On pluricanonical boundedness of varieties of general type

We present a new proof of a theorem of Chen and Jiang: for any integer $n>1$, there is a constant Kₙ>0 such that every smooth projective n-fold X with vol(X)>Kₙ has either the stable birational $2$-canonical map or a MᶜKernan fibration. This amends a gap in the original proof. As a direct application of our method, we improve a former boundedness theorem of Lacini and prove that for any integer $r>1$ and n≥ 1, r-canonical maps of n-folds of general type have birationally bounded fibers. This gives an affirmative answer to a question posed by Chen and Jiang in 2014.