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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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