Let μ be a non-negative measure with finite mass given by $\phi \left(d{d}^c\psi \right)ⁿ$, where ψ is a bounded plurisubharmonic function with zero boundary values and $\phi \in L^q\left(\left(d{d}^c\psi \right)ⁿ\right)$, φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.

The Banach hyperbolicity and disc-convexity of complex Banach manifolds and their relations are investigated.

We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain ${\Omega }_n=\zeta \in {ℂ}^n:\varrho \left(\zeta \right)>0$, $\varrho \left(\zeta \right)=Im\left({\zeta }_1\right)-{\left|{\zeta }^{\text{'}}\right|}^2$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the ${\varrho }^{\alpha }$-Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of ${\Omega }_n$. We build an anti-holomorphic embedding of ${\Omega }_n$ in the complex projective Hilbert space $ℂ\mathbb{P}\left(H_{\alpha }^2\left({\Omega }_n\right)\right)$ and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes....

A dual space of the Tjurina algebra attached to a non-quasihomogeneous unimodal or bimodal singularity is considered. It is shown that almost every algebraic local cohomology class, belonging to the dual space, can be characterized as a solution of a holonomic system of first order differential equations.

We extend the results obtained in our previous paper, concerning quasiregular polynomials of algebraic degree two, to the case of polynomial endomorphisms of ℝ² whose algebraic degree is equal to their topological degree. We also deal with some other classes of polynomial endomorphisms extendable to ℂℙ².

We characterize intrinsically two classes of manifolds that can be properly embedded into spaces of the form ${\mathbb{P}}^N\setminus {\mathbb{P}}^{N-q}$. The first theorem is a compactification theorem for pseudoconcave manifolds that can be realized as $\overline{X}\setminus (\overline{X}\cap {\mathbb{P}}^{N-q})$ where $\overline{X}\subset {\mathbb{P}}^N$ is a projective variety. The second theorem is an embedding theorem for holomorphically convex manifolds into ${\mathbb{P}}^1\times {ℂ}^N$.

In this paper we show that a 1-convex (i.e., strongly pseudoconvex) manifold $X$, with 1- dimensional exceptional set $S$ and finitely generated second homology group $H_2(X,\mathbb{Z})$, is embeddable in ${ℂ}^m\times ℂ{\mathbb{P}}_n$ if and only if $X$ is Kähler, and this case occurs only when $S$ does not contain any effective curve which is a boundary.

The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ($C^{\infty }$) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...

The paper is concerned with the relations between real and complex topological invariants of germs of real-analytic functions. We give a formula for the Euler characteristic of the real Milnor fibres of a real-analytic germ in terms of the Milnor numbers of appropriate functions.