The quantization of vector bundles is defined. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. (Coadjoint orbits are symplectic spaces with a transitive, semisimple symmetry group.) In preparation for the main result, the quantization of coadjoint orbits is discussed in detail. This subject should not be confused with the quanti...

To every oriented tree we associate vector bundle problems. We define semistability concepts for these vector bundle problems and establish the existence of moduli spaces. As an important application, we obtain an algebraic construction of the moduli space of holomorphic triples.

Let E 1, . . . , Ek and E be natural vector bundles defined over the category Mf m of smooth oriented m–dimensional manifolds and orientation preserving local diffeomorphisms, with m ≥ 2. Let M be an object of Mf m which is connected. We give a complete classification of all separately continuous k–linear operators D : Γc(E1M) × . . . × Γc(EkM) → Γ(EM) defined on sections with compact supports,...

Using Fujita-Griffiths method of computing metrics on Hodge bundles, we show that for every semi-ample vector bundle E on a compact complex manifold, and every positive integer k, the vector bundle SE ⊗ detE has a continuous metric with Griffiths semi-positive curvature. If E is ample, the metric can be made smooth and Griffiths positive.

We define a new symmetry for morphisms of vector bundles, called triality symmetry, and compute Chern class formulas for the degeneracy loci of such morphisms. In an appendix, we show how to canonically associate an octonion algebra bundle to any rank 2 vector bundle.

Abstract. One describes, using a detailed analysis of Atiyah–Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constru...