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 $K_n>0$ such that every smooth projective $n$-fold $X$ with $\operatorname{vol}(X)>K_n$ has either the stable birational $2$-canonical map or a M$^c$Kernan 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 $n\geq 1$, $r$-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.