这篇未发表的论文是发表在《复分析和数字几何》的同名文章的副本（通过第5节的发展和微小的更正完成），来自Kiselmanfest的论文集，2006年，Acta Universitatis Upsaliensis，C。Organisation och Historia，86，Uppsala University Library（2009），147-157.
英文摘要：
This unpublished paper is a copy (completed by a development of section 5 and by minor corrections) of the article with the same title published in: Complex Analysis and Digital Geometry, Proceedings from the Kiselmanfest, 2006, Acta Universitatis Upsaliensis, C. Organisation och Historia,86, Uppsala University Library (2009), 147-157.
We study residue currents of the Bochner--Martinelli type using their relationship with Mellin transforms of residue integrals. We present the structure formula for residue currents associated with monomial mappings: they admit representations as sums of products of residue currents, principal value currents and hypergeometric functions.
Using methods from the theory of residue currents we provide asymptotic expansions of certain divergent integrals on complex manifolds. We express the coefficients in these expansions with the conjugate Dolbeault residue, introduced by Felder and Kazhdan, and define a new residue which we call the Aeppli residue.
For a Cohen-Macaulay ideal of holomorphic functions, we construct by elementary means residue currents whose annihilator is precisely the given ideal. We give two proofs that the currents have the prescribed annihilator, one using the theory of linkage, and another using an explicit division formula involving these residue currents to express the ideal membership.
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.
Given two ideals $\mathcal{I}$ and $\mathcal{J}$ of holomorphic functions such that $\mathcal{I} \subseteq \mathcal{J}$, we describe a comparison formula relating the Andersson-Wulcan currents of $\mathcal{I}$ and $\mathcal{J}$. More generally, this comparison formula holds for residue currents associated to two generically exact Hermitian complexes together with a morphism between the complexes. One application of the comparison formula is a generalization of the transformation law for Coleff-Herrera products to Andersson-Wulcan currents of Cohen-Macaulay ideals. We also use it to give an ana…
We show that a global holomorphic section of $\mathscr{O}(d)$ restricted to a closed complex subspace $X \subset \mathbb{P}^n$ has an interpolant if and only if it satisfies a set of moment conditions that involves a residue current associated with a locally free resolution of $\mathscr{O}_X$. When $X$ is a finite set of points in $\mathbb{C}^n \subset \mathbb{P}^n$ this can be interpreted as a set of linear conditions that a function on $X$ has to satisfy in order to have a polynomial interpolant of degree at most $d$.
Given an ideal $\mathcal{J}$ on a complex manifold, Andersson and Wulcan constructed a current $R^\mathcal{J}$ such that the annihilator of $R^\mathcal{J}$ is $\mathcal{J}$, generalizing the duality theorem for Coleff-Herrera products. We describe a way to generalize this construction to ideals on singular varieties.
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 Poincaré-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.
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 Poincaré-Lelong formula.
In this work we prove a Brunella-Khanedani-Suwa variational type residue theorem for currents invariant by holomorphic foliations. As a consequence, we give conditions for the leaves of a singular holomorphic foliation to accumulate in the intersection of the singular set of the foliation with the support of an invariant current.
We introduce the ring of partial differential operators with constant coefficients and commensurate time lags (we use the terminology D$Δ$ operators from now) initially defined by H. Glüsing-Lürßen for ordinary $DΔ$ operators and investigate its cohomological properties. Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $DΔ$ equations.
