  For a local system and a function on a smooth complex algebraic variety, we
give a proof of a conjecture of M. Kontsevich on a formula for the vanishing
cycles using the twisted de Rham complex of the formal microlocalization of the
corresponding locally free sheaf with integrable connection having regular
singularity at infinity. We also prove its local version, which may be viewed
as a natural generalization of a result of E. Brieskorn in the isolated
singularity case. We then generalize these to the case of the de Rham complexes
of regular holonomic D-modules where we have to use the tensor product with a
certain sheaf of formal microlocal differential operators instead of the formal
completion.
