Number of singular points of an annulus in $\mathbb{C}^2$

TL;DR

This paper proves that an algebraic annulus in complex two-space without self-intersections can have at most three cuspidal singularities, using advanced inequalities and bounds.

Contribution

It establishes a new upper bound on the number of cuspidal singularities for algebraic annuli in ^2, combining geometric inequalities with singularity theory.

Findings

Maximum of three cuspidal singularities for such annuli

Application of Bogomolov-Miyaoka-Yau inequality in singularity bounds

Use of Milnor number bounds to constrain singularities

Abstract

Using Bogomolov-Miyaoka-Yau inequality and a Milnor number bound we prove that any algebraic annulus $\mathbb{C}^*$ in $\mathbb{C}^2$ with no self-intersections can have at most three cuspidal singularities.