# Tagged Questions

For questions on vector bundles.

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### Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
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### 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 ...
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### Null-correlation and Tango bundles on $\mathbb{P}^3$

Let $V$ be a four-dimensional complex vector space and $\mathbb{P}^3=\mathbb{P}(V)$. There are two interesting bundles $N$ and $T$ on $\mathbb{P}^3$, both of rank 2, called respectively a null-...
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### Condition for a complex vector bundle to be holomorphic?

Suppose that $(E,\pi,M)$ is a complex vector bundle. I've seen it suggested in a few places that if $E$ and $M$ are complex manifolds, and $\pi$ is a holomorphic map, then $(E,\pi,M)$ is in fact a ...
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### 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 $E$ ...
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### 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 ...
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### 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 ...
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### Exterior algebra as a complex of vector bundles and an element in relative K-theory

There is a description of relative $K$-theory in terms of complexes of vector bundles. The following is an excerpt from Atiyah's book. Let $V$ be a complex vector space and consider the exterior ...
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### Motivation for equivalence of Tautological Line Bundle and ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$

I am currently reading a book on Algebraic Geometry and both the Tautological Line Bundle $L \to \mathbb{P}_k^n$ and the blow up ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$ are defined set ...
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### $E$-orientation of a closed manifold induce $E$-orientation of normal bundle: passage in the proof.

I'm trying to follow Kochman's proof of the well-known result For a ring spectrum $E$, and closed manifold $M^n$ together with an embedding in $\mathbb{R}^{n+k}$, the following are equivalent: ...
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### Compact Vertical Cohomology and Euler Class of CP1

First of all, please excuse my English. I'm not native Englsih speaker, so you will see so many grammer mistakes, I just only hope that my mistkae wouldn't effect what I want to mean. Hi, recently I'...
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### Pull back and intersection of divisors

Let $X$ be a smooth projective variety over complex numbers. Let $Z$ be a smooth closed subscheme of $X$. Let $L$ be a very ample line bundle on $X$. Then $L|_Z=E$ is a very ample line bundle on $Z$. ...
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### Hermitian Structure on $\mathcal{O}(1)$ line bundle over $\mathbb{P}^{n}$

I'm reading through Huybrecht's section on Vector Bundles. In general, he notes that for a (holomorphic) line bundle $L$ with $s_{1}, \ldots, s_{k}$ globally generating (holomorphic) sections, we can ...
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### Normal Space of an orbit at a point

Let $X$ be a curve of genus $g$. If $d>n(2g-1)$,then for any vector bundle $E$ of rank $n$ and degree $d$ over $X$ has $H^1(X,E)=0$ and $E$ is generated by its global sections. For such a vector ...
Let F be a torsion free sheaf of rank $n+4$ over $\mathbb{P}^1$ which fits in the SES $0\longrightarrow\mathcal{O}_\mathbb{P^1}(-3)^{n+2}\longrightarrow F\longrightarrow\mathcal{O}_{\mathbb{P}^1}(-1)^... 0answers 35 views ### Transition function of line bundle Let$X$be a smooth algebraic curve over$\mathbb C$and fix a closed point$p\in X$. I want to calculate the transition functions of the line bundle associated the sheaf$\mathcal O(p)$. I know ... 0answers 31 views ### Pullback of a semistable sheaf to a product is semistable? Let$X$be a smooth projective variety over$\mathbb{C}$. Let$L$be an ample line bundle on$X$. Let$F$be a$\mu_L$semistable rank 2 vector bundle on$X$(semistability in the sense of Mumford-... 0answers 110 views ### Pullback along the Frobenius morphism Let$X$be a scheme over a finite field$\mathbb{F}_q$and let$F : X \to X$be the absolute Frobenius morphism. If$\mathcal{L}$is an invertible$\mathcal{O}_X$-module, then$F^*(\mathcal{L}) \cong \...
Let $M$ be a $n$-smooth manifold, and $\pi :E\to M$ a rank $r$ complex vector bundle on $M$. Let $J:E\to E$ be a bundle morphism that is diagonalizable, i.e. for every $p\in M$ the automorphism \$J_p :...