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 $X$.

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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 \...
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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}\...
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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 ...
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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_{...
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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(...
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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 ...
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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^...
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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, ...
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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 ...
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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 $\...
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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 ...
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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,...
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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-...
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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 ...
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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 $L_{|_y}$ ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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^...
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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 ...
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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 ...
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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 ...
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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 $\eta$ ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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]...
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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 ...
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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 ...
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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(...
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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 $E$?
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2answers
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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 $\mathbb R^k$...
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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}) \...
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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 ...
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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 ...
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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)\...
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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 ...
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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}^{...
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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 $p^{-1}N_x\cong N_x\times F$ ...
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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 ...
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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 ...
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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 $...
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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(...
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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 ...

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