On the Geometry of Holomorphic Curves and Complex Surface

Abstract We investigate the geometry of holomorphic curves and complex surfaces from a singularity theory viewpoint. We show that with the choice of the holomorphic metric, the families of functions and mappings that measure the contact between curves or surfaces with model objects are holomorphic. This allows the application of singularity theory techniques to pick up the geometry that is invariant under translations and complex rotations.