Fractional Hamiltonian monodromy from a Gauss–Manin monodromy

International audienceFractional Hamiltonian monodromy is a generalization of the notion of Hamiltonian monodromy, recently introduced by Nekhoroshev, Sadovskií, and Zhilinskií, C. R. Acad. Sci. Paris, Ser. 1 335, 985 2002; Ann. Henri Poincare 7, 1099 2006 for energy-momentum maps whose image has a particular type of nonisolated singularities. In this paper, we analyze the notion of fractional Hamiltonian monodromy in terms of the Gauss–Manin monodromy of a Riemann surface constructed from the energy-momentum map and associated with a loop in complex space which bypasses the line of singularities. We also prove some propositions on fractional Hamiltonian monodromy for 1:−n and m:−n resonant systems