Birational boundedness of low dimensional elliptic Calabi-Yau varieties with a section

TL;DR

This paper proves finiteness results for families of low-dimensional elliptic Calabi-Yau varieties with a section, showing that their geometric and Hodge-theoretic structures are limited in number, especially in dimensions up to five.

Contribution

It establishes the birational boundedness of certain elliptic Calabi-Yau varieties with a section in dimensions up to five, extending understanding of their classification.

Findings

Finitely many families of elliptic Calabi-Yau manifolds with a section in dim ≤ 5.

Boundedness of the Hodge diamond configurations for these manifolds.

Log birational boundedness of specific klt pairs in dimension up to 4.

Abstract

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim(Y)\leq 5$ and $Y$ is not of product-type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of klt pairs $(X, \Delta)$ with $K_X+\Delta$ numerically trivial and not of product-type, in dimension at most $4$.