Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations
Geometric quantization requires to construct a (``prequantum'') line bundle \(L\) over a symplectic manifold \((M, \omega)\) and to choose a polarization \(P\) on \(M\). An important detail is to study the cohomology groups of the leafwise complex with respect to \(P\) with values in \(L\). If \(P\) is a Kähler polarization, index and Lefschetz theorems are applicable. The author initiates a program to present similar theorems for more general polarizations. To that end he considers polarizations that satisfy certain non-negativity and reducibility conditions. For compact symplectic manifolds he can then show that the above-mentioned complex has finitedimensional cohomology groups. Moreover, a Lefschetz theorem and a fixed point theorem can be proven for this case.
scientific article; zbMATH DE number 1016974