Singularities of linear systems and boundedness of Fano varieties

We study log canonical thresholds (also called global log canonical threshold or alpha-invariant) of R-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds; that is, given a natural number d and a positive real number epsilon, the set of Fano varieties of dimension d with epsilon-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which, in particular, answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case.