The holomorphic Gauss Parametrization

TL;DR

This paper provides a local parametrization of holomorphic hypersurfaces in complex spaces with constant nullity, extending the classical Gauss parametrization from real to complex geometry with applications.

Contribution

It introduces a complex analogue of the Gauss parametrization for holomorphic hypersurfaces with constant index of nullity, offering new insights and tools in complex differential geometry.

Findings

Derived a local parametric description of holomorphic hypersurfaces

Extended the classical Gauss parametrization to complex hypersurfaces

Presented applications in complex Euclidean and projective spaces

Abstract

We give a local parametric description of all holomorphic hypersurfaces in complex Euclidean and projective spaces with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization.