Residue currents of holomorphic morphisms

Mats Andersson

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arXiv · math/0511241

Residue currents of holomorphic morphisms

Mats Andersson

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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.

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Mats Andersson. 2005-11-09. Residue currents of holomorphic morphisms. https://arxiv.org/abs/math/0511241

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