TAILIEUCHUNG - Đề tài " 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. | Annals of Mathematics Moduli space of principal sheaves over projective varieties By Tom as G omez and Ignacio Sols Annals of Mathematics 161 2005 1037 1092 Moduli space of principal sheaves over projective varieties By Tomas Gomez and Ignacio Sols To A. Ramanathan in memoriam Abstract 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 0 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 0 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 . filtrations with E E_j_1 and Ei Ej c Ej jvv have Y PEi rk E - Pe rk Ei X 0 where Peì is the Hilbert polynomial of Ej. 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. Introduction Let X be a smooth projective variety of dimension n over C with a very ample line bundle Ox 1 and let G be a connected algebraic reductive group. A principal GL R C -bundle over X is equivalent to a vector bundle of rank R. If X is a curve the moduli space was constructed by Narasimhan and Seshadri N-S Sesh . 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 Gi Ma Si . Ramanathan Ra1 Ra2 Ra3 defined a notion 1038 TOMAS GOMEZ AND IGNACIO SOLS of stability for .

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