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Corollary 8. The canonical orders of the noncanonical varieties in Lemma 6 are uniformly bounded.

Lemma 7. Let u:W→X be a projective birational morphism from a smooth projective fourfold to a normal projective variety. A pseudo-effective u -exceptional real divisor on W is effective.

Lemma 6. Suppose that a projective klt fourfold X , with KX∼_ 0 , has a finite quasi-\'etale cover P→X in which P is one four-dimensional irreducible Calabi--Yau or irreducible holomorphic symplectic factor as above. Then the following …

Proposition 4. Fix an integer 1≤d≤4 , a DCC set I⊂[0,1]∩ , and a positive integer p . There exists a positive integer ℓ=ℓ(d,I,p) with the following property. Let Z be a normal projective complex variety of dimension d with a rationally …

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Choose a smooth projective resolution u:W→X exhibiting a negative ordinary discrepancy. Its canonical divisor satisfies K_W=u^*K_X+E∼_ E, where E is exceptional with a negative coefficient. By Lemma 7, K_W is not pseudo-effective. The uniruledness criterion of implies that W is uniruled. Its maximal rationally connected quotient consequently has dimension less than four, and its base is non-uniruled . If that base had positive dimension, the induced rational map from X would contradict Lemma 6(i). Thus W is rationally connected. Apply Proposition 4 directly to (X,0) , with zero nef data. Its hypotheses are precisely klt singularities, K_X∼_ 0 , and a rationally connected smooth resolution, all of which have been checked. Its uniform integer bounds the canonical order.

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