Birational automorphisms and alpha-invariants

Complex singularity exponent is a local invariant of a holomorphic function defined by the square-integrability of fractional powers of the function. Explicitly this invariant was introduced by Alexander Varchenko. Log canonical thresholds of $\mathbb{Q}$-divisorsm were
introduced by Vyacheslav Shokurov in his work on 3-fold log flips. These numbers are algebraic counterparts of complex singularity exponents.
For a given Fano variety $X$, it useful to consider the infimum of log canonical thresholds of all effective $\mathbb{Q}$-divisors
numerically equivalent to the anticanonical divisor of $X$. This infimum is called a global log canonical threshold of $X$. This number is an algebraic counterpart of the so-called
alpha-invariant introduced by Gang Tian to prove the existence of Kaehler-Einstein metrics on some Fano manifolds. I will discuss the role played by global log canonical thresholds
in birational geometry.