Semistable principal bundles. I: Characteristic zero.
The moduli space of semistable principal \(G\)-bundles on a smooth projective curve over a field of characteristic zero was first constructed in the PhD thesis of \textit{A. Ramanathan}, published only twenty years later [Proc. Indian Acad. Sci. 106, 301--328 (1996; Zbl 0901.14007) and Proc. Indian Acad. Sci. 106, 421--449 (1996; Zbl 0901.14008)]. An alternative proof was given by \textit{G. Faltings} [J. Algebr. Geom. 2, No. 3, 507--568 (1993; Zbl 0790.14019)]. In the paper under review, the authors give a different approach, which is intended to be generalized to positive characterictic [this work is done in part II, \textit{V. Balaji} and \textit{A. J. Parameswaran}, Transf. Groups 8, 3--36 (2003; Zbl 1084.14013)]. The main tool is the proof of a semistable reduction theorem which reduces the proof to the well-known case of vector bundles. Let \(A\) be a discrete valuation ring with residue field \(k\) and function field \(K\). The problem is to extend any semistable principal \(G_{K}\)-bundles over \(X\_{K}\) to a semistable principal \(G_A\)-bundle over \(X_A\). The authors choose a faithful representation \(G\hookrightarrow \text{SL}_n\) and interpret the principal bundle as \( \text{SL}_n\)-bundle together with a reduction of the structure group. This bundle can be extended to \(X_A\) with the added property of semistability (up to base change, one may need a finite cover of \(A\)). The reduction of the structure group is given by a section, whose extension is the main technical difficulty.
scientific article; zbMATH DE number 1868091
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