  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.
