2003/06/11 by Vasile Brinzanescu, Vasile Brînzănescu, Brinzanescu, Vasile +2 · 1 citation
Mathematics · #14D22 #14F05 #14J27 #14J60 #32J15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #math.AG #math.CV #msc:14D22 #msc:14F05 #msc:14J27 #msc:14J60 #msc:32J15
paper · pdf · doi:10.48550/arxiv.math/0306191
15 pages, shortened version, corrections were made
openalex publication_date 2003/06/11 · arxiv created 2003/09/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-K" ahler elliptic surfaces.