Global log-canonical thresholds for Fano varieties

Ivan Cheltsov (Edinburgh)

Complex singularity exponent is a local invariant of a holomorphic function defined by the square-integrability of fractional powers of the function.
Log canonical thresholds of Q-divisors 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 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 and show how to compute this number in some cases.

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