Moduli space of principal sheaves over projective varieties

• 成果类型: Article
• 署名作者: Gómez, T; Sols, I
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.161.1037
• 发表日期: 2005
• 页码: 1037-1092
• 关键词: unitary vector-bundles

摘要

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, psi), 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 psi is an isomorphism between E vertical bar(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(perpendicular to) = E-i-1 and [E-i, E-j] subset of E-i+j(VV), have Sigma(P-Ei rk E - P-E rk E-i) (<=) 0, where P-Ei is the Hilbert polynomial of E-i. 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 dim X = 1, our notion of (semi)stability is equivalent to Ramanathan's notion.