  Hamiltonian quantization of an integral compact symplectic manifold M depends
on a choice of compatible almost complex structure J. For open sets U in the
set of compatible almost complex structures and small enough values of Planck's
constant, the Hilbert spaces of the quantization form a bundle over U with a
natural connection. In this paper we examine the dependence of the Hilbert
spaces on the choice of J, by computing the semi-classical limit of the
curvature of this connection. We also show that parallel transport provides a
link between the action of the group Symp(M) of symplectomorphisms of M and the
Schrodinger equation.
