Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture

[2505.15295] Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Algebraic Geometry arXiv:2505.15295 (math) [Submitted on 21 May 2025 ( v1 ), last revised 10 Apr 2026 (this version, v2)] Title: Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture Authors: Lie Fu , Zhiyuan Li , Teppei Takamatsu , Haitao Zou View a PDF of the paper titled Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture, by Lie Fu and 2 other authors View PDF HTML (experimental) Abstract: We investigate in this paper the so-called pointed Shafarevich problem for families of primitive symplectic varieties. More precisely, for any fixed pointed curve $(B, 0)$ and any fixed primitive symplectic variety $X$, among all locally trivial families of $\mathbb{Q}$-factorial and terminal primitive symplectic varieties over $B$ whose fiber over $0$ is isomorphic to $X$, we show that there are only finitely many isomorphism classes of generic fibers. Moreover, assuming semi-ampleness of isotropic nef divisors, which holds true for all hyper-Kähler manifolds of known deformation types, we show that there are only finitely many such projective families up to isomorphism. These results are optimal since we can construct infinitely many pairwise non-isomorphic (not necessarily projective) families of smooth hyper-Kähler varieties over some pointed curve $(B, 0)$ such that they are all isomorphic over the punctured curve $B\backslash \{0\}$ and have isomorphic fibers over the base point $0$. Comments: 36 pages; final version; to appear in Peking Mathematical Journal Subjects: Algebraic Geometry (math.AG) MSC classes: 14J42 (Primary), 14D10, 14D23, 32Q45, 14D07 Cite as: arXiv:2505.15295 [math.AG] (or arXiv:2505.15295v2 [math.AG] for this version) https://doi.org/10.48550/arXiv.2505.15295 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1007/s42543-026-00122-9 Focus to learn more DOI(s) linking to related resources Submission history From: Haitao Zou [ view email ] [v1] Wed, 21 May 2025 09:25:27 UTC (41 KB) [v2] Fri, 10 Apr 2026 17:54:34 UTC (43 KB) Full-text links: Access Paper: View a PDF of the paper titled Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture, by Lie Fu and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2025-05 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from

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