Hilbert modules and complex analytic fibre spaces

We describe the ‘‘eigenbundle’’ (localization bundle) of Hilbert modules over bounded symmetric domains and show that the fibres are closely related to Kähler geometry of certain flag manifolds defined in terms of Jordan algebras. This gives an intriguing relationship between Hilbert space operators, coherent module sheaves and geometry of Jordan manifolds. The resulting complex-analytic fibre spaces are stratified of arbitrary depth r .