Boundedness of polarized log Calabi–Yau fibrations with bounded bases

We investigate the boundedness problem for log Calabi–Yau fibrations whose bases and general fibers are bounded. We prove that the total spaces of log Calabi–Yau fibrations are bounded in codimension one after fixing some natural invariants, which confirms a conjecture of Birkar and Hacon. We also prove that the total spaces are bounded if, in addition, the irregularity of the general fibers vanishes. Then we apply our results to the boundedness problem for stable minimal models and fibered Calabi–Yau varieties.