A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator

Abstract

We study the complex hypersurfaces f : M (n) ! C n+1 which together with their transversal bundles have the property that around any point of M there exists a local section of the transversal bundle inducing ar-parallel anti-complex shape opera- tor S. We give a class of examples of such hypersurfaces with an arbitrary rank of S from 1 to (n=2) and show that every such hypersurface with positive type number and S6 0 is locally of this kind, modulo an ane isomorphism of C n+1 .