Residue currents of holomorphic morphisms

Abstract

Given a generically surjective holomorphic vector bundle morphism $f\colon E\to Q$, $E$ and $Q$ Hermitian bundles, we construct a current $R^f$ 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 $R^fϕ=0$, 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 $H^p$-corona problem.