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**Generalized vector bundles on curves.**
*(English)*
Zbl 0908.14011

G. Harder and M. S. Narasimhan [Math. Ann. 212, 215-248 (1975; Zbl 0324.14006)] (and independently D. Quillen) have constructed a canonical flag of subbundles on any vector bundle on a complete smooth algebraic curve over a field. This flag measures how far away from semistability the vector bundle is. Influenced by this D. Grayson and U. Stuhler have studied the corresponding situation over number fields. Here one is looking at lattices in some euclidean or hermitean vector space. It turns out that one has again a canonical filtration by a flag of sublattices. If one compares both situations more carefully one sees that the second is in a certain sense more general. This leads one to consider what we call generalized vector bundles in this paper. The idea is to replace locally or better over the completion of the local rings of the curve the lattices given by the stalks of the locally free sheaf associated to the vector bundle by the real valued norms given by these lattices on the associated vector spaces.

It turns out that the Harder-Narasimhan-filtration exists also in this more general context. This is done in section two of this paper and we could follow for this more or less completely the exposition by D. R. Grayson [in: Algebraic \(k\)-theory, Proc. Conf., Oberwolfach 1980, Part I, Lect. Notes Math. 966, 69-90 (1982; Zbl 0502.14004)]. We have included in this section a study of the H-N-filtration where the vector bundle is deformed in a family. The main motivation for us to consider this kind of generalisation comes from reduction theory of the general linear group.

It turns out that the Harder-Narasimhan-filtration exists also in this more general context. This is done in section two of this paper and we could follow for this more or less completely the exposition by D. R. Grayson [in: Algebraic \(k\)-theory, Proc. Conf., Oberwolfach 1980, Part I, Lect. Notes Math. 966, 69-90 (1982; Zbl 0502.14004)]. We have included in this section a study of the H-N-filtration where the vector bundle is deformed in a family. The main motivation for us to consider this kind of generalisation comes from reduction theory of the general linear group.

### MSC:

14H60 | Vector bundles on curves and their moduli |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |