Submanifold and holonomy (II)

After surveying, as an introduction, some results related to submanifolds and holonomy we will speak about some joint work with Sergio Console and Antonio Di Scala. Namely, we proved a Berger type theorem for the normal holonomy of complex submanifolds of the complex projective space (also for the complex Euclidean space). Namely, for a full and complete complex projective submanifold M, are equivalent:

(i) The normal holonomy is not transitive (i.e. different from U(n), since it is an s-representation).
(ii) M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible hermitian symmetric space.
(iii) M is extrinsic symmetric (last two equivalences are well known).

The methods in the proof rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space) and complex geometry.