  Given Y a non-compact manifold or orbifold, we define a natural subspace of
the cohomology of Y called the narrow cohomology. We show that despite Y being
non-compact, there is a well-defined and non-degenerate pairing on this
subspace. The narrow cohomology proves useful for the study of genus zero
Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford
stack, one can define a quantum D-module on the narrow cohomology of Y. This
yields a new formulation of quantum Serre duality.
