A Berger type normal holonomy theorem for complex submanifolds

TL;DR

This paper establishes a Berger type theorem for the normal holonomy groups of complex submanifolds in projective and Euclidean spaces, characterizing their actions and geometric structures.

Contribution

It extends Berger's theorem to the normal holonomy of complex submanifolds, linking non-transitive cases to complex orbits of Hermitian symmetric spaces.

Findings

Normal holonomy acts transitively on the sphere in Euclidean space.

Non-transitive normal holonomy corresponds to complex orbits of symmetric spaces.

Normal holonomy is generic for irreducible submanifolds in Euclidean space.

Abstract

We prove a Berger type theorem for the normal holonomy group (i.e., the holonomy group of the normal connection) of a full complete complex submanifold of the complex projective space. Namely, if the normal holonomy does not act transitively, then the submanifold 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 the complex Euclidean space 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 projective case) and basic facts of complex submanifolds.