Fano $T$-varieties of complexity one and the existence of Kaehler-Einstein metrics on them

We address the question of the existence of Einstein-Kahler metrics on Fano varieties. Up to now there exist explicit criteria only for special classes of Fano varieties. For example, toric Fano varieties admit such metric if and only if the barycenter of the corresponding polytope coincides with the origin. Smooth del Pezzo surfaces are Kahler-Einstein if and only if their automorphism group is reductive. In this talk we look for the case of varieties admitting a torus action of complexity one. These varieties admit an explicit combinatorial description, which was introduced by Timashev and Altmann/Hausen independently. We give a sufficient criterion for the existence of a Kaehler-Einstein metric using this description and formulate some open problems in this context.