On the geometry of holomorphic curves and complex surface

TL;DR

This paper explores the geometry of holomorphic curves and complex surfaces using singularity theory, introducing new invariants like $C$-curvature and $C$-torsion, and providing geometric insights into complexified contact families.

Contribution

It develops a framework to analyze holomorphic curves and surfaces via singularity theory, defining new geometric invariants and linking complex contact families to classical geometric concepts.

Findings

Introduction of $C$-curvature and $C$-torsion as geometric invariants.

Application of singularity theory to complexified contact families.

Establishment of geometric meaning for complex contact families in $ ext{R}^3$.

Abstract

We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that measure the contact between curves or surfaces and model objects become holomorphic. This allows the application of singularity theory, yielding analogues of classical results from the real case. Our approach enables the definition of geometric invariants of curves, which we call the $C$-curvature and $C$-torsion, as well as surface invariants such as the $C$-principal curvature and $C$-Gaussian curvature. It also gives geometric meaning to the complexification of of the families measuring contact of analytic surfaces in $\mathbb R^3$ with lines, planes and spheres.