Doubling chains on complements of algebraic hypersurfaces

Abstract

A doubling chart on an $n$-dimensional complex manifold $Y$ is a univalent analytic mapping $\psi:B_1\to Y$ of the unit ball in $\mathbb{C}^n$, which is extendible to the (say) four times larger concentric ball of $B_1$. A doubling covering of a compact set $G$ in $Y$ is its covering with images of doubling charts on $Y$. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney's ball coverings of a domain $W\subset {\mathbb R}^n$, introduced in [17] (compare [9]). We study doubling chains in the complement $Y=\mathbb{C}^n\setminus H$ of a complex algebraic hypersurface $H$ of degree $d$ in $\mathbb{C}^n$, and provide information on their length and other properties. Our main result is that any two points $v_1,v_2$ in a distance $\delta$ from $H$ can be joined via a doubling chain in the complement $Y=\mathbb{C}^n\setminus H$ of length at most $c_1\log (\frac{c_2}{\delta})$ with explicit constants $c_1,c_2$ depending only on $n$ and $d$. As a consequence, we obtain an upper bound on the Kobayashi distance in $Y$, and an upper bound for the constant in a doubling inequality for regular algebraic functions on $Y$. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on $Y$.