  We introduce a new class of pseudodifferential operators, called Heisenberg
semiclassical pseudodifferential operators, to study the space of sections of a
power of a line bundle on a compact manifold, in the limit where the power is
large. This class contains the Bochner Laplacian associated with a connection
of the line bundle, and when the curvature is nondegenerate, its resolvent and
some associated spectral projections, including generalized Bergman kernels.
