Residue Currents of Holomorphic Morphisms

Given a generically surjective holomorphic vector bundle morphism f : E → Q, E and Q Hermitian bundles, we construct a current R 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 Rφ = 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 H-corona problem.