Boundedness of polarized log Calabi-Yau fibrations with bounded bases
Xiaowei Jiang, Junpeng Jiao, Minzhe Zhu
TL;DR
The paper addresses boundedness questions for polarized log Calabi–Yau fibrations with bounded bases, proving total spaces are bounded in codimension one after fixing invariants, and bounded outright when the general fibers have vanishing irregularity. The authors develop a framework using polarized log Calabi–Yau structures, Birkar’s moduli spaces for polarized CY pairs, Ambro’s moduli/b-divisor theory, and MMP in families to obtain log boundedness, log birational boundedness, and eventual codimension-one boundedness. They further extend from finite coefficient sets to arbitrary coefficients and treat special cases with vanishing irregularity, leveraging nef/big moduli divisors and maximal variation to bound moduli maps. As applications, they deduce boundedness results for stable minimal models and fibered Calabi–Yau varieties, contributing to moduli theory and inductive strategies in the minimal model program.
Abstract
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. 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.