On an efficient induction step with Nklt(X,D)
  Applying the effective induction on Nklt(X,D) developed by Hacon, Mckernan and Takayama, Todorov proved that the 5-canonical map is birational for projective 3-folds V with Vol(V) sufficiently large, which was recently improved by Di Biagio in loosing the volume constraint. The observation is that the least efficient induction step can be studied in an alternative way, which allows to assert Todorov's statement for $vol(V)>12^3$. The 4-dimensional analog is also given in this note. The idea works well in all dimensions. 
Meng Chen
math.AG