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Moduli spaces for principal bundles in arbitrary characteristic. (English) Zbl 1163.14008
The moduli spaces of principal \(G\)-bundles on smooth varieties \(X\) defined over algebraically closed fields of characteristic zero were constructed by different methods by T. Gomez and I. Sols [Ann. Math. (2) 161, No. 2, 1037–1092 (2005; Zbl 1079.14018)] and by A. Schmitt [Int. Math. Res. Not. 2002, No. 23, 1183–1209 (2002; Zbl 1034.14017)]. The present paper generalizes these results to arbitrary characteristics for connected semisimple algebraic groups \(G\). This is done by using the basic idea of A. Schmitt and rewriting his construction to suit positive characteristics. The authors fix a faithful representation \(\rho: G \to GL(V), r=\) dim \(V\), \(\rho(G) \subset SL(V)\). They define a pseudo \(G\)-bundle as a pair \((A, \sigma)\) where \(A\) is a torsionfree \({\mathcal O}_X\)-module of rank \(r\) with a trivial determinant and \(\sigma\) is a section of \(\operatorname{Hom} (A, V^{\vee}\otimes {\mathcal O}_X)//G\). If \(U\) is the maximum open set where \(A\) is locally free and \(\sigma (U) \subset \text{Isom} (V\otimes {\mathcal O}_U, A^{\vee}\mid_U)\), then the pseudo \(G\)-bundle is called a singular principal \(G\)-bundle. Over \(U\), the singular principal \(G\)-bundle gives a principal \(G\)-bundle.
The authors construct quasi-projective moduli spaces of singular principal \(G\)-bundles as open subsets of projective moduli spaces of \(\delta\)-semistable pseudo \(G\)-bundles, \(\delta \in \mathbb{Q}[t]\). Semistable reduction theorem for singular principal \(G\)-bundles is proved under the assumptions that \(\rho\) is of low separable index or \(G\) is an adjoint group and \(\rho\) is an adjoint representation of low height. The authors also recover Heinloth’s semistable reduction theorem over curves. The moduli spaces of singular principal \(G\)-bundles are projective under these assumptions. Over smooth complex curves, these moduli spaces are the same as the moduli spaces of principal \(G\)-bundles constructed by A. Ramanathan in his thesis in 1976.

MSC:
14D20 Algebraic moduli problems, moduli of vector bundles
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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