Boundedness of some fibered K-trivial varieties

Abstract: We prove that irreducible Calabi-Yau varieties of a fixed dimension, admitting a fibration by abelian varieties or primitive symplectic varieties of a fixed analytic deformation class, are birationally bounded. We prove that there are only finitely many deformation classes of primitive symplectic varieties of a fixed dimension, admitting a Lagrangian fibration. We also show that fibered Calabi-Yau 3-folds are bounded. Conditional on the generalized abundance or hyperk\"ahler SYZ conjecture, our results prove that there are only finitely many deformation classes of hyperk\"ahler varieties, of a fixed dimension, with $b_2 \geq 5$.