GEOMETRIC-QUANTIZATION OF SYMPLECTIC-MANIFOLDS WITH RESPECT TO REDUCIBLE NONNEGATIVE POLARIZATIONS

The leafwise complex of a reducible non-negative polarization with val ues in the prequantum bundle on a prequantizable symplectic manifold i s studied. The cohomology groups of this complex is shown to vanish in rank less than the rank of the real part of the non-negative polariza tion. The Bohr-Sommerfeld set for a reducible non-negative polarizatio n is defined. A factorization theorem is proved for these reducible no n-negative polarizations. For compact symplectic manifolds, it is show n that the above complex has finite dimensional cohomology groups, mor eover a Lefschetz fixed point theorem and an index theorem for these n on-elliptic complexes is proved. As a corollary of the index theorem, we deduce that the cardinality of the Bohr-Sommerfeld set for any redu cible real polarization on a compact symplectic manifold is determined by the volume and the dimension of the manifold.