Einstein-Kähler forms, Futaki invariants and convex geometry on toric Fano varieties
The author studies the problem of classifying all smooth n-dimensional toric Fano manifolds X (Kaehler manifolds with effective almost homogeneous algebraic group action) that admits an Einstein-Kähler form \(\omega\), i.e. \(\omega\) with \(\sqrt{-1}{\bar \partial}\partial \log \det g(\omega)=\omega.\) The Fubini-Study form on \({\mathbb{P}}^ n({\mathbb{C}})\) is a typical Einstein-Kaehler form. It is not known whether or not \({\mathbb{P}}^ 2({\mathbb{C}})\) with three points blown up admits an Einstein-Kähler form. The paper gives a survey of recent progress on the mentioned problem. In particular, an explicit description of Einstein-Kähler metrics for Sakane-Koiso's examples is given.
scientific article; zbMATH DE number 4080318