On pluricanonical boundedness of varieties of general type

In this paper, we present a new proof of a theorem of Chen and Jiang: for any integer [Formula: see text], there is a constant [Formula: see text] such that every smooth projective [Formula: see text]-fold [Formula: see text] with [Formula: see text] 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 [Formula: see text] and [Formula: see text], [Formula: see text]-canonical maps of [Formula: see text]-folds of general type have birationally bounded fibers. This gives an affirmative answer to a question posed by Chen and Jiang in 2014.