Log canonical thresholds of smooth Fano threefolds

I. A. Cheltsov ; K. A. Shramov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 63 (2008) no. 5, pp. 859-958

Résumé

The complex singularity exponent is a local invariant of a holomorphic function determined by the integrability of fractional powers of the function. The log canonical thresholds of effective $\mathbb{Q}$-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 the so-called $\alpha$-invariant of Tian; in the latter case, it is the global log canonical threshold of the Fano variety, which is the infimum of log canonical thresholds of all effective $\mathbb{Q}$-divisors numerically equivalent to the anticanonical divisor. An appendix to this paper contains a proof that the global log canonical threshold of a smooth Fano variety coincides with its $\alpha$-invariant of Tian. The purpose of the paper is to compute the global log canonical thresholds of smooth Fano threefolds (altogether, there are 105 deformation families of such threefolds). The global log canonical thresholds are computed for every smooth threefold in 64 deformation families, and the global log canonical thresholds are computed for a general threefold in 20 deformation families. Some bounds for the global log canonical thresholds are computed for 14 deformation families. Appendix A is due to J.-P. Demailly.