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3
votes
3answers
232 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 ...
1
vote
1answer
134 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
125 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
58 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
92 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
175 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
157 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
116 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
91 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
312 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
128 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
248 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
90 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
151 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
343 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
234 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
246 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
169 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
201 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
126 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
212 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
306 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
160 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
186 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
912 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
111 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
256 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
129 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
350 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
147 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
157 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 ...
3
votes
2answers
188 views

Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle

I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up. Let $X$ be a ...
5
votes
3answers
469 views

Hartshorne-Serre's correspondence in higher codimension

There's a well-known correspondence (traditionally called Hartshorne-Serre) between codimension 2 smooth subvarieties $S\subset X$ of a smooth algebraic variety $X$ and certain rank two vector bundles ...
1
vote
1answer
310 views

Computing relative Lie algebra cohomology (as appears in Borel-Weil-Bott theorem)

Suppose $G$ is a complex Lie group, $P$ a Borel subgroup, $E$ a representation of $P$ that induces a vector bundle ${\cal E}$ over $G/P$. The general version of Borel-Weil-Bott theorem, as stated in ...
0
votes
1answer
175 views

Connections on tangent bundles and double tangent bundles

This can be viewed as a sequel to my previous question on double tangent bundle. Where I learned that the double tangent bundle $TTM$ is not natural diffeomorphic to $\oplus^3 TM$. Recently, I also ...
5
votes
1answer
202 views

Looking for a special rank 2 vector bundle

Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$. By Riemann-Roch theorem, $$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$ Question: For which $g$, there is such ...
3
votes
2answers
100 views

A question on a projective bundle $\mathbb{P}(L\oplus \mathcal{O}_X)$

Let $X$ be a complex manifold and $L$ a line bundle on it. Define $Y:=\mathbb{P}(L\oplus \mathcal{O}_X)$ be the projective bundle over $X$. Here is a statement I don't understand: The summands $L$ ...
0
votes
1answer
116 views

A version of implicit function theorem when sections are not everywhere smooth?

Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$ a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section ...
1
vote
0answers
141 views

Can one always extend a smooth section defined on a non compact submanifold to the whole manifold, provided it extends continuously to the closure?

Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$ (without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed. Suppose $s: X ...
0
votes
2answers
398 views

Recommended books/lecture notes for vector bundle on algebraic curve

I am going to enroll in a ceminar with the topic "vector bundle on algebraic curve". Except Algebraic Geometry(which I think GTM 52 by Hartshone is the main source), which topic I should prepare in ...
1
vote
2answers
203 views

Deforming to decompose vector bundles

After edit: How do we show that for every (holomorphic) vector bundle over a curve, is it possible to deform it to another one which is decomposable (into line bundles)? Before edit: I am not sure ...
1
vote
1answer
125 views

Non-(stable)-triviality of the tautological bundles

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/396217/7110/ The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold ...
2
votes
2answers
173 views

Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles?

As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 ...