Doubling chains on complements of algebraic hypersurfaces

TL;DR

This paper investigates the properties of doubling chains in the complement of algebraic hypersurfaces in complex space, providing bounds on their length, implications for Kobayashi distance, and doubling inequalities for algebraic functions.

Contribution

It introduces explicit bounds on the length of doubling chains in hypersurface complements, linking geometric properties to function theory and complex analysis.

Findings

Doubling chain length is at most logarithmic in inverse distance to the hypersurface.

Upper bounds on Kobayashi distance in hypersurface complements.

Lower bounds for doubling chain length based on specific functions' doubling constants.

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 $Y$.