A Berger type normal holonomy theorem for complex submanifolds

Abstract

We prove a Berger type theorem for the normal holonomy \({\Phi^\perp}\) (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space \({\mathbb{C} P^n}\). Namely, if \({\Phi^\perp}\) does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of \({\mathbb{C}^n}\) the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the \({\mathbb{C} P^n}\) case) and basic facts of complex submanifolds.