For questions on vector bundles.

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9
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0answers
213 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
8
votes
0answers
85 views

What's Griffiths' ampleness in modern language?

When reading Griffiths' paper "Hermitian differential geometry, Chern classes and positive vector bundles", I found his definition of ampleness: Let $X$ be a compact complex manifold, $E\to X$ a ...
6
votes
0answers
154 views

Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
6
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0answers
349 views

understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
6
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0answers
366 views

Orientability of the total space of a vector bundle over an oriented manifold

Let $M$ be a (smooth) manifold of dimension $n$, and let $\pi : E \to M$ be a (smooth) vector bundle of rank $r$. If I choose a connection on $E$, then I obtain a decomposition of $T E$ as $V E ...
5
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59 views

Is there a natural ring structure on $\operatorname{Pic}(\mathbb{CP}^1)$?

The set of isomorphism classes of holomorphic line bundles on a complex manifold $X$ is a group under tensor product. This group is called the Picard group and is denoted $\operatorname{Pic}(X)$. We ...
5
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0answers
110 views

Not taking holomorphic bundles for granted

When we say something to the effect of, "Consider a holomorphic bundle $V$ on a complex manifold $X$...", we are saying that the transition functions of $V$ are holomorphic functions with respect to ...
5
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0answers
72 views

Determining an explicit line bundle over surface

The following is a explicitly defined complex line bundle $E\to\Sigma$ over a closed surface: View $\Sigma$ as a subset of $\mathbb{R}^3$ and consider its Gauss map $n:\Sigma\to S^2$ given by ...
5
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69 views

How many sections of a vector bundle send a point outside a divisor?

Let $\pi:E\to X$ be a holomorphic vector bundle on a complex algebraic variety $X$, and assume $E$ has nonzero global sections; fix a divisor $D\subset E$ and a point $P\in X$. I have the vague ...
5
votes
0answers
122 views

How should we think of 'differences' of vector bundles?

Given a Hausdorff space $X$, the set of all vector bundles of finite dimensions $Vect(X)$ (up to isomorphism) with respect to the direct sum is a monoid. We can use the group completion construction ...
5
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163 views

Complex vector bundles with real transition functions

After doing a bit of playing around (I think) I was able to show that the map $\operatorname{id}\otimes\ \psi : \Omega^{p,q}(X, E) \to \Omega^{p,q}(X, \bar{E})$, where $\psi $ is the conjugation map ...
5
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0answers
205 views

de Rham Cohomology of Non-Flat Bundle

Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$. If $E$ ...
5
votes
0answers
79 views

How to understand the coordinate transition map in $E\otimes E'$?

This question seems embarrassingly basic. Let $X$ be some manifold and $E,E'$ two vector bundles over $X$ with rank $n$ and $n'$ respectively. Now assume to possible refinement we have $E,E'$ defined ...
5
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0answers
134 views

Alternate pullback bundle construction

If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is $$\phi^* F := \{(m,f) \in M ...
5
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0answers
103 views

Decomposition of vector bundles over a CW complex

Let $X$ be CW complex having only cells up to dimension $n$. I want to prove that every $m$-dimensional vector bundle $E$ over $X$ decomposes as a sum $E\cong A\oplus B$ where $A$ is a ...
4
votes
0answers
25 views

Why is it called *adjunction* formula?

Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy $$ \omega_X \big|_Y = ...
4
votes
0answers
116 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
4
votes
0answers
116 views

What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$. Suppose there is a surjection: $E\longrightarrow ...
4
votes
0answers
67 views

Pullback of indecomposable bundles on an elliptic curve

I consider an elliptic curve $\mathcal C$ over $\mathbb{C}$ and the multiplication by $[n]$ map on the curve. Then I consider an indecomposable vector bundle $E$ on $C$. What can I say of the ...
4
votes
0answers
71 views

Sections of endomorphisms of a vector bundle

The following could be rather silly question but I haven'd found it stated explicitly; from the other side, it seems to me, that this fact is used often without comments. The problem is the ...
4
votes
0answers
89 views

Definition of the $\Bbb C^*$-weight of a line bundle

I'm new to geometric invariant theory and am unsure about a definition. Let $X$ be a smooth projective-over-affine variety equipped with a $\Bbb C^*$ action and let $Z$ be the fixed locus of this ...
4
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0answers
99 views

Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
4
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0answers
59 views

Holonomy group as a quotient group of $\text{GL}(k,\mathbb{R})$

Currently, I'm reading the script of a Global Analysis lecture at my university. There, we look at a real vector-bundle $(E,\pi,M)$ of rank $k$ with a given connection $\nabla$ and define the ...
4
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0answers
151 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
3
votes
0answers
34 views

Vector Bundle Structure on $\sqcup_{p\in M}\mathcal L(T_pM, T_{f(p)}N)$.

Let $f:M\to N$ be a smooth map between smooth manifolds. Is there a natural way to give a smooth vector bundle structure to $\bigsqcup_{p\in M} \mathcal L(T_pM, T_{f(p)}N)$. where $\mathcal L(V, ...
3
votes
0answers
48 views

Confusion over common criterion for a line bundle on a scheme to be trivial?

Suppose you have a line bundle $L\to X$ on a scheme $X$. I'll denote by $\sigma\colon X\to L$ to be the zero section. Why is it that this bundle is trivial iff there is another section $s\colon X\to ...
3
votes
0answers
50 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
3
votes
0answers
51 views

Do homotopic transition functions define isomorphic bundles (on smooth manifolds)?

It is fairly known that given a cover $\{U_i\}_{i \in \mathbb{N}}$ of some smooth manifold $M$ together with smooth transition functions $g_{\alpha \beta} \colon U_{\alpha} \cap U_{\beta} \rightarrow ...
3
votes
0answers
36 views

Computing cohomology of the sheaf $\mathcal{End}(T_{\mathbb{P}^2})$ restricted to a curve

Characteristic of the basic field is zero in this question. Let $E \subset \mathbb{P}^2$ be a smooth elliptic curve. Let $\mathcal{F}$ be the vector bundle on $E$ obtained as restriction to $E$ of the ...
3
votes
0answers
33 views

How many charts?

My question is rather vaque but I hope that You will feel what I would like to know. A manifold (say: real, $n$-dimensional) is something which locally looks like an open set in $\mathbb{R}^n$. A rank ...
3
votes
0answers
120 views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
3
votes
0answers
107 views

measurable function induces a measurable bundle

Greetings I am preparing a work on bundles and I found this statement Let $V$ a topological vector space with $\dim V=n$ and $(E,\pi,M)$ a vector bundle continuous over $M$ (compact space). If ...
3
votes
0answers
87 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
0answers
49 views

Total space of a geometric vector bundle

Suppose that $X$ is a scheme with a $B$-action, and that $\lambda: B \to \mathbb{C}^*$ is a character of $B$. We then have a geometric vector bundle $\pi: X \times^B k \to X/B$, where $B$ acts on $X ...
3
votes
0answers
232 views

Grassmannian bundle: any good reference?

I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far ...
3
votes
0answers
114 views

Triviality of (equivariant) holomorphic vector bundles

Let $G$ the 1-dimensional diagonalisable linear complex analytic group $\mathbb C^*$. We suppose that $G$ acts linearly on $\mathbb C^n$ with positive weights. Set $X=\mathbb C^n -\lbrace 0 \rbrace$. ...
3
votes
0answers
79 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...
3
votes
0answers
58 views

Maps between total spaces of holomorphic vector bundles

I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles. Let me outline a situation that is a bit more concrete, to help focus ...
3
votes
0answers
100 views

Triviality of the tangent space of an abelian variety

The tangent bundle of an abelian variety $A/K$ is trivial. Mumford's proof of this fact goes something like this: given the (non-zero) value of a section in a fiber, one can determine the value in all ...
3
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0answers
115 views

A question of extension of vector bundles.

Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong ...
3
votes
0answers
57 views

Questions on a sub-bundle of $\mathbb R^3\setminus \{0\}$

Let us consider spherical coordinates $(r,\theta,\phi)$ on $\mathbb R^3$ and the manifold $M:=\mathbb R^3 \setminus \{0\}$. Let us consider the 1-form on $M$ $$ \omega = zdz ...
3
votes
0answers
62 views

Second order equations on manifolds

In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let's write $$v(u,e)=((u,e), (a(u,e),b(u,e)).$$ It is said that $v$ is a second order equation ...
3
votes
0answers
124 views

Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
2
votes
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 ...
2
votes
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
votes
0answers
22 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 ...
2
votes
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 ...
2
votes
0answers
59 views

Formula for Stiefel-Whitney of tensor product

I need help for solving Ex. 7C from 'Characteristic Classes' by Milnor/Stasheff: The exercise asks to find a formula for the (total) Stiefel-Whitney class of $\xi^m\otimes\eta^n$ over a paracompact ...
2
votes
0answers
31 views

Connection and reduction of the structure group

I am writing a memoir about gauge theory. I have trouble with a small proof which should be simple and have the feeling that I am missing something obvious. I want to show that the set of connections ...
2
votes
0answers
63 views

Examples of vector bundles

I know three examples of interesting vector bundles (not taking into consideration what one gets from them using algebraic operations): Tangent bundle of a smooth manifold Tautological bundle over ...