\(d_ f\)-cohomology of Lagrangian foliations (Q1121577)

\(d_ f\) is the notation used for the exterior differential along the leaves of a foliation, and the \(d_ f\)-cohomology of differential forms yields resolutions for various interesting sheaves. After discussing this subject, particularly in the case of Lagrangian foliations, some simple computational examples are given. The second part of the paper discusses various obstructions which belong to the \(d_ f\)-cohomology such as: 1) The linearity obstruction for a foliation with affine leaves and, particularly, for a Lagrangian foliation. A characterization of the cotangent bundles appears if this obstruction vanishes. 2) The obstructions to the existence of an affine transversal distribution. 3) The obstruction to the existence of a symplectic form which makes a given foliation Lagrangian. A formula giving all the symplectic forms of a cotangent bundle for which the fibers form a Lagrangian foliation also results here.