2005/11/09 by Mats Andersson, Andersson, Mats
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #math.CV
paper · pdf · doi:10.48550/arxiv.math/0511241
arxiv created 2005/11/09 · openalex publication_date 2005/11/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a generically surjective holomorphic vector bundle morphism f\colon E→ Q, E and Q Hermitian bundles, we construct a current Rf with values in \Hom(Q,H), where H is a certain derived bundle, and with support on the set Z where f is not surjective. The main property is that if ϕ is a holomorphic section of Q, and Rfϕ=0, then locally fψ=ϕ has a holomorphic solution ψ. In the generic case also the converse holds. This gives a generalization of the corresponding theorem for a complete intersection, due to Dickenstein-Sessa and Passare. We also present results for polynomial mappings, related to M Noether's theorem and the effective Nullstellensatz. The construction of the current is based on a generalization of the Koszul complex. By means of this complex one can also obtain new global estimates of solutions to fψ=ϕ, and as an example we give new results related to the Hp-corona problem.