Lemma 14 ([An unbounded-index obstruction]). Let F_i be projective of positive bounded dimension and let G_i be ample -Cartier rational divisors such that (F_i,R(G_i)) is klt and -(K_Fi +R(G_i))∼_IiGi, I_i>0, I_i⟶∞. Let C_i be the …

The contraction exists by semiampleness of the infinity divisor. One can also construct it by the finite fibre-power map to the ordinary compactified bundle for a very ample Cartier multiple of G_i , contracting its zero section and then Stein factorizing. The bundle identity gives K_Y+(1-I_i)E_0+(1+I_i)D∼_ 0. The pair with subboundary (1-I_i)E_0 is sub-klt by ; the discrepancy of the contracted zero section is I_i>0 . This proves the assertions about the cone and its canonical divisor. Near infinity the discrepancy formula reads, for u=v(D)>0 , A_C(v)=A_ F,R(G) (w)+u+def(v), u≡w(G). If held, apply Lemma 1 ([Comparison along long parameter intervals]) over a point to the nonincreasing family with discrepancies A_C-tv(D) , boundary tD , and nef part (I_i+1-t)D on a range tending to infinity. The phase-free ratio hypothesis is exactly. The comparison lemma would make this family klt on every fixed bounded parameter range, whereas at t=1 the prime D has discrepancy zero. This is a contradiction.