Moduli Space of Principal Sheaves over Projective Varieties

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’s 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, ψ), 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 ψ 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 i = E−i−1 and [Ei, Ej ] ⊂ 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’s notion. To A. Ramanathan, in memoriam