# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### (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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### 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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### 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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### 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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### 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 ...
### 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 ...