Stable tensors and moduli space of orthogonal sheaves

TL;DR

This paper develops a framework for moduli spaces of orthogonal, symplectic, and tensor sheaves on smooth projective varieties, extending stability notions and compactifying the moduli space to include sheaves with singularities.

Contribution

It introduces a natural notion of semistability for orthogonal bundles and constructs their moduli space, including compactifications via orthogonal and symplectic sheaves, and generalizes to semistable tensors.

Findings

Constructed moduli space of semistable orthogonal sheaves.

Extended stability notions to orthogonal and symplectic sheaves.

Developed GIT-based construction for moduli of tensors.

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.