For questions on vector bundles.

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3
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1answer
51 views

Is $T\mathbb{C}\mathbb{P}^n$ globally generated?

A vector bundle $E\to X$ is globally generated if there exists global holomorphic sections $s_1,\dots,s_n$ such that $E_x$ is spanned by $s_1(x),\dots,s_n(x)$ for all $x\in X$. Consider the ...
2
votes
1answer
32 views

Under what type of transformations are characteristic classes and characteristic numbers of a manfold invariant?

When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they ...
2
votes
1answer
20 views

What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers?

I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that ...
3
votes
2answers
71 views

Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
3
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0answers
48 views

Motivation for equivalence of Tautological Line Bundle and ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$

I currently reading a book on Complex Algebraic Geometry and both the Tautological Line Bundle $L \to \mathbb{P}_k^n$ and the blow up ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$ are defined set ...
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0answers
65 views
+50

Bundle isomorphism $\Phi : TM \oplus T^*M \to T(T^*M)$.

Prove that there is a bundle isomorphism $\Phi : TM \oplus T^*M \to T(T^*M)$ which identifies the summand $T^*M$ with the vertical vectors. If $\omega_{can}$ is the canonical symplectic ...
5
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0answers
64 views

Definition of cotangent and conormal bundle

I have read the following definition of cotangent bundle: Let $X$ be a $n$-dimensional smooth algebraic variety. For any $p\in X$ there exist a neighbourhood $U_{p}\subseteq X$ and functions (...
0
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1answer
50 views

Pullback of global sections?

Let $X$, $Y$ be abelian varieties and let $f:X\to Y$ be a morphism. They told me that we can define the pullback $f^*s$ of a global section $s\in\Gamma (L)$ where $L$ is an ample line bundle on $Y$, ...
0
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0answers
22 views

Global holomorphic vector field on a two-sphere

I'm sure this question has been asked before... Till today, I thought that one cannot define a global holomorphic vector field on a two-sphere due to the hairy ball theorem. However, here's an ...
1
vote
1answer
33 views

Degree of filtered vector bundle

Suppose I have the sheaf $\mathscr{M}$ defined by $$0\to \mathscr{M}\to \mathscr{O}_{\mathbf{P}^r}^{r+1}\to \mathscr{O}_{\mathbf{P}^r}(1)\to 0 $$ that is, $\mathscr{M}\simeq \Omega_{\mathbf{P}^r}^1(1)...
3
votes
1answer
40 views

Cohomology class of zero set of a section

Say we have a rank $r$ smooth vector bundle $E\to X$ and a smooth section $s:X\to E$ of it. Shortly after the 30 min mark in this video Joe Harris defines the $r$th Chern class of this bundle to be ...
1
vote
1answer
37 views

Vector subbundle and frame field relation

Question: Let $E \to M $ be a vector bundle of rank $k$. Suppose that for each $p \in M $ we are given a subspace $E'_p$ of $E_p$ and consider the set $\displaystyle E' = \bigcup_{p \in M} E'_p $....
2
votes
1answer
85 views

Cohomologies with line bundle vs. coherent coefficients

I recently learned in a lecture that the derived category of a smooth variety is generated/spanned by (complexes of) locally free sheaves. (Unfortunately I haven't been able to find a more precise ...
0
votes
0answers
67 views

How to show $Hom(V,V)\rightarrow Hom(V_x,V_x)$ is injective, V being semi-stable

Let $V$ be a semi-stable vector bundle over a smooth irreducible projective curve of genus $g\geq 2$. Let $x\in X$. How do we show that the canonical map $Hom(V,V)\rightarrow Hom(V_x,V_x)$ which ...
13
votes
2answers
987 views

Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
0
votes
1answer
56 views

Relation between projective equivalence and linear equivalence of divisors

For the whole question I'll be working in $\mathbb{P}^n_{\mathbb{C}}$ and assume that everything is smooth. We say that two sets $U,V\subseteq \mathbb{P}^n$ are projectively equivalent if there ...
4
votes
1answer
71 views

Computing Curvature of a Connection (Dirac Monopole)

I'm trying to verify some computations in a paper I'm reading and am feeling a little lost. In particular I haven't been able to properly compute curvature of a connection acting on a line bundle. ...
3
votes
1answer
90 views

Splitness of quotient sequence

Let $A, B, C$ be holomorphic vector bundles over some complex manifold $X$. Let $A', B', C'$ be sub bundles, respectively. Suppose that we have short exact sequences: $$0 \rightarrow A \rightarrow B \...
1
vote
1answer
62 views

The non-existense of the fine moduli scheme of vector bundles. Why?

The reference I am using is this enter link description here. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a ...
3
votes
1answer
88 views

Show that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$

Let $M, N$ be compact manifolds and $\Omega^*$ its algebra exterior. How to prove that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$? I thought about the function $f(\omega,...
1
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0answers
52 views

Chern classes of a double cover

Let $X$ be a compact complex surface and let $D$ be a double cover of $X$. Let $\pi:D\to X$ be the double cover map (a 2:1) map. If $E$ is a vector bundle (rank at least 2) on $X$ with $c_1(E) = A$ ...
1
vote
1answer
50 views

Are there characteristic classes for symplectic vector bundles?

Given a real vector bundle, say the tangent bundle of a manifold, some obstructions to this being the underlying real bundle of a complex bundle come from characteristic classes. These can prove the ...
1
vote
1answer
70 views

Whitney sum bundle vs. direct product bundle [closed]

The question is simple: are Whitney sum bundle and direct product bundle the same? When are they different? PS: I have realized my mistake: let $E_1 \to B$ and $E_2 \to B$ be two bundles, their ...
0
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0answers
37 views

What does “factoring out an (group) action $\tau$ of a group $G$ acting on some set $E$” mean?

I am reading a survey article where they define the following objects: $\Gamma:=\mathbb{Z}^{n}$ seen as a group of translations. $\mathbb{T}:=\mathbb{R}^{n}/\Gamma$ is the $n$-dimensional ...
3
votes
1answer
43 views

Isomorphic modules of sections imply isomorphic bundles

For reference this is about a part of question 3-F in Characteristic Classes by Milnor and Stasheff, which discusses the $C(B)$-module structure of the space of sections of a topological vector bundle ...
5
votes
1answer
65 views

Short Exact Sequence of Vector Bundles

Just wish to clarify, is it true that in order to show some vector bundles (over the same space) fit into a short exact sequence we just need to check that their fibers fit into a short exact sequence ...
0
votes
1answer
21 views

On the zero in the fibre of a vector bundle.

Let $X$ be a differentiable manifold, let $\{U_i \mid i\in I\}$ be an open cover of $X$, let $\{g_{ij}:U_i\cap U_j\to\mathbb{R}^r\}$ be a set of differentiable maps satisfying the cocycle condition, ...
0
votes
0answers
56 views

Roots of canonical line bundles that are not necessarily square roots

I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root. But what about further roots? ...
0
votes
2answers
64 views

Why does a tangent bundle have dimension 2n instead of n?

Let $n=dim(T_pM)$ for every $p\in M$, where $M$ is a smooth manifold. I understand that specifying $p$ is not enough to determine an element of $TM$, but what if do we specify only $v\in T_pM\subset ...
1
vote
1answer
19 views

Extending a section to a basis of sections in a trivial bundle.

Suppose we have a manifold $M$ of dimension $m$ and let $E$ be the rank $n$ trivial vector bundle on $M$. Let $\Gamma$ be a section of the trivial bundle $E$. My question is, are there some conditions ...
2
votes
1answer
51 views

Exact sequences of bundles and orientations

If we have an exact sequence of finite-dimensional vector spaces $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow0$$ then an orientation of any two induces an orientation of the third. I have ...
0
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0answers
19 views

Linear connection in a vector bundle in terms of the vertical projection

Let $\pi:E \rightarrow M$ a vector bundle. Can we define a linear connection as a connection $\Phi \in \Omega^1(E,VE)$ such that the horizontal projection $p_H = \text{id}_{TE} - \Phi$ is a vector ...
0
votes
2answers
55 views

Is multiplication by a Stiefel-Whitney class an injective map?

I have a doubt: In cohomology, when you multiply by a Stiefel-Whitney class is it always an injective map? For example: is $$H^{j-1}(X)\xrightarrow{\smile\ w_1}H^{j}(X)$$ always injective? Thanks!
3
votes
2answers
103 views

Kernel of $\omega^\#$ is $k$-dimensional

Let $M$ be a smooth manifold with coordinates $\{q^i\}_{i=1}^n$ .The variables $(q^i,p_i)$ are coordinates on the cotangent space $\Omega=T^*M$. Any cotangent space carries a natural one-form $\tilde{\...
1
vote
0answers
28 views

Proving that every tangent bundle is direct summand of a trivial bundle.

I am trying to prove that there exists a formal immersion from a manifold $M$ of dim $m$ into $\mathbb{R}^{2m}$. Formal immersion is just an injective bundle map from $TM$ into $T\mathbb{R}^{2m}$ that ...
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0answers
15 views

Definition of isomorphism classes of vector bundles with reduced structure group

I'm looking for a definition of the notion of isomorphism between vector bundles $E$ and $F$, over the same base $X$, whose structure groups have been reduced from $GL(n)$ to some subgroup $G\subseteq ...
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0answers
39 views

Global sections of dual of the universal bundle on the Grassmanian

Let $G=G(k, V)$ be the Grassmanian of $k$-dimensional subspaces of the $n$th dimensional vector space $V$, regarded as a smooth algebraic variety over $\mathbb{C}$. Denote with $S$ the tautological (...
0
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0answers
16 views

Isotypic component of a vector bundle

Suppose that $E \to B$ is a vector bundle with fibre $F$. Let $G$ be a finite group and assume that $F$ has the structure of a $G$-module. Let $\chi$ be another $G$-module. I have come across a ...
2
votes
0answers
23 views

When can a vector field be “lifted” to a spinor field with preservation of continuity?

Suppose we are given a vector field $\xi ^a (x)$ on some region of Minkowski spacetime which is null everywhere, $$\xi^a(x) \xi_a (x) =0.$$ For every point of our region we can choose a spinor $\...
0
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0answers
6 views

Using partition of unit to extend paths on vector bundles in time dependent sections.

Let $\pi: A\longrightarrow M$ be a vector bundle and $a: I\longrightarrow A$ be a path where $I=[0, 1]$. I would like to show there are time dependent section $\alpha: I\times M\longrightarrow A$ ...
2
votes
1answer
52 views

Understanding algebraic operations on vector bundles

Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$. In particular what manifold structure do we give ...
3
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0answers
29 views

$E$-orientation of a closed manifold induce $E$-orientation of normal bundle: passage in the proof.

I'm trying to follow Kochman's proof of the well-known result For a ring spectrum $E$, and closed manifold $M^n$ together with an embedding in $\mathbb{R}^{n+k}$, the following are equivalent: ...
2
votes
1answer
36 views

Is a bundle morphism which restricts to homeomorphisms of the fibers a bundle isomorphism?

If $f$ is such a map between total spaces (assume a common base space) then it is a bijective and continuous and the inverse will be fiber preserving so that all we would need to prove is that the ...
2
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0answers
33 views

Cylinder as trivial fiber bundle with fiber $S^1$?

In prepartion for the example of a Mobius strip, a cylinder is often taken as a first example of a trivial fiber bundle with fiber $I$ and total space $S^1\times I$. However, it seems to me that we ...
0
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0answers
26 views

Showing that Killing vector fields form a vector space without introducing connection

I'm trying to show that the sum of two killing vector fields is again a vector field without introducing a connection and just using the smooth structure on a manifold. Let $X,Y$ be Killing vector ...
1
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0answers
48 views

Fundamental class for tangent bundle

For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
2
votes
1answer
33 views

Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle?

Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)? \begin{equation} End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} \text{Mat}_n(\...
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0answers
35 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
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0answers
23 views

Normal vector field for an immersion

I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ...
2
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0answers
53 views

Exterior derivative for functions with values in a parallelizable manifold

In Sharpe's text on Cartan geometry, he explains in section 1.5 on page 52 how to define an exterior derivative for maps into a parallelizable manifold $N$. Let $f: M \to N$ be a smooth map, and $\...