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Birational boundedness of low-dimensional elliptic Calabi–Yau varieties with a section
- Part of
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- Journal:
- Compositio Mathematica / Volume 157 / Issue 8 / August 2021
- Published online by Cambridge University Press:
- 21 July 2021, pp. 1766-1806
- Print publication:
- August 2021
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- Article
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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.
