These theorems generalize to complete intersections. The base of the fibration in this case is the intersection of the maximal dual balanced polyhedra for the corresponding hypersurfaces (we have to choose them in a mutually general position).

From a different point of view the base is dual to a maximal mixed lattice subdivision of the Newton polyhedra of the participating equations. The primitive pieces for complete intersections are products of the primitive pieces for hypersurfaces. Sturmfels’ generalization [14] of the patchworking technique allows to produce in this case the Lagrangian lifts of the base cycles.

This generalization will be the subject of a future paper.