1 1 Ju n 20 03 Holomorphic rank - 2 vector bundles on non - Kähler elliptic surfaces

The existence problem for vector bundles on a smooth compact complex surface consists in determining which topological complex vector bundles admit holomorphic structures. For projective surfaces, Schwarzenberger proved that a topological complex vector bundle admits a holomorphic (algebraic) structure if and only if its first Chern class belongs to the Neron-Severi group of the surface. In contrast, for non-projective surfaces there is only a necessary condition for the existence problem (the discriminant of the vector bundles must be positive) and the difficulty of the problem resides in the lack of a general method for constructing non-filtrable vector bundles. In this paper, we close the existence problem in the rank-2 case, by giving necessary and sufficient conditions for the existence of holomorphic rank-2 vector bundles on non-Kähler elliptic surfaces. Such a surface X admits a holomorphic fibration, whose general fibre is an elliptic curve, and one can construct two objects that encode the holomorphic type of a rank-2 vector bundle on X over each smooth fibre of the fibration: an effective divisor on a ruled surface (the quotient the relative Jacobian of X by an involution), called the graph of the bundle, and its double cover on the relative Jacobian, called the spectral cover of the bundle. The proofs are based on a careful study of these two objects. 2000 Mathematics Subject Classification. Primary: 14J60; Secondary: 14D22, 14F05, 14J27, 32J15.