Moduli space of principal sheaves over projective varieties

Abstract

Let G be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal G-bundle on a Riemann surface and constructed a projective moduli space of such objects. We generalize Ramanathan?fs notion and construction to higher dimension, allowing also objects which we call semistable principal G-sheaves, in order to obtain a projective moduli space: a principal G-sheaf on a projective variety X is a triple (P,E,?O), where E is a torsion free sheaf on X, P is a principal G-bundle on the open set U where E is locally free and ?O is an isomorphism between E|U and the vector bundle associated to P by the adjoint representation. We say it is (semi)stable if all filtrations E. of E as sheaf of (Killing) orthogonal algebras, i.e. filtrations with E?Ui = E.i.1 and [Ei,Ej ] ?? E ?E?E i+j , have
(PEi rkE . PE rkEi) () 0, where PEi is the Hilbert polynomial of Ei. After fixing the Chern classes of E and of the line bundles associated to the principal bundle P and characters of G, we obtain a projective moduli space of semistable principal G-sheaves. We prove that, in case dimX = 1, our notion of (semi)stability is equivalent to Ramanathan?fs notion.