On Stein Neighborhood Bases, Parametric Splitting and Bumpings of Finite Type Domains

Abstract

In this thesis we investigate various topics within the field of several complex variables. The following four papers constitute the scientific contribution of the thesis:

Paper 1: On Newton Diagrams of Plurisubharmonic Polynomials (Joint with Berit Stensønes) In this paper we construct a polynomial whose properties rule out a prima facie approach to bumping in $\mathbb{C}^3$.

Paper 2: A Parameter Version of Forstnerič's Splitting Lemma In this paper we show that the biholomorphic maps obtained from Forstnerič's splitting can be chosen to depend continuously on a parameter, provided the original maps and domains do.

Paper 3: A Homogeneous Function Constant Along The Leaves Of A Foliation In this paper we construct a real-valued function that is both homogeneous and constant along the leaves of a foliation. We also discuss how this relates to the problem of ``bumping out'' certain pseudoconvex domains of finite type.

Paper 4: An Example on $s$-H-Convexity in $\mathbb{C}^2$ (Joint with Berit Stensønes) In this paper we construct a bounded (pseudoconvex) domain $\Omega$ in $\mathbb{C}^2$ with boundary of class $\mathcal{C}^{1,1}$, such that $\overline{\Omega}$ has a Stein neighborhood basis, but is not $s$-H-convex for any real number $s\geq{1}$.