  In this paper we study associative algebras with a Poisson algebra structure
on the center acting by derivations on the rest of the algebra. These
structures, which we call Poisson fibred algebras, appear in the study of
quantum groups at roots of 1 and related algebras, as well as in the
representation theory of affine Lie algebras at the critical level. Poisson
fibred algebras lead to a generalization of Poisson geometry, which we develop
in the paper. We also take up the general study of noncommutative spaces which
are close to enough commutative ones so that they contain enough points to have
interesting commutative geometry. One of the most striking uses of our
noncommutative spaces is the quantum Borel-Weil-Bott Theorem for quantum sl_q
(2) at a root of unity, which comes as a calculation of the cohomology of
actual sheaves on actual topological spaces.
