  In this paper we develop a theory of curvature (resp. multiplicity) invariant
for tensor products of full Fock spaces and also for tensor products of
symmetric Fock spaces. This is an attempt to find a more general framework for
these invariants and extend some of the results obtained by Arveson for the
symmetric Fock space, by the author and Kribs for the full Fock space, and by
Fang for the Hardy space over the polydisc. To prove the existence of the
curvature and its basic properties in these settings requires a new approach
based on noncommutative Berezin transforms and multivariable operator theory on
polyballs and varieties, as well as summability results for completely positive
maps. The results are presented in the more general setting of regular
polyballs.
