A new approximation method for geodesics on the space of K\"ahler metrics using complexified symplectomorphisms and Gr\"obner Lie series

Abstract

It has been shown that the Cauchy problem for geodesics in the space of K\"ahler 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 Gr\"obner 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 their first twelve terms, calculated with the help of Mathematica. This leads to approximate geodesics which hit the boundary of the space of K\"ahler metrics in finite geodesic time. For quantum mechanical applications, one is interested also on the non-K\"ahler polarizations that one obtains by crossing the boundary of the space of K\"ahler structures. Properties of the approximate geodesics and its extensions are also studied using Mathematica.