Boundedness of some fibered K-trivial varieties

TL;DR

This paper establishes boundedness results for certain fibered K-trivial varieties, including Calabi-Yau and hyperk"ahler varieties, under specific conditions and conjectures, advancing classification in algebraic geometry.

Contribution

It proves birational boundedness of fibered Calabi-Yau and primitive symplectic varieties of fixed dimension and finiteness of deformation classes under certain conditions.

Findings

Fibered Calabi-Yau varieties of fixed dimension are birationally bounded.

Finitely many deformation classes of primitive symplectic varieties admit Lagrangian fibrations.

Fibered Calabi-Yau 3-folds are bounded.

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 $b_2 \geq 5$.