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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 ...

0

Perhaps I am misunderstanding the question, but it seems to me that $\text{deg}(V)=\text{deg}(\text{det}(V))=\text{deg}(L)$, so we are just asking if for each $n$ there is a (virtual) vector bundle $[V']$ of rank $n$ and degree $\text{deg}(L')$. But $\mathcal{O}_X^{\oplus(n-1)}\oplus L'$ works, right?

0

I will not answer your question, but tell you something about the holomorphic structure of a Fuchsian representation. I guess you are aware of the paper "Twisted harmonic maps..." of Donaldson (cited in Hitchin's paper). The thing is that if you start with a Fuchsian representation, your corresponding twisted harmonic map is just the developing map of you ...

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