A new approximation method for geodesics on the space of Kähler metrics

The Cauchy problem for (real analytic) geodesics in the space of Kähler metrics with a fixed cohomology class on a compact complex manifold M can be effectively reduced to the problem of finding the flow of a related Hamiltonian vector field X_H , followed by analytic continuation of the time to complex time. This opens the possibility of expressing the geodesic ω _t in terms of Gröbner Lie series of the form exp (√(-1) tX_H)(f) , for local holomorphic functions f . The main goal of this paper is to use truncated Lie series as a new way of constructing approximate solutions to the geodesic equation. For the case of an elliptic curve and H a certain Morse function squared, we approximate the relevant Lie series by the first twelve terms, calculated with the help of Mathematica. This leads to approximate geodesics which hit the boundary of the space of Kähler metrics in finite geodesic time. For quantum mechanical applications, one is interested also on the non-Kähler polarizations that one obtains by crossing the boundary of the space of Kähler structures.