# Tagged Questions

For questions on vector bundles.

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### 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 (...
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### 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 $Y$, ...
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### 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 ...
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### 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 ...
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### 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. ...
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### Definition of Levi-Civita connection map?

Does anyone know definition of Levi-Civita connection map that defined as $TTM\to TM$. I would appreciate if you could give a good reference. Thanks.
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### How many distinct flat connections are there on a flat bundle?

Given a flat smooth vector bundle (i.e. with constant transition functions), how many distinct flat connections could we put on it? If the flat connection is not unique, is it unique up to gauge ...
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### Normal bundle associated to $\mathbb{R}P^n\hookrightarrow \mathbb{R}P^{n+1}$ is the tautological line bundle

I'm interested in proving the fact of the title and I was following the reasoning in page 8 here (at the end). There is a step which are totally unclear to me, namely the identification of the normal ...
Let $U$ be a topological space, and let $U \times \mathbb{R}^n$ be the trivial bundle. Let $\varphi: U \times \mathbb{R}^n \to U \times \mathbb{R}^n$ be a bundle map, but assume we don't know whether ...
### Chern character of canonical line bundle over $\mathbb{CP}$
Let $H \to \mathbb{CP}$ be the canonical line bundle over $\mathbb{CP}=S^2$. Then from the text Vector Bundles and K-theory by Hatcher, given the chern character, $ch$, and first chern class $c_1(H)$,...