A note on the singularities of residue currents of integrally closed ideals

Given a free resolution of an ideal $\mathfrak a$ of holomorpic functions there is an associated residue current $R$ that coincides with the classical Coleff-Herrera product if $\mathfrak a$ is a complete intersection ideal and whose annihilator ideal equals $\mathfrak a$. In the case when $\mathfrak a$ is an Artinian monomial ideal, we show that the singularities of $R$ are small in a certain sense if and only if $\mathfrak a$ is integrally closed.Comment: 9 page

Stabilization of monomial maps

A monomial (or equivariant) selfmap of a toric variety is called stable if its action on the Picard group commutes with iteration. Generalizing work of Favre to higher dimensions, we show that under suitable conditions, a monomial map can be made stable by refining the underlying fan. In general, the resulting toric variety has quotient singularities; in dimension two we give criteria for when it can be chosen smooth, as well as examples when it cannot.Comment: To appear in Michigan Math.

Residue currents and cycles of complexes of vector bundles

We give a factorization of the cycle of a bounded complex of vector bundles in terms of certain associated differential forms and residue currents. This is a generalization of previous results in the case when the complex is a locally free resolution of the structure sheaf of an analytic space and it can be seen as a generalization of the classical Poincar\'e-Lelong formula.Comment: 18 page

Residue currents and fundamental cycles

We give a factorization of the fundamental cycle of an analytic space in terms of certain differential forms and residue currents associated with a locally free resolution of its structure sheaf. Our result can be seen as a generalization of the classical Poincar\'e-Lelong formula. It is also a current version of a result by Lejeune-Jalabert, who similarly expressed the fundamental class of a Cohen-Macaulay analytic space in terms of differential forms and cohomological residues.Comment: 24 page