  The toric manifolds in question were invented by Bott and studied by
Grossberg and Karshon under the name "Bott towers". Interest in them comes from
their relation to characters of semisimple Lie groups and geometric
quantization. We offer another construction of them, introduce a family
symplectic structures as curvatures of line bundles, work out canonical
coordinates (action-angle variables), derive positivity criteria for the
curvature, and determine the range of the moment map. A character of the torus
group is introduced as an integral in the spirit of Kirillov's character
formula and shown to equal the sum over the integral points in the moment
polytope by methods from convex analysis and equivariant cohomology. The result
is compared with an explicit calculation of a path integral, based on a
definition devised for the occasion.
