The exponential map of the complexification of the group of analytic Hamiltonian diffeomorphisms

Let (M, omega, J) be a Kahler manifold and 1C = Ham(M, omega) its group of Hamiltonian symplectomorphisms. Complexifications of 1C have been introduced by Semmes and then Donaldson which are not groups, only "formal Lie group" in a precise sense. However, it still makes sense to talk about the exponential map in the complexification. In this note we give a geometric construction of the exponential map (for small time), in case the initial data are real-analytic. (A more general analytic description has been given by Semmes.) The construction is motivated by, but does not use, semiclassical analysis and quantum coherent states. We use this geometric construction to solve the equations of motion in several basic examples and recapture a case already considered in the physics community where the quantum analogue of our system is explicitly solvable, showing a potential relation to non-Hermitian quantum mechanics. Finally, in the case of geodesics in the space of Kahler metrics on a Kahler manifold originally studied variously by Mabuchi, Semmes and Donaldson, we derive an infinitesimal obstruction to the completeness of Mabuchi geodesic rays in the space of smooth Kahler metrics.