Principal bundles over a projective scheme (Q2781390)

The paper deals with faithfully flat principal \(G\)-bundles on a projective scheme \(X\), \(G\) being a connected reductive algebraic group defined over an algebraically closed field of characteristic zero. The author defines a principal \(G\)-bundle to be (semi)stable if and only if the vector bundle associated to it by adjoint representation is (semi)stable. It is shown that there exists a quasi-projective coarse moduli scheme of stable principal \(G\)-bundles. \textit{A. Ramanathan} has constructed (projective) moduli spaces of (semi)stable principal \(G\)-bundles on a smooth curve and proposed a (semi)stability condition in the higher dimensional case. The author's definition of stability is stronger than Ramanathan's definition even over smooth curves. NEWLINENEWLINENEWLINERecently, projective moduli spaces containing (as open sets) moduli spaces of principal \(G\)-bundles on smooth varieties have been constructed by \textit{T. L. Gomez} and \textit{I. Sols} [``Projective moduli space of semistable principal sheaves for a reductive group'' (http://front.math.ucdavis.edu./ math.AG/0112096)] and by \textit{H. W. A. Schmitt} [Int. Mat. Res. Not. 2002, No. 23, 1183--1209 (2002; Zbl 1034.14017)] using different methods.