Geodesics in the space of Kähler metrics

Abstract

Let (X,ω) be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set H0 of Kähler forms cohomologous to ω has the natural structure of an infinite-dimensional Riemannian manifold. We address the question of whether any two points in H0 can be connected by a smooth geodesic and show that the answer, in general, is “no.”