Residue Currents and their Annihilator Ideals

Doctoral thesis, 2007

This thesis presents results in multidimensional residue theory. From a generically exact complex of locally free analytic sheaves $\mathcal C$ we construct a vector valued residue current $R^\mathcal C$, which in a sense measures the exactness of $\mathcal C$. If $\mathcal C$ is a locally free resolution of the ideal (sheaf) $J$ the annihilator ideal of $R^\mathcal C$ is precisely $J$. This generalizes the Duality Theorem for Coleff-Herrera products of complete intersection ideals and can be used to extend several results, previously known for complete intersections. We compute $R^\mathcal C$ explicitly if $\mathcal C$ is a so called cellular resolution of an Artinian monomial ideal $J$, and relate the structure of $R^\mathcal C$ to irreducible decompositions of $J$. If $\mathcal C$ is the Koszul complex associated with a set of generators $f$ of the ideal $J$ the entries of $R^\mathcal C$ are the residue currents of Bochner-Martinelli type of $f$, which were introduced by Passare, Tsikh and Yger. We compute these in case $J$ is an Artinian monomial ideal and conclude that the corresponding annihilator ideal is strictly included in $J$, unless $J$ is a complete intersection. We also define products of residue currents of Bochner-Martinelli type, generalizing the classical Coleff-Herrera product, and show that if $f$ defines a complete intersection the product of the residue currents of Bochner-Martinelli type of subtuples of $f$ coincides with the residue current of Bochner-Martinelli type of $f$.