On the boundedness of elliptic Calabi-Yau 4-folds

In this work, we settle the boundedness of a vast class of elliptic Calabi-Yau 4-folds.

In particular, we show that any elliptic Calabi-Yau 4-fold that is not crepant to the quotient of a product $Y \times E$, where $E$ is an elliptic curve and $Y$ a Calabi-Yau 3-fold, belongs to finitely many algebraic families.

In particular, the statement applies whenever the elliptic fibration of the Calabi-Yau 4-fold is not isotrivial.

We also provide partial evidence for the boundedness of the middle Betti number of Calabi-Yau 3-folds and study the index of fibered $K$-trivial 4-folds.