Questions tagged [vector-bundles]
For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.
1,948
questions
2
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0answers
24 views
Complexified bundle $\xi \otimes \mathbb{C}$ is isomorphic to its conjugate $\overline{\xi \otimes \mathbb{C}}$?
The following appears in Milnor's Characteristic classes, Chapter 15, and I don't fully understand what's going on.
Let $\xi$ be a real vector bundle.Milnor claims that the complexification $\xi \...
1
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0answers
32 views
Stokes's Theorem with singularities on projective line
Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
0
votes
0answers
10 views
When is a symplectic connection a symplectomorphism?
Let $(M, \omega)$ be a symplectic manifold and $\nabla$ a symplectic connection, i.e. an element of $\Omega^1 (M, \operatorname{End} (TM)).$ Denote by $\operatorname{End}_{\omega} (TM)$ a bundle of ...
0
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0answers
13 views
The same generalized Gauss map
Let $f:M^{n}\rightarrow \mathbb{R}^{m}$ an isometric immersion of a simply connected Riemannian manifold such that
$\Phi\in \Gamma(\mbox{End}(TM))$ is a Codazzi tensor on $M^{n}$, that is, $$(\nabla_{...
2
votes
0answers
48 views
Is Atiyah's periodicity Theorem related to splitting principle?
I assumed all vector bundles are over complex number. Let $V$ be a vector bundle over $X$. Then $P(V)$ denotes the projectivization of fibers of $V-0$ as a projective space bundle over $X$. Denote $K(...
1
vote
1answer
37 views
Check stability of extension bundle
Let $X$ be a compact complex surface.
This definition is from Donaldson, Kronheimer: The Geometry of 4-Manifolds, p. 209:
Definition: A holomorphic $SL(2,\mathbb{C})$ bundle $E$ over $X$ is called ...
1
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1answer
37 views
A complex vector bundle $E$ is a holomorphic vector bundle iff $(\overline{\partial^E})^2=0$ help with proof?
Ok so I am following a set of notes on complex differential geometry and there is a theorem that says the following:
If $E$ is a complex vector bundle over a complex manifold and $\overline{\partial^...
4
votes
1answer
109 views
Geometric visualization of tangent bundles/sheaves and normal cones/bundles
In algebraic geometry it is customary to identify locally free sheaves on say a scheme $X$ and vector bundles over the same scheme.
So say I have a nice scheme $X$ (you can assume it to be a variety, ...
1
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1answer
60 views
Construction of a resolution for a coherent sheaf
Let $\mathcal{S}$ be a coherent sheaf over a complex manifold $M$. How do I construct a resolution of $\mathcal{S}$ by holomorphic vector bundles? Is this construction "unique"? Are the answers the ...
2
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0answers
44 views
$ \mathrm{ch}(F(A)) = \mathrm{ch}(F(B)) \implies \mathrm{ch}(A) = \mathrm{ch}(B) $ for autoequivalences?
Let $\mathcal{C} \subset D^b(X)$ be a subcategory of the derived category of coherent sheaves on a smooth projective variety $X$. Let $F : \mathcal{C} \to \mathcal{C}$ be an autoequivalence. Let $\...
0
votes
1answer
36 views
Levi-Civita-Connection on $2$-Form
Let $(M, g)$ be a Riemann manifold and $$\nabla^{LC}: \Gamma(M,TM) \to \Gamma(M, T^*M \otimes TM)$$ the Levi-Civita connection over the tangential bundle $p:TM \to M$.
Since in general for arbitrary ...
1
vote
1answer
27 views
Under surjective submersion, pushforward of a $k$-form is smooth
Let $\pi:P\to M$ be a principal $G$-bundle with some connection $\omega\in\Omega^1(P,\mathfrak g)$. Let $\psi\in \Omega^k(P)$ such that $\psi_{p\cdot g}\big(r_{g*}(v_1),...,r_{g*}(v_k)\big)=\psi_p(v_1,...
0
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0answers
6 views
Functoriality of twisted de Rham Cohomology
Let $f: M\to N$ be a smooth map of finite-dimensional manifolds, and let $E\to N$ be a flat vector bundle over $N$. Consider the pullback bundle $f^*E\to M$ over $M$ and consider the twisted de Rham-...
1
vote
0answers
67 views
Complex structure on the tangent space to normal bundle
Consider a compact submanifold $X\hookrightarrow Y$ and its normal bundle $N\to X$:
$$0\to TX\to TY|_X \to N \to 0$$
There is a diffeomorphism which identifies $N$ with a tubular neighborhood $\tilde ...
2
votes
1answer
36 views
Fibers of a line bundle
Let $C$ be a smooth curve over complex numbers and $x\in C$ be a point. Consider the line bundle $L=\mathcal{O}_C(x)$. I am confused with the following. Since $L$ is a line bundle, the fiber $L_{|_y}$ ...
1
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0answers
43 views
Symmetric deformation of vector bundles
Let $E_1$ and $E_2$ be two non-isomorphic extensions of vector bundles $M$ by $N$ on an algebraic variety $X$. Assume $E_1\cong E_2$. Is it possible to deform $E_1$ to $E_2$ in a symmetric way? More ...
1
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0answers
49 views
What is the degree of a vector bundle?
I heard that the degree of a vector bundle encodes the number of "twists" of the bundle.
I heard also that it is roughly the difference between number of roots and number of poles of a function in a ...
0
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0answers
7 views
Examples of points in the total space of a fiber bundle
What are some examples of notation for points in the total space of a fiber bundle?
I think I have found one variant for vector bundles in this question about trivializations (something like $v^i \...
0
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0answers
34 views
Variation of the (Chern) curvature with respect to the metric
Let $E\rightarrow X$ be a holomorphic vector bundle, for any Hermitian metric $h$ on $E$ we denote by $F_h$ the curvature of the Chern connection associated to $h$. Fix a metric $h_0$ and consider a ...
0
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0answers
26 views
Two different interpretations of connections over vector bundles as an affine space
The following is not exactly a question but a consideration that needs to be completed, and I wish such a completion can be provided as an answer to this post. I am writing here such result of a ...
0
votes
1answer
51 views
Fiber from $S^1$ to $\mathbb{R} P^1$ that does not admit a section.
Consider the map $q:S^1 \to \mathbb{R} P^1$ given by $v \mapsto \mathbb{R} v$. I want to show that this map does not admit sections. I've previously shown that $S^1$ is diffeomorphic to $\mathbb{R} P^...
2
votes
0answers
41 views
Vector bundle with flat connection over simply connected manifold is trivial
I am trying to finish a proof of the statement in the title. Let $M$ be a simply connected smooth manifold and $E \twoheadrightarrow M$ be a vector bundle with a flat connection. Since the bundle is ...
1
vote
1answer
35 views
Category of vector bundles with connections
Vector bundles with connections over the same manifold $M$ make up a category. Indeed, let $E, E' \twoheadrightarrow M$ be vector bundles with connections $\nabla$ and $\nabla'$. A morphism between ...
0
votes
0answers
37 views
Fiber of a vector bundle at a point on a smooth curve
So I'm confused here, and I can't find any satisfactory definitions online for this... So in this text that I am going through, it says the following:
For a vector bundle V on a smooth curve C and ...
0
votes
0answers
19 views
Naturality axiom for Stiefel-Whitney Classes
In Milnor and Stasheff's Characteristic Classes the "Naturality" axiom for Stiefel-Whitney classes is defined as follows:
If $f : B(\xi) \to B(\eta)$ is covered by a bundle map from $\xi$ to $\eta$ ...
0
votes
0answers
19 views
Conditions for extending a restricted vector bundle of an embedded submanifold to the whole manifold
Suppose $(\pi,E,M)$ is a vector bundle (total space $E$, base space $M$) and that $S \subset M$ is an embedded submanifold. If $E$ has positive rank, show that every smooth section of $E|_S$ (the ...
1
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1answer
43 views
(lectures) Reference request for Vector bundles and characteristic classes
Are there some available online lectures for first year graduate course on vector bundles and characteristic classes?
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0answers
5 views
Hitchin fibration and twisted connections
Let $p$ be a positive prime.
Let $X,Y$ be smooth projective curves over a scheme $S$ of characteristic $p$, and let $\pi:X\times_S Y \to X$ be the first projection.
Let $\mathcal E$ be a vector ...
0
votes
0answers
21 views
Canonical connection on $\mathcal{A}\times X$
Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
0
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0answers
16 views
Compute the obstruction classes of the tangent bundle of $\mathbb{R}\mathrm{P}^2$
I want to compute the obstruction classes of the tangent bundle of $\mathbb{R}\mathrm{P}^2$.
I learnt obstruction classes mainly from the books on characteristic classes by Milnor. The 2nd ...
0
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0answers
36 views
Bass-Quillen conjecture for non-affine case
Bass-Quillen conjecture expects that any vector bundle on $U\times \mathbb{A}^1$, to be extended from $U$. Here $U$ is a regular affine scheme. Being affine is an essential part of the conjecture, you ...
0
votes
1answer
32 views
Injective morphisms of locally free sheaves and fiberwise injectivity of vector bundles
Let's fix a smooth integral algebraic variety $X$ over $\mathbb C$. If $\mathscr E$ is a locally free sheaf on $X$, then at each closed point $x\in X$ we have a complex vector space $E_x:=\mathscr E_x\...
1
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1answer
50 views
Euler class of tautological line bundle over $\mathbf{HP}^n$.
Let $e$ denote the Euler class of the tautological line bundle $\gamma^n$ over $\mathbf{HP}^n$. My question is how to determine the pairing
$$ \langle e,\ [\mathbf{HP}^n]\rangle$$
with $[\mathbf{HP}^n]...
0
votes
1answer
48 views
Pushforward of tangent bundle. Can it be $0$?
Let $f:X\to Y$ a smooth flat map between, integral projective varieties such that $\dim(Y)<\dim(X)$. Let's assume by semplicity that everything is over $\mathbb C$. So we are in the situation of "a ...
2
votes
2answers
74 views
Is it true that $\Gamma(\Lambda(T^*M)) \cong \Lambda(\Gamma(T^*M))$?
Well, the question is in the title.
Is it true, that given a smooth manifold $M$, the following isomorphism holds:
$$
\Gamma(\Lambda(T^*M)) \cong \Lambda(\Gamma(T^*M))
$$
$\Gamma$ - smooth sections ...
2
votes
1answer
48 views
$C^\infty(M)$-linear maps are local
Very often in mathematical physics literature I've heard a fact that probably (if I've understood it correctly) translates as follows.
Let $M$ be a differentiable manifold, and denote by $C^\infty(...
0
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0answers
36 views
Chern Character of the dual bundle
Let $(E,\nabla)$ be a complex vector bundle with connection. Is there a formula for the Chern character of the dual vector bundle $(E^*,\nabla^*)$ in terms of the chern character/classes of $E$?
2
votes
2answers
39 views
Stiefel manifold as a fibre bundle over another Stiefel manifold.
I want to show $V_n(\mathbb R^k)$ is a fibre bundle over $V_m(\mathbb R^k)$ were $n>m$
Here $V_n(\mathbb R^k)$ is the Stiefel manifold, that is the set of all $n$ orthogonal frames in $\mathbb R^k$...
2
votes
1answer
57 views
Vector bundle is Manifold
I want to show that the total space of a vector bundle $E \overset{\pi}{\rightarrow} M$ is itself a manifold. It is easy to construct charts by just composing the bundle maps $\pi^{-1}(U_{\alpha}) \...
1
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1answer
54 views
Motivation for Quasi Coherent Sheaf
I have some background in vector bundles in the context of differential geometry and I have seen how vector fields form a module over smooth functions on a smooth manifold.
Recently I came across ...
0
votes
1answer
74 views
identification vertical bundle and fiber product
Let $\pi:E\rightarrow M$ a vector bundle. I've seen that a vertical vector field on E is a section of $\pi_1:E\times_M E\rightarrow E$, but I thought that it was a section of $\pi|_{VE}:VE\rightarrow ...
2
votes
1answer
42 views
Two vector bundles on $\mathbb{CP}^1$
Fix any integer $n$. Let $a$ and $b$ be two different integers such that $a+b \neq n$. Consider two vector bundles $E_{a,n-a} = \mathcal{O}(a)\oplus\mathcal{O}(n-a)$ and $E_{b,n-b} = \mathcal{O}(b)\...
1
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1answer
36 views
Taking directional derivatives of smooth sections of vector bundles
Let $E \xrightarrow{ \pi} M$ be a smooth vector bundle. If $E$ is the trivial bundle $ M \times \mathbb{R}^k$ then we can define directional derivatives of a section $\sigma \in \Gamma(E)$ (without ...
4
votes
1answer
67 views
Why does the vanishing of this cohomology say that?
Consider the following proposition found in the article: Stable Vector Bundles of Rank 2 on $\mathbb{P}^{3}$ - Hartshorne.
Proposition 3.1. Let $\mathscr{E}$ be a rank 2 bundle on $\mathbb{P}^{...
0
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1answer
32 views
A continuous function is a bundle with the same fiber as another bundle.
I have the following problem.
Let $p:Y\to X$ be a bundle with fiber $F$. This means that for each $x\in X$, there is an open neighborhood $N_x$ of $x$ and a homeomorphism
$p^{-1}N_x\cong N_x\times F$ ...
5
votes
1answer
49 views
Visualizing $|\mathcal{B}\mathbb{Z}| \simeq S^1$.
The classifying space of the integer group $\mathbb{Z}$ can be defined as the geometric realization of the underlying groupoid $\mathcal{B}\mathbb{Z}$.
To unwind, $\mathcal{B}\mathbb{Z}$ is simply ...
1
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1answer
32 views
Fibered category over $\operatorname{Aff}/\Bbb A^1$
Let $X$ is be smooth projective curve over an algebraically closed field $k$.
Let $S$ be a scheme and let $S \xrightarrow[]{t}\Bbb A^1$ be a global function. Let $\mathcal C_S$ be the category whose ...
3
votes
1answer
51 views
Vector bundles as Hilbert-C*-modules
In Example II.7.1.7 (iii) (p. 139) from Bruce Blackadar's "Operator Algebras - Theory of C*-Algebras and von Neumann Algebras" we have the following setup:
Let $V$ be a complex vector bundle of rank $...
1
vote
1answer
42 views
Confusion with the Tangent bundle of product manifolds
Let's suppose $M$ and $N$ are manifolds.
Here 1 one states that
\begin{equation}
T(M \times N) \cong \pi_{M}^{*}(T M) \times \pi_{N}^{*}(T N)
\end{equation}
But in my notes I find
\begin{equation}
T(...
4
votes
0answers
84 views
Chern character of $\mathrm{ch}(i_* \mathcal{O}_L)$ for $i: L \hookrightarrow X$ a line
$\newcommand{\oh}{\mathcal{O}} \newcommand{\ch}{\mathrm{ch}} \newcommand{\td}{\mathrm{td}}$Let $X$ be a Fano threefold of index $1$ and degree $d$ with ample class $H$, so $K_X=-H$ and $H^3=d$, and ...
