Complex plane curves in the ball
This paper is devoted to the study of topological properties of complex curves in the standard ball in \({\mathbb{C}}^ 2\). The author enlarges the already quite developed theory by generalizing many basic tools (such as the Maslov-Arnold index for Legendre curves in the sphere, the Bennequin invariant, the Rokhlin quadratic form, and the classical Plücker formula for plane curves), and by applying this to the various possible positions of a complex curve in the standard ball. His main result is a statement on the monodromy of smooth families of elliptic curves in the standard ball in \({\mathbb{C}}^ 2\), parametrized by the circle \(S^ 1\). It says that if the boundaries of the fibres of such a family are contained in the 3-sphere \(S^ 3\) and form a self-isotopy of a knot with trivial Arf invariant, then the monodromy of the fibres has a non-trivial integer invariant class. The author's approach can also be applied to the study of the singularities of complex plane curves in the standard ball. In this special situation, the author's contribution to the topological theory of curve singularities consists in a (local) result stating that certain degenerations of plane complex singularities do not occur, although the general Milnor theory would not exclude them. The whole treatise shows how powerful and far-reaching the author's topological analysis of plane curves in a ball is, in particular compared to the general theory of complex plane curves.
topological properties of complex curves in the standard ball
monodromy of smooth families of elliptic curves
scientific article; zbMATH DE number 4110832
scientific article