The presence of focus-focus singularities in semi-toric integrables Hamiltonian systems is one of the reasons why there cannot exist global Action-Angle coordinates on such systems. At focus-focus critical points, the LiouvilleArnold-Mineur theorem does not apply. In particular, the affine structure of the image of the moment map around has non-trivial monodromy. In this article, we establish that the singular behaviour and the multi-vauedness of the Action integrals is given by a complex logarithm. This extends a previous result by San Vũ Ngo . c to any dimension. We also calculate the monodromy matrix for these systems.
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