# On pluricanonical boundedness of varieties of general type

## TL;DR

This paper provides a new proof of a theorem relating the volume of varieties of general type to their canonical maps or fibrations, and improves boundedness results for their canonical maps.

## Contribution

It offers a new proof of Chen and Jiang's theorem, amends a gap in the original proof, and extends boundedness results for r-canonical maps of varieties of general type.

## Key findings

- Established a volume threshold for stable birational 2-canonical maps or fibrations.
- Improved boundedness results for r-canonical maps of varieties of general type.
- Confirmed that r-canonical maps have birationally bounded fibers for all r>1.

## Abstract

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.