## New answers tagged vector-bundles

5

This discussion seems to involve two holomorphic structures on the smooth manifold $T S^2$.
Structure 1 Identify $S^2$ with $\mathbb{CP}^1$. Place a complex structure on the real tangent bundle to $\mathbb{CP}^1$ by using the complex structure on $\mathbb{CP}^1$. This is the structure Alex Degtyarev's answer refers to. As Alex says, this is the line bundle ...

1

$TS^2\to S^2$ has Euler class $2$; ergo, the line bundle in question is $\mathcal{O}_{\Bbb P^1}(2)$. It's holomorphic sections form a vector space of dimension $3$; they can be thought of as degree $\le2$ polynomials on $\Bbb C$.
Added in proof: All this is independent of the holomorphic structure on $TS^2$ assuming that there is a fibration $TS^2\to S^2$. ...

2

I've taken the liberty of removing this claim about the Vandermonde determinant from all of the relevant Wikipedia articles I could find, listed below for convenience.
Euler class
Pontryagin class
Vandermonde polynomial
Characteristic class
Splitting principle
Alternating polynomial

8

The category of maps from a test object $T$ to a quotient stack $[X/G]$ has the following general form. Objects are pairs $(P, f)$, where $P$ is a $G$-torsor over $T$, and $f: P \to X$ is a $G$-equivariant map. Morphisms $(P,f) \to (P',f')$ are torsor isomorphisms $g: P \to P'$ satisfying $f = f' g$. Here, $X$ is the vector representation $\mathbf{O}^n$, ...

19

I'll use the definition of stack as a (weak) functor from the category of schemes to that of groupoids (as opposed to the definition as a fibered category over the category of schemes).
The prestack associated to the action of $GL_n$ on $\mathbb A_n$ is, by definition, given by
$$
X \mapsto \left\{\begin{matrix}\text{Objects: maps $s:X \to \mathbb ...

3

The canonical homomorphism $\varphi : p^*p_*E\rightarrow E$ induces on each fiber $F_z$ $(z\in\mathbb{P}^1)$ the evaluation map $H^0(F_z, E_{|F_z})\otimes \mathscr{O}_{F_z}\rightarrow E_{|F_z}$. By your hypothesis this is an isomorphism, so $\varphi $ is an isomorphism.

2

If $V$ is a vector bundle of rank $n$, the corresponding universal algebra $A$ which makes $V$ trivial (i.e. $V \otimes A \cong A^n$), or equivalently the algebra of the corresponding $\mathrm{GL}_n$-torsor, is given by
$$A = \mathrm{Sym}(V^n) \otimes_{\mathrm{Sym}(\Lambda^n V)} \mathrm{Sym}^{\mathbb{Z}}(\Lambda^n V).$$
Here, we define ...

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