I hope that this is the appropriate place for asking about a step I don't understand in a proof which I think is due to a lack of knowledge. This is a step in Drinfeld-Simpson's paper: ``$B$ structures on $G$-bundles and Local Triviality" in which they showed that under some nice conditions, every $G$-bundle on a curve admits a $B$-reduction. Let me get to the specific statement.

Suppose that $X$ is a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{P}$ be a $G$-bundle on $X$ where $G$ is a linear algebraic group over $k$.

Now, some notation: let $B$ be a Borel of $G$, $\alpha_i: H \rightarrow G_m$ a simple root. If $\mathcal{E}$ is a $B$-bundle then I can produce the $G_m$-bundle associated to $\mathcal{E}$ and call its degree $deg_i(\mathcal{E})$ just as a line bundle over a smooth projective curve.

The problem set up: suppose I now have another $G$-bundle $\mathcal{P}'$ and (1) I have a trivialization of both $\mathcal{P}$ and $\mathcal{P}'$ on an open subscheme of $X$ and I have chosen $h$ an isomorphism between them and (2) a $B$-reduction on $\mathcal{P}$ (with corresponding $B$-bundle $\mathcal{E}$) which induces a $B$-reduction on $\mathcal{P}'$ with corresponding $B$-bundle $\mathcal{E}'$.


why does there exist a number $c$ such that $-c < deg_i(\mathcal{E}) - deg_i(\mathcal{E}') < c$ depending on the singularities of $h$? (as claimed)

I suppose I don't see how these numerical data relate to one another.

  • $\begingroup$ Is this a Riemann roch argument, perhaps by tensoring the two resulting line bundles? $\endgroup$ – Yosemite Sam Jan 6 '15 at 9:57

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