Boundedness of Log Terminal Fano Pairs of Bounded Index

A fundamental problem in classifying varieties is to determine natural subsets whose moduli is bounded. The difficulty of this problem is partially measured by the behaviour of the canonical class of the variety. Three extreme cases are especially of interest: either the canonical class is ample, that is the variety is of general type, or the canonical class is trivial, for example the variety is an abelian variety or Calabi-Yau, or minus the canonical class is ample, that is the variety is Fano. Let us first suppose that the variety is smooth. It is well known that varieties of general type do not form a bounded family unless one fixes some invariants. For example a curve of genus g is of general type iff g ≥ 2. Typically even varieties with trivial canonical class do not form bounded families; for example the moduli space of abelian varieties has infinitely many components corresponding to the type of polarisation. Even though the same is true of K3 surfaces, to the best of the author’s knowledge it is not known whether the family of Calabi-Yau threefolds is bounded. The picture for smooth Fano varieties however is much brighter; indeed there is a complete classification up to dimension three. Such a classification in general seem unfeasible, but on the other hand it was proved by Kollár, Miyaoka and Mori [15] that smooth Fano varieties of fixed dimension form a bounded family. One reason for focusing on the three extreme cases is that roughly speaking it is expected that up to birational equivalence, any variety is either of general type or admits a fibration to another variety, whose fibres are either Fano or have trivial canonical class. However one can only achieve this birational factorisation if one allows singularities. For example for a surface of general type one needs to contract all the −2-curves on the surface to make the canonical divisor ample. Unfortunately it is no longer true that singular Fano varieties form a bounded family. For example cones over a rational normal curve of degree d form an unbounded family of Fano surfaces; to form a bounded family one needs to impose some restrictions on the singularities. One natural restriction to impose is that