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

The Cauchy problem for (real analytic) geodesics in the space of Kahler 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 $$\omega _t$$ in terms of Grobner Lie series of the form $$\exp (\sqrt{-1} \, tX_H)(f)$$, 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 Kahler metrics in finite geodesic time. For quantum mechanical applications, one is interested also on the non-Kahler polarizations that one obtains by crossing the boundary of the space of Kahler structures.