Boundedness of Spherical Fano Varieties

Classically, G. Fano proved that the family of (smooth, anticanonically embedded) Fano 3-dimensional varieties is bounded, and moreover provided their classification, later completed by V.A. Iskovskikh, S. Mukai and S. Mori. For singular Fano varieties with log terminal singularities, there are two basic boundedness conjectures: Index Boundedness and the much stronger ǫ-lt Boundedness. The ǫ-lt Boundedness was known only in two cases: in dimension 2 [Ale94] and for toric varieties [BB93]. In this paper we prove it for a significantly less “elementary” class, that of spherical varieties. In addition to an argument adapted from the toric case, the proof contains quite a few new twists. In Section 5, we introduce a new invariant of a spherical subgroup H in a reductive group G which measures how nice an equivariant Fano compactification of G/H there exists.