Boundedness of elliptic Calabi–Yau threefolds

We show that elliptic Calabi-Yau threefolds form a bounded family.We also show that the same result holds for minimal terminal threefolds of Kodaira dimension 2, upon fixing the rate of growth of pluricanonical forms and the degree of a multisection of the Iitaka fibration.Both of these hypotheses are necessary to prove the boundedness of such a family.Contents 1. Introduction 1 2. Preliminaries 5 3. Finiteness of models for elliptic Calabi-Yau fiber spaces 17 4. Deforming divisors in a family 29 5. Interludium: an example 31 6.Elliptic threefolds 34 7. Proof of the main results 50 References 51