Geodesic
The first main theorem on complements: from global to local
Izvestiya. Mathematics , Tome 65 (2001) no. 6, pp. 1169-1196

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The purpose of this paper is to explain and generalize the methods of [24] (see also [18] and [19]). We establish that for local Fano contractions the existence of complements can be reduced to the existence of complements for projective Fano varieties of smaller dimension. 
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     author = {Yu. G. Prokhorov and V. V. Shokurov},
     title = {The first main theorem on complements: from global to local},
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Yu. G. Prokhorov; V. V. Shokurov. The first main theorem on complements: from global to local. Izvestiya. Mathematics , Tome 65 (2001) no. 6, pp. 1169-1196. http://geodesic.mathdoc.fr/item/IM2_2001_65_6_a3/