A Berger type normal holonomy theorem for complex submanifolds

Abstract

DOI: 10.1007/s00208-010-0597-0 Abstract. We prove Berger type theorems for the normal holonomy ©? (i.e.,the holonomy group of the normal connection) of a full complete complexsubmanifold M both of Cn and of the complex projective space CPn. Namely,(1) for Cn, if M is irreducible, then ©? acts transitively on the unit sphereof the normal space;(2) for CPn, if ©? does not act transitively, then M is the complex orbit,in the complex projective space, of the isotropy representation of anirreducible Hermitian symmetric space.The methods in the proofs rely heavily on the singular data of appropriateholonomy tubes (after lifting the submanifold to the complex Euclidean space,in the CPn case) and basic facts of complex submanifolds.