Stable Tensor Fields and Moduli Space of Principal G-sheaves for Classical Groups

Let X be a smooth projective variety over C. Let G be the group O(r,C), or Sp(r,C). We find the natural notion of semistable principal G-bundle and construct the moduli space, which we compactify by considering also principal G-sheaves, i.e., pairs (E,φ), where E is a torsion free sheaf on X and φ is a symmetric (if G is orthogonal) or antisymmetric (if G is symplectic) bilinear form on E. If G is SO(r,C), then we have to consider triples (E,φ, ψ), where ψ is an isomorphism between det(E) and OX such that det(φ) = ψ . More generally, we consider semistable tensor fields, i.e., multilinear forms on a torsion free sheaf, and construct their projective moduli space using GIT. LetX be a smooth projective variety of dimension n over C. A principal GL(r,C)bundle overX is equivalent to a vector bundleE of rank r. IfX is a curve, the moduli space was constructed by Mumford, Narasimhan and Seshadri. If dim(X) > 1, to obtain a projective moduli space we have to consider also torsion free sheaves, and this was done by Gieseker, Maruyama and Simpson. Let G be the orthogonal group O(r,C) or symplectic group Sp(r,C). A principal G-bundle is equivalent to a pair (E,φ), where E is a vector bundle, and φ : E ⊗ E −→ OX is a non-degenerate morphism. If G = O(r,C), then det(E)⊗2 = OX and φ is symmetric. If G = Sp(r,C), then det(E) = OX and φ is antisymmetric. If G is the special orthogonal group SO(r,C), then a principal G-bundle is equivalent to a triple (E,φ, ψ), where E is a vector bundle, φ is a morphism as before, and ψ : det(E) −→ OX is an isomorphism such that det(φ) = ψ2 (this equation means that for all points x ∈ X, if we choose a basis for the fiber Ex, the determinant of the matrix associated to φ at x is equal to the square of the scalar associated to ψ at x). To obtain a projective moduli space we have to consider also semistable principal G-sheaves (definitions 5.1 and 6.1), allowing E to be a torsion free sheaf (and then requiring φ to be nondegenerate only on the open subset of X where E is locally free). We say that a subsheaf F of E is isotropic if φ|F⊗F = 0. A principal G-sheaf is called stable (resp. semistable) if for all proper isotropic subsheaves F of E PF + PF⊥ ≺ PE (resp. ), where PF is the Hilbert polynomial of F , F ⊥ is the sheaf perpendicular to F with respect to the form φ, and as usual, the inequality between polynomials P1 ≺ P2 (resp. ) means that P1(m) < P2(m) (resp. ≤) for m ≫ 0 (see section 5 for precise definitions). Date: 11 November 2001. Mathematical Subject Classification: Primary 14D22, Secondary 14D20. 1