The aim of this paper is to study the Łojasiewicz exponent of c-holomorphic mappings. After introducing an order of flatness for c-holomorphic mappings we give an estimate of the Łojasiewicz exponent in the case of isolated zero, which is a generalization of the one given by Płoski and earlier by Chądzyński for two variables.

We find a relation between the vanishing of a globally defined residue current on ${\mathbb{P}}^n$ and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther’s theorem, for which we also obtain a generalization. Furthermore there are some connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions.

In this paper a new class of multi-valued mappings (multi-morphisms) is defined as a version of a strongly admissible mapping, and its properties and applications are presented.

Let $\Omega$ be a complex analytic manifold of dimension $n$ with a hermitian metric and $C^{\infty }$ boundary, and let $\square =\overline{\partial }{\overline{\partial }}^{*}+{\overline{\partial }}^{*}\overline{\partial }$ be the self-adjoint $\overline{\partial }$-Neumann operator on the space $L_{0,q}^2\left(\Omega \right)$ of forms of type $(0,q)$. If the Levi form of $\partial \Omega$ has everywhere at least $n-q$ positive or at least $q+1$ negative eigenvalues, it is well known that $\mathrm{Ker}$$\square$ has finite dimension and that the range of $\square$ is the orthogonal complement. In...

We consider the (characteristic and non-characteristic) Cauchy problem for a system of constant coefficients partial differential equations with initial data on an affine subspace of arbitrary codimension. We show that evolution is equivalent to the validity of a principle on the complex characteristic variety and we study the relationship of this condition with the one introduced by Hörmander in the case of scalar operators and initial data on a hypersurface.

We show that the projections of the pluripolar hull of the graph of an analytic function in a subdomain of the complex plane are open in the fine topology.

We consider the solution operator $S{ℱ}_{\mu ,(p,q)}\to L^2{\left(\mu \right)}_{(p,q)}$ to the $\overline{\partial }$-operator restricted to forms with coefficients in ${ℱ}_{\mu }=\left\{ff\text{is}\text{entire}\text{and}{\int }_{{ℂ}^n}{\left|f\left(z\right)\right|}^2\mathrm{d}\mu \left(z\right)<\infty \right\}$. Here ${ℱ}_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in ${ℱ}_{\mu }$, $L^2\left(\mu \right)$ is the corresponding $L^2$-space and $\mu$ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\overline{\partial }$. This solution operator will have the property $Sv\perp {ℱ}_{(p,q)}\forall v\in {ℱ}_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators...

The Serre problem is solved for fiber bundles whose fibers are two-dimensional pseudoconvex hyperbolic Reinhardt domains.