Holomorphic vector bundles on non-algebraic surfaces

Andrei Teleman (1) , Matei Toma (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 Universität Osnabrück - Osnabrück University

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces.

Mots clés: Complex surface; Holomorphic bundle; Vector bundle; Non algebraic surface; Donaldson invariants

Domaines: Variables complexes [math.CV]; Topologie géométrique [math.GT]; Géométrie différentielle [math.DG]

Andrei Teleman, Matei Toma. Holomorphic vector bundles on non-algebraic surfaces. Comptes Rendus. Mathématique, 2002, 334 (5), pp.383-388. ⟨10.1016/S1631-073X(02)02278-1⟩. ⟨hal-00881747⟩