Abstract. Given a compact Kähler manifold (X, ω 0 ), according to Mabuchi, the set H 0 of Kähler forms cohomologous to ω 0 has the natural structure of an infinite-dimensional Riemannian manifold. We address the question whether points in H 0 can be joined by a geodesic, and strengthening the finding of [LV], we show that this cannot always be done even with a certain type of generalized geodesics. As in [LV], the result is obtained through the analysis of a Monge-Ampère equation.
We show how ideas from [4] and [5] to prove the Suita conjecture can be adapted to give a proof of the Ohsawa-Takegoshi extension theorem with sharp estimates.
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