Moduli spaces for principal bundles in large characteristic

Abstract

This paper addresses the problem of constructing and compactifying moduli spaces of stable principal G -bundles over smooth projective schemes for an arbitrary reductive group G in both zero and nonzero characteristic. This involves refining the methods of the third author [Transform. Groups 9 (2004), no. 2, 167–209; Int. Math. Res. Not. 2004, no. 62, 3327–3366;] and creating a unified formulation of the results of these papers and a paper of the first and fourth authors [Ann. of Math. (2) 161 (2005), no. 2, 1037–1092;]. The paper also complements a previous article by all four authors [Adv. Math. 219 (2008), no. 4, 1177–1245]. Particular features of the paper are the inclusion of non-semisimple groups and non-faithful representations of G , a new approach to obtaining the semistable reduction theorem in characteristic zero and in large positive characteristic and a construction of the moduli space of decorated sheaves over projective varieties in arbitrary characteristic. The first main theorem states that, in characteristic zero and in large positive characteristic, coarse moduli spaces for (semi)stable singular principal G -bundles can be defined using faithful representations (under some conditions) and exist as projective schemes. This covers many known cases, for example when G is semisimple, for G=GL(V) , also for G one of the classical groups O r (k) , SO r (k) and Sp r (k) provided the characteristic is not 2 and for groups of adjoint type. In order to use a non-faithful representation ρ , singular principal G -bundles must be replaced by principal ρ -sheaves. The second main theorem states that the corresponding coarse moduli spaces exist as quasi-projective schemes provided the base scheme is a curve or the characteristic is zero. Both types of moduli space can be considered as compactifications of the moduli space of slope-stable rational principal G -bundles.