Abstract
We show that for each fixed dimension $d \geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical class is bounded and the elliptic fibration admits a rational section. This case builds on an analogous boundedness result for the set of rationally connected log Calabi–Yau pairs with bounded torsion index. In dimension $3$, we prove the more general statement that the set of $\epsilon \textrm{-lc}$ pairs $(X,B)$ with $-(K_X + B)$ nef and rationally connected $X$ is bounded up to isomorphism in codimension one.
Funding Statement
C.B. was supported by a grant of the Leverhulme Trust and a grant of the Royal Society.
G.D. was supported by the NSF under grants numbers DMS-1702358 and DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester.
R.S. kindly acknowledges support from Churchill College, Cambridge, the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 842071, and from the “Programma per giovani ricercatori Rita Levi Montalcini”.
Citation
Caucher Birkar. Gabriele Di Cerbo. Roberto Svaldi. "Boundedness of elliptic Calabi–Yau varieties with a rational section." J. Differential Geom. 128 (2) 463 - 519, October 2024. https://doi.org/10.4310/jdg/1727712887
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