For questions on vector bundles.

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1answer
28 views

Rank $n$ vector bundle with $n$ pointwise linearly independent sections is trivial

I have an $n$-dimensional vector bundle $E \to X$ and sections $s_1, \dots, s_n : X \to E$ such that for any $x \in X$, the elements $s_1(x), \dots, s_n(x)$ are linearly independent in the vector ...
3
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1answer
39 views

Symbol of the differential operator on vector bundles

Suppose that we have a differential operator $D:C^{\infty}(\mathbb{R}^n) \to C^{\infty}(\mathbb{R}^n)$ of the form $(Df)(x)=\sum_{|\alpha| \leq k}a_{\alpha}(x)\frac{\partial^{|\alpha|}f}{\partial ...
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2answers
68 views

Triviality of a vector bundle is an open condition

Is the following statement true (and if not, are there additional assumptions that make it true?) A vector bundle on a variety which is trivial if restricted to a closed subvariety is trivial on ...
4
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1answer
50 views
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0answers
7 views

Defining an Ehresmann Connection Using Linear Connection

In some places that I have seen, given the covariant derivative $\nabla$ of a linear connection on a vector bundle $\pi: E\rightarrow M$ (dim(M)=k), an Ehresmann connection is defined in the following ...
0
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1answer
38 views

injective morphism between line bundles on curves

Let $X$ be a smooth projective, irreducible, curve, $\mathcal{L}$ be an invertible sheaf on $X$ and $\mathcal{L}' \subset \mathcal{L}$, an invertible subsheaf. Is $\deg(\mathcal{L}') \le ...
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0answers
11 views

Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
3
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1answer
47 views

Transformation behavior of connection on vector bundle.

Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be ...
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4answers
72 views

commuting property of connections and bundle homomorphisms

I have the following situation: $E,F$ are (smooth) vector bundles over a smooth manifold $M$. Assume we are given connections $\nabla^E,\nabla^F$ on $E,F$ and a homomorphism of $M$-bundles ...
4
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1answer
50 views

Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
3
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0answers
28 views

Module of vector fields reflexive.

While studying tensor fields in classical differential geometry, I found it crucial (at least for some approach) to know that for a smooth $n$-manifold $M$, its $C^{\infty}(M)$-module $\Gamma(M,T_M)$ ...
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0answers
18 views

Classification of rank $\geq 2$vector bundles over Grassmanians

Are there classification results of higher rank (complex) vector bundles over (complex) Grassmanian manifolds? For example, we know that line bundles are in correspondence with the $H^2(G)$, the ...
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0answers
10 views

If $\xi$ is a vector bundle and $f^{*}(\xi)$ is the bundle induced by $f$ then $f^{*}(\xi)$ is orientable.

Let $\xi = \pi : E \rightarrow X$ be a vector bundle and $f: Y \rightarrow X$ a continuous map. Let $E' \subset Y \times E$ be the set of all $(y,e)$ with $f(y) = \pi(e)$. We can define a vector ...
2
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1answer
29 views

Is there a natural Dolbeault operator on a almost holomorphic vector bundle?

For vector bundles $(\pi: V \rightarrow M )$ over a complex manifold, there is a notion of holomorphicity that can be defined in two equivalent ways : $V$ is a complex manifolds and $\pi:V ...
0
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1answer
24 views

Prove that $s: B \rightarrow E$ is a section ( vector bundles)

I'm very very unfamiliar with vector bundles, so maybe this question is quite trivial. Let $\pi : E \rightarrow B$ be a vector bundle and $s: B \rightarrow E$ a map sending each $p \in B$ to the $0$ ...
1
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1answer
19 views

construction of linear independent local section in vector bundle [duplicate]

Let $X_1,\ldots,X_k$ be lineary independent local sections of a vector bunlde $V$, $rank(V)>k$, defined on some open set $U$. Can I construct a local section $X_{k+1}$ on a (maybe smaller) open ...
1
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0answers
55 views

Proof of the formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
3
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1answer
60 views

Explaining the definition of vector bundles

Recall the definition of a vector bundle: Let $M$ be a topological space. A $k$-dimensional vector bundle over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E \to ...
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1answer
30 views
0
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1answer
31 views

Why are compact complex manifolds Liouville?

I know this is true but strangely can't find references. Also, consider the trivial $n$-bundle over any connected compact manifold, does Liouville imply that all holomorphic sections are constant? ...
0
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1answer
44 views

$\bigwedge^k T^*M$ is a $\binom{n}{m}$-dimensional Subbundle of $\bigotimes^k T^*M$.

I am trying to prove the following: Let $M$ be a smooth manifold. Then $\bigwedge^k T^*M$ is a smooth subbundle of dimension $\binom{n}{k}$ of $\bigotimes^kT^*M$. To do this, I think the ...
5
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1answer
76 views

Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism.

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
2
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1answer
106 views

Graded projective modules and vector bundles on projective varieties

Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}(S)$. Let $E$ be a vector bundle over $S$. Is $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ a graded ...
0
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1answer
32 views

(Co)Tangent bundle of Cone manifold

Given a Riemannian manifold $(M,\bar{g})$, we can construct the Riemannian cone manifold $(C(M), g )$ as follows. Topologically, $C(M)$ is $M \times \mathbb{R}_{>0}$. We equip this with the ...
2
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2answers
105 views

On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let $Y$ be a submanifold of $M$ and let $(-)^0$ denote the ...
3
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1answer
60 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
0
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2answers
56 views

Endomorphisms in an exact sequence of vector bundles

Let X be a smooth projective variety over $\mathbb{C}$. And suppose we have an exact sequence of vector bundles over $X$. $\qquad\qquad\qquad\qquad\qquad 0\longrightarrow A\longrightarrow ...
2
votes
1answer
26 views

Exact sequence of vector bundles

I'am working on a shorter proof of a theorem but to manage it I need to know if a lemma is true. Conjecture: Given a manifold $M$ and an short exact sequence of vector bundles $$ 0 \rightarrow E' ...
4
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1answer
67 views

When do the Nakano identities hold?

In "Complex Geometry" by Huybrechts, he states the following version of the Nakano identity on page $240$: Let $X$ be a Kähler manifold and $(E,h)$ a holomorphic hermitian vector bundle on $X$. ...
0
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1answer
19 views

Obstruction to the splitness of an exact sequence of holomorphic vector bundles

This question was asked here, but I think there will not be an answer, so let me recopy the question on Mathoverflow. In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma ...
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0answers
24 views

Obstruction to the splitness of an exact sequence of holomorphic vector bundles

In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma (3.A.3) on the page 153, section Royden's extension lemma, whose statement is Every exact sequence of holomorphic vector ...
2
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0answers
29 views

Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
3
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1answer
40 views

Is the total space of a vector bundle over an irreducible scheme irreducible?

Let $X$ be an irreducible scheme over $\mathbb{C}$ and let $F$ be a locally free sheaf of rank $r$ on $X$. Is the total space $Y$ of the associated vector bundle to $F$, $Y=Spec(Sym(F^{\vee}))$, ...
1
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1answer
238 views

Is a sub-bundle of a vector bundle a vector bundle?

Could anyone please help me with this question? (1) Let $(E, p, B)$ be a vector bundle where $E$ is the total space, $B$ is the base, and $p$ is the structure map, that is, $p:E\to B$. Now suppose ...
3
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1answer
32 views

Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don't really ...
3
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1answer
36 views

Vanishing of the first Chern class of a complex vector bundle

Suppose that $E\to M$ is a $\mathbb{C}^n$-bundle with a metric. This is equivalent to saying that there exists a chart $\{U_\alpha\}$ of $M$ and $\phi_{\alpha,\beta} \colon U_\alpha\cap U_\beta\to ...
3
votes
1answer
53 views

Metric on Homogeneous Space $G/H$

For simplicity, assume $G$ is compact and semi-simple Lie group, and $H$ is a closed subgroup of $G$. Therefore the homogeneous space is reductvie, say $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ where ...
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0answers
31 views

spin structures definition

One way to define a spin structure on a $SO(n)$-bundle $p:E\to B$ is to require that there is an element $\sigma\in H^1(E;Z_2)$ such that restricted to each fiber gives a generator of ...
1
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0answers
15 views

Tangent Bundle of Product Manifold

Suppose $M,N$ are manifolds, and consider the product $M\times N$. From this answer, I know that: $T_{(m,n)}(M \times N) \cong T_m M \oplus T_n N $ Can we conclude that $T(M\times N) \cong T(M) ...
2
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0answers
26 views

Reduction to the special orthogonal group

It is well known that an $SL_n$-bundle $E$ on an algebraic curve $X$ is self dual (i.e $E\cong E^*$) iff it is an $SO_n$-bundle However, I can't see why, because the isomorphism $E\cong E^*$ means ...
2
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1answer
30 views

Applying $\bar \partial$ to a tensor product of holomorphic vector bundles bundles

Reading Griffiths and Harris, it is stated as an "obvious" fact that for a tensor product $E \otimes E'$ of holomorphic vector bundles, $(D_E \otimes 1 + 1 \otimes D_{E'})'' = \bar \partial$ where ...
3
votes
1answer
157 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
4
votes
1answer
48 views

To Reconcile Two Different Descriptions of the Dual Bundle

$\newcommand{\mc}{\mathcal}$ Let $\pi:E\to M$ be a smooth vector bundle with typical fibre a $k$-dimensional vector space $\mc V$. There are (at least) two ways to construct the dual bundle of $E$. ...
3
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2answers
79 views

triviality of tensor product of vector bundles

Let $\xi$ be a $O(n)$-bundle with fibre $\mathbb{R}^n$. Let $\xi\otimes \mathbb{C}$, $\xi\otimes \mathbb{H}$ be complex vector bundles and quaternionic vector bundles. If $\xi$ is not a trivial ...
3
votes
1answer
23 views

Vector fields spanning the tangent bundle at each point

Let $M$ be a smooth manifold of dimension $n.$ It is well known that it is not allways possible to find such global vector fields $X_1,\ldots, X_n$ which would be a basis for the tangent space at each ...
2
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0answers
23 views

Quotient space of equivariant vector bundle.

Let $p: P \to B$ be a principal $G$-bundle, and $\pi : E \to P$ a vector bundle with action of $G$ on $E$ such that $G$ acts by vector bundle isomorphisms and $\pi$ is equivariant. Is it always the ...
1
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1answer
28 views

Trivialization of vector bundles

Is it true that any $R^n$-bundle over a space (say a simplicial space) of dimension $k<n$ is trivial? It seems to me any $R^n$-bundle ,for $n>1$, over $S^1$ is trivial. But I cannot figure out ...
2
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0answers
19 views

The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\mc}{\mathcal} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\pr}{pr} \newcommand{\at}{\big|} \DeclareMathOperator{\GL}{GL}$ Let $\pi:E\to N$ be a smooth vector bundle over a smooth ...
0
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0answers
30 views

Rank 2 vector bundle V and its k$_V$

This question is from the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle ,Ex4.6 Suppose o$\le$a$\le$b$\le$c are three integers, show that there exist vector bundle V which fit ...
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1answer
25 views

Relating holomorphic sections of a line bundle to holomorphic functions on the line bundle

I am in the following situation: $X$ is a complex manifold and $L$ a holomorphically trivial line bundle over $X$. My objective is to understand the space of holomorphic functions $\mathcal{O}(L)$ on ...