Log canonical thresholds of smooth Fano threefolds

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 effective Q-divisors on normal algebraic varieties are algebraic counterparts of complex singularity exponents. For a Fano variety, these invariants have global analogues. In the former case, it is so-called α-invariant of Tian. In the latter case, it is a global log canonical threshold of the Fano variety, which is the infimum of log canonical thresholds of all effective Q-divisors numerically equivalent to the anticanonical divisor. The appendix to this paper contains the proof that the global log canonical threshold of a smooth Fano variety coincides with its α-invariant. The purpose of this paper is to compute global log canonical thresholds of smooth Fano threefolds (altogether there are 105 deformation families of such threefolds). We compute global log canonical thresholds of every smooth threefold in 64 deformation families, and we compute global log canonical thresholds of general threefold in 20 deformation families. For 14