Bounded klt complements for Fano contractions

For fixed dimension d and positive rational ϵ, we prove that every ϵ-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits, near each closed base point, a klt complement of index bounded only by d and ϵ. Over ℂ, we also obtain monotone klt complements for Fano type pairs with nef anti-log-canonical divisor and coefficients in a fixed finite rational set. This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.

We prove the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over ℂ. For an ϵ-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, a Cartier divisor through any prescribed base point can be chosen with pullback log canonical threshold bounded below in terms of the dimension and ϵ alone.