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In this paper we find that the asymptotic nonlinear Maslov index defined onthe universal cover of the group of all contact Hamiltonian diffeomorphisms ofthe standard 2n-1 dimensional contact sphere is a quasimorphism. Then we showour main result: Let M be standard n-1 complex projective space. We prove thatthe value of the pull back of the asymptotic nonlinear Maslov index to theuniversal cover of the group of Hamiltonian diffeomorphisms of M, restricted toelements supported in sufficiently small open subsets of M, equals the Calabiinvariant of these elements up to a multiplication by a constant.

THE NONLINEAR MASLOV INDEX AND THE CALABI HOMOMORPHISM