Mok's characteristic varieties and the normal holonomy group

TL;DR

This paper classifies the normal holonomy groups of complex submanifolds in complex space forms, linking them to Hermitian s-representations and characteristic varieties, and characterizes submanifolds with reducible holonomy as joins of projective submanifolds.

Contribution

It completes the classification of normal holonomy groups for complex submanifolds, connecting them to Mok's characteristic varieties and describing geometric structures with reducible holonomy.

Findings

Irreducible non transitive holonomies are Hermitian s-representations.

Constructs submanifolds with prescribed holonomy groups.

Characterizes submanifolds with reducible holonomy as joins of projective submanifolds.

Abstract

In this paper we complete the study of the normal holonomy groups of complex submanifolds (non nec. complete) of Cn or CPn. We show that irreducible but non transitive normal holonomies are exactly the Hermitian s-representations of [CD09, Table 1] (see Corollary 1.1). For each one of them we construct a non necessarily complete complex submanifold whose normal holonomy is the prescribed s-representation. We also show that if the submanifold has irreducible non transitive normal holonomy then it is an open subset of the smooth part of one of the characteristic varieties studied by N. Mok in his work about rigidity of locally symmetric spaces. Finally, we prove that if the action of the normal holonomy group of a projective submanifold is reducible then the submanifold is an open subset of the smooth part of a so called join, i.e. the union of the lines joining two projective submanifolds.