A criterion for the canonical bundle of a 3-fold to be ample (Q1078262)

We show that the (anti-)canonical bundle of a complete normal 3-fold with only canonical singularities is ample if and only if it is numerical positive. If we put a stronger assumption \(\kappa (X)>0\) as P. M. H. Wilson did in his original proof of the theorem, then the assertion is an immediate consequence of Kawamata's base point free theorem. The essential difficulty to prove the above theorem is to exclude the worst possible case \(\kappa (X)=0\), which we overcome using the classification of normal surfaces due to F. Sakai in connection with the abundance conjecture.