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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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