Stable tensors and moduli space of orthogonal sheaves

Abstract

Let X be a smooth projective variety over C. We find the natural notion of semistable orthogonal bundle and construct the moduli space, which we compactify by considering also orthogonal sheaves, i.e. pairs (E,\phi), where E is a torsion free sheaf on X and \phi is a symmetric nondegenerate (in the open set where E is locally free) bilinear form on E. We also consider special orthogonal sheaves, by adding a trivialization \psi of the determinant of E such that det(\phi)=\psi^2 ; and symplectic sheaves, by considering a form which is skewsymmetric. More generally, we consider semistable tensors, i.e. multilinear forms on a torsion free sheaf, and construct their projective moduli space using GIT.