The sheaf SF(L) of germs of sections of a line bundle L on a manifold X covariant constant with respect to a flat connection defined for vectors in a complex subbundle F of the tangent bundle has a resolution by differential forms defined on F with values in L provided F satisfies the integrability conditions of the complex Frobenius theorem. This includes as special cases the de Rham and Dolbeault resolutions. If there is a free S' action on X whose generator is tangent to F, let Y be the subset of X where parallel transport in L around the Si orbits is trivial. It is shown that the cohomology groups of SF(L) depend only on the restriction of SF(L) to Y. This is used to obtain a spectrum for a periodic Hamiltonian flow with generator in a polarisation. In the case of a classical harmonic oscillator this spectrum is found to be the same as that of the quantum mechanical oscillator.