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

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 $\dim(Y)\leq 5$ and $Y$ is not product-type. As consequence, we obtain possibilities for the Hodge diamond such manifolds. The result follows from log birational boundedness klt pairs $(X, \Delta)$ $K_X+\Delta$ numerically trivial product-type, dimension at most $4$.