  We study the leading term of the holonomy map of a perturbed plane polynomial
Hamiltonian foliation. The non-vanishing of this term implies the
non-persistence of the corresponding Hamiltonian identity cycle. We prove that
this does happen for generic perturbations and cycles, as well for cycles which
are commutators in Hamiltonian foliations of degree two. Our approach relies on
the Chen's theory of iterated path integrals which we briefly resume.
