  We introduce the concept of Bergman bundle attached to a hermitian manifold
X, assuming the manifold X to be compact - although the results are local for a
large part. The Bergman bundle is some sort of infinite dimensional very ample
Hilbert bundle whose fibers are isomorphic to the standard L${}^2$ Hardy space
on the complex unit ball; however the bundle is locally trivial only in the
real analytic category, and its complex structure is strongly twisted. We
compute the Chern curvature of the Bergman bundle, and show that it is strictly
positive. As a potential application, we investigate a long standing and still
unsolved conjecture of Siu on the invariance of plurigenera in the general
situation of polarized families of compact K{\"a}hler manifolds.
