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

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 Kawamata log terminal pairs$(X, \Delta )$with$K_X+\Delta$numerically trivial and not of product type, in dimension at most four.