Torsor über Kurve/Starke Harder-Narasimhan Filtration/Affinitätskriterium/Fakt/en

Let ${}C$ denote a smooth projective curve over an algebraically closed field ${}K$ and let ${}E$ be a vector bundle over ${}C$ together with a cohomology class ${}c\in H^{1}(C,{\mathcal {S}})$. Let

${}E_{1}\subset E_{2}\subset \ldots \subset E_{t-1}\subset E_{t}=F^{e*}(E)\,$

be a strong Harder-Narasimhan filtration. We choose ${}i$ such that ${}E_{i}/E_{i-1}$ has degree ${}\geq 0$ and that ${}E_{i+1}/E_{i}$ has degree ${}<0$. We set ${}{\mathcal {Q}}=F^{e*}(E)/E_{i}$.

Then the following are equivalent.

1. The torsor ${}T(c)$ is not an affine scheme.
2. Some Frobenius power of the image of ${}F^{e*}(c)$ inside ${}H^{1}(X,{\mathcal {Q}})$ is ${}0$.