Global log canonical thresholds of Fano varieties and their applications

Abstract: A global log-canonical threshold of a Fano variety X is a numerical invariant that measures, roughly speaking, how singular a divisor D on X can be provided that the degree of D (whatever it means) is comparatively small. It is worth mentioning that the first definition of this invariant was given by G.Tian in the analytical settings, but there also exists a purely algebro-geometrical description. Global log-canonical thresholds have various applications, in particular, to birational geometry (birational rigidity and related questions) and differential geometry (existence of Kahler--Einstein metrics). I will survey the definitions and main properties of this invariant and give examples of typical computations.