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

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  21 July 2021

Gabriele Di Cerbo  and
Roberto Svaldi
Gabriele Di Cerbo
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ08540, USAdicerbo@math.princeton.edu
Roberto Svaldi
Affiliation:
EPFL, SB MATH-GE, MA B1 497 (Bâtiment MA), Station 8, CH-1015Lausanne, Switzerlandroberto.svaldi@epfl.ch
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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.

Keywords

Calabi–Yau varietieslog Calabi–Yau pairsboundedness of algebraic varietieselliptic fibrations

MSC classification

Primary: 14J32: Calabi-Yau manifolds
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14J10: Families, moduli, classification 14J81: Relationships with physics
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 8 , August 2021 , pp. 1766 - 1806
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Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

G.D.C. is partially supported by the Simons Foundation and NSF grant DMS-1702358. Most of this work was completed during several visits of R.S. to Columbia University. He would like to thank Columbia University for the hospitality and the nice working environment. He would also like to thank MIT where he was a graduate student and UCSD where he was a visitor when part of this work was completed. He kindly acknowledges financial support from NSF research grant nos 1200656 and 1265263 and Churchill College, Cambridge. During the final revision of this work he was supported by funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 307119.

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