# On quint-canonical birationality of irregular threefolds

## TL;DR

This paper proves that for complex smooth projective threefolds of general type with positive irregularity, the m-canonical map is birational for all m ≥ 5, advancing understanding of their birational geometry.

## Contribution

It establishes the birationality of the m-canonical map for all m ≥ 5 on irregular threefolds of general type, a significant step in their classification.

## Key findings

- m-canonical map is birational for all m ≥ 5
- extends known results to irregular threefolds of general type
- contributes to the classification theory of algebraic threefolds

## Abstract

Let $X$ be a complex smooth projective threefold of general type. Assume $q(X)>0$. We show that the $m$-canonical map of $X$ is birational for all $m\geq 5$.