10
votes

Accepted

### Triviality of vector bundles on affine open subsets of affine space

For your final question, the answer is that all vector bundles over $U$ are trivial.
Sketch of proof: Let $R=k[x_1,\ldots,x_n]$.
Let $L_1, L_2,\ldots, L_m$ be the equations of hyperplanes in $R$. ...

1
vote

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### Existence of rigid objects in the derived category of a smooth projective variety

I am writing up as one answer the comments by @Johan, by @Libli, and by myself. If either of them prefers to write an answer, I am happy to delete this answer.
Let $A$ be an Abelian variety. For ...

Community wiki

1
vote

### Stability of sheaves of non-constant rank

The rank of a coherent sheaf is defined as its rank at the general point (equivalently, as the rank on a dense open subset where the sheaf is locally free). So, yes, the definition applies.

1
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### Why is it useful to study vector bundles?

Calculus is extended from vector spaces to manifolds by way of the tangent bundle. This is a very specific kind of vector bundle. The Cartan calculus of differential forms is built over it as well as ...

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