Abstract
We show Shokurov's complements conjecture holds for surfaces. More precisely, we show the existence of (ϵ,n)-complements for (ϵ,R)-complementary surface pairs when the coefficients of boundaries belong to a DCC set. For higher dimensional varieties, we show that Shokurov's complements conjecture implies the global index conjecture, Shokurov's index conjecture, and Shokurov's conjecture on Fano fibrations.
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