On the irrationality of moduli spaces of K3 surfaces
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- by Daniele Agostini, Ignacio Barros and Kuan-Wen Lai;
- Trans. Amer. Math. Soc. 376 (2023), 1407-1426
- DOI: https://doi.org/10.1090/tran/8830
- Published electronically: November 9, 2022
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Abstract:
We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus $g$. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon >0$ and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree $10$. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated Hodge theoretically with certain cubic fourfolds, Gushel–Mukai fourfolds, and hyperkähler fourfolds.References
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Bibliographic Information
- Daniele Agostini
- Affiliation: Universität Tübingen, Fachbereich Mathematik, Auf der Morgenstelle 10, 72076 Tübingen, Germany; and Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22, 04103 Leipzig, Germany
- MR Author ID: 1135681
- Email: daniele.agostini@uni-tuebingen.de, daniele.agostini@mis.mpg.de
- Ignacio Barros
- Affiliation: Departement Wiskunde-Informatica, Universiteit Antwerpen, Middelheimcampus, Middelheimlaan 1, 2020 Antwerp, Belgium
- MR Author ID: 1276538
- ORCID: 0000-0002-7729-9413
- Email: ignacio.barros@uantwerpen.be
- Kuan-Wen Lai
- Affiliation: Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53121 Bonn, Germany
- MR Author ID: 1231887
- ORCID: 0000-0002-0071-5781
- Email: kwlai@math.uni-bonn.de
- Received by editor(s): March 11, 2022
- Received by editor(s) in revised form: August 9, 2022
- Published electronically: November 9, 2022
- Additional Notes: The second and third authors were supported by the ERC Synergy Grant ERC-2020-SyG-854361-HyperK.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 1407-1426
- MSC (2020): Primary 14E08, 14J28
- DOI: https://doi.org/10.1090/tran/8830
- MathSciNet review: 4531679