A Finiteness Theorem for Elliptic Calabi-Yau Threefolds

TL;DR

This paper proves that, up to birational equivalence, only finitely many elliptic Calabi-Yau threefolds with certain singularities and fibrations to rational surfaces exist, extending previous results on their Euler characteristics.

Contribution

It establishes a finiteness theorem for families of elliptic Calabi-Yau threefolds with specific properties, strengthening prior results on their invariants.

Findings

Finite number of elliptic Calabi-Yau threefold families up to birational equivalence.

Finiteness applies to those with trivial canonical class and factorial terminal singularities.

Extends previous results on Euler characteristics of such threefolds.

Abstract

We prove that up to birational equivalence, there exists only a finite number of families of Calabi-Yau threefolds (i.e. a threefold with trivial canonical class and factorial terminal singularities) which have an elliptic fibration to a rational surface. This strengthens a result of B. Hunt that there are only a finite number of possible Euler characteristics for such threefolds.