  In this article we prove a rigidity theorem for lagrangian singularities by
studying the local cohomology of the lagrangian de Rham complex that was
introduced in math.AG/0002083. The result can be applied to show the rigidity
of all open swallowtails of dimension greater or equal than two. In the case of
lagrangian complete intersection singularities the lagrangian de Rham complex
turns out to be perverse. We also show that lagrangian complete intersection in
dimension greater than two cannot be regular in codimension one.
