On irregular Sasaki-Einstein metrics in dimension 5

Sasakian geometry can be seen as the odd-dimensional counterpart of Kaehler geometry. Indeed, a "regular" Sasakian manifold $M$ is a circle bundle over some Kaehler manifold $Z$. In this situation the Sasakian geometry of $M$ and the Kaehler geometry of $Z$ are closely related to each other. For example the problem of finding a Sasaki–Einstein metric on $M$ is equivalent to the problem of finding a Kaehler–Einstein metric on $Z$. However, in the so-called "irregular" case this approach breaks down. On the other hand, one also obtains a new tool in this situation: a torus action of higher rank. In this talk I will explain how to make use of this new tool in order to prove the existence or non-existence of irregular Sasaki–Einstein metrics on certain 5-manifolds.