On the cohomology of a polarization which contains the generators of a free torus action

Abstract

In [Trans. Am. Math. Soc. 230 (1977) 235], Rawnsley studies a strongly integrable subtangent bundle F on a smooth manifold M which contains the generators of a free circle action on M and a line bundle L→ M with a flat F-connection ∇. It is proved that the cohomology groups of the sheaf S F of sections of L which are covariantly constant along F can be injectively mapped in the cohomology groups of the restriction of S F on the Bohr–Sommerfeld set Y of the action on M. In the present paper, we discuss similar results for free torus actions as well as some consequences in the context of the geometric quantization of a symplectic manifold with a Hamiltonian action of a torus T k .