Unpolarized Shafarevich conjectures for hyper-Kähler varieties

[2203.10391] Unpolarized Shafarevich conjectures for hyper-Kähler varieties Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2203.10391 (math) [Submitted on 19 Mar 2022 ( v1 ), last revised 10 Apr 2026 (this version, v2)] Title: Unpolarized Shafarevich conjectures for hyper-Kähler varieties Authors: Lie Fu , Zhiyuan Li , Teppei Takamatsu , Haitao Zou View PDF HTML (experimental) Abstract: The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higher-dimensional analogs of K3 surfaces, Y. André proved the Shafarevich conjecture for hyper-Kähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-Kähler varieties in a given deformation type. We also discuss the cohomological generalization of the Shafarevich conjecture by replacing the good reduction condition by the unramifiedness of the cohomology, where our results are subject to a certain necessary assumption on the faithfulness of the action of the automorphism group on cohomology. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-Kähler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach to these results is a uniform Kuga--Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest. Comments: 43 pages. Comments welcome. This paper supersedes arXiv:2201.03350 and arXiv:2201.00482 . Accepted version; to appear in Algebraic Geometry Subjects: Algebraic Geometry (math.AG) ; Number Theory (math.NT) MSC classes: 14G35, 14J28, 14J42, 11G15 Cite as: arXiv:2203.10391 [math.AG] (or arXiv:2203.10391v2 [math.AG] for this version) https://doi.org/10.48550/arXiv.2203.10391 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Algebraic Geometry 13 (2026), no. 5, 598-638 Related DOI : https://doi.org/10.14231/AG-2026-018 Focus to learn more DOI(s) linking to related resources Submission history From: Haitao Zou [ view email ] [v1] Sat, 19 Mar 2022 20:51:32 UTC (51 KB) [v2] Fri, 10 Apr 2026 15:37:56 UTC (45 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2022-03 Change to browse by: math math.NT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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