Thresholds, valuations, and K-stability

I will present joint work with Harold Blum, in which we se valuations to study the singularities of the linearseries defined by a big line bundle L on a smooth complex projective variety X. More precisely, we study the stability threshold of L, a generalization of an invariant introduced by Fujita and Odaka, as well as the more commonly known ``alpha invariant" of L. When X is a Fano manifold, we show that the stability threshold detects K-(semi)stability of X. As aconsequence, the existence of a Kaehler-Einstein metric on X can be detected by the singularities of anticanonical Q-divisors.