zbMATH — the first resource for mathematics

Vector bundles with a Frobenius structure on the punctured unit disc. (English) Zbl 1074.14028
The paper under review deals with the classification of \(\sigma\)-vector bundles on the punctured unit disc \(\dot{D} \subset \mathbb{C}\), where \(\mathbb{C}\) is a complete non-archimedian valued algebraically closed field of positive characteristic \(p\). A \(\sigma\)-bundle on \(\dot{D}\) is a vector bundle together with a fixed isomorphism with its pull-back under the arithmetic Frobenius. For \(r\) and \(d\) relatively prime, there exist \(\sigma\)-bundles denoted \({\mathcal F}_{d,r}\) of rank \(r\), which are quite easy to construct.
The main result of the paper (Theorem 11.1) states that every \(\sigma\)-bundle is isomorphic to the direct sum of such indecomposable bundles. This result resembles the classification of vector bundle over an elliptic curve by Atiyah, but the \(\sigma\)-line bundles here are just given by \({\mathcal F}_{d,1}\), so isomorphic with \(\mathbb{Z}\).

14H60 Vector bundles on curves and their moduli
14G22 Rigid analytic geometry
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
Full Text: DOI