Heinz Spindler

Das Spektrum torsionsfreier Garben I

In this paper we study torsion free sheaves of arbitrary rank on protective spaces. These sheaves naturally occur in the closure of the moduli spaces of stable vector bundles. We generalize some of the techniques and results of Hartshorne [3], [4] to torsion free sheaves. Applications will be given in another paper.

Holomorphic Vector Bundles and the Geometry of ℙ n

In this section we shall establish the notation and assemble the most important facts about the cohomology of projective spaces with coefficients in an analytic coherent sheaf. Then we shall recall the definition of the Chern classes of a vector bundle and for holomorphic bundles we shall interpret them in some cases as the dual classes of appropriate submanifolds.

Stability and moduli spaces

In this section we introduce the crucial concept of stability of holomorphic vector bundles over ℙ n . We begin by collecting together several theorems about torsion-free, normal and reflexive sheaves which we shall need later. Then we define stable and semistable torsion-free sheaves in the sense of Mumford and Takemoto and compare this concept of stability with that of \(\mathcal{O}_{\mathbb{P}n} \) (1)-stability as introduced by Gieseker and Maruyama. In a final section we investigate the stability of a number of bundles with which we became acquainted in earlier sections.