
160 views

Does the suspension isomorphism $K_1(A) \to K_0(SA)$ descend from a more refined invariant?

If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension ...
1
vote
0answers
54 views

Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties: $$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...