The membership problem for polynomial ideals in terms of residue currents

We find a relation between the vanishing of a globally defined residue current on $\P^n$ and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max N\"other's theorem, and we also obtain a generalization of that theorem. There are also connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions

Residue currents of holomorphic morphisms

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 $\phi$ is a holomorphic section of $Q$, and $R^f\phi=0$, then locally $f\psi=\phi$ has a holomorphic solution $\psi$. 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 $f\psi=\phi$, and as an example we give new results related to the $H^p$-corona problem

Integral representation with weights II, division and interpolation

Let $f$ be a $r\times m$-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic $\psi$ such that $\phi=f\psi$, provided that $\phi$ is holomorphic and annihilates a certain residue current with support on the set where $f$ is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case $r=1$

A residue criterion for strong holomorphicity

We give a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety. This generalizes a result by A Tsikh for the case of a reduced complete intersection

Global Koppelman formulas on (singular) projective varieties

Let i\colon X\to \Pk^N be a projective manifold of dimension $n$ embedded in projective space \Pk^N, and let $L$ be the pull-back to $X$ of the line bundle \Ok_{\Pk^N}(1). We construct global explicit Koppelman formulas on $X$ for smooth $(0,*)$-forms with values in $L^s$ for any $s$. %The formulas are intrinsic on $X$. The same construction works for singular, even non-reduced, $X$ of pure dimension, if the sheaves of smooth forms are replaced by suitable sheaves \A_X^* of $(0,*)$-currents with mild singularities at $X_{sing}$. In particular, if s\ge \reg X -1, where \reg X is the Castelnuovo-Mumford regularity, we get an explicit %%% representation of the well-known vanishing of $H^{0,q}(X, L^{s-q})$, $q\ge 1$. Also some other applications are indicated

The ∂ˉ\bar{\partial}-equation on a non-reduced analytic space

Let $X$ be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of $\overline{\partial}$-equation on $X$ and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves $\mathcal{A}_X^q$ of $(0,q)$-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf $\mathscr{O}_X$. Our construction is based on intrinsic semi-global Koppelman formulas on $X$.Comment: v2: Some changes from the review proces