Finiteness of Pointed Families of Symplectic Varieties: A Geometric Shafarevich Conjecture

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}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Q}$$\end{document}

-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\}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B\backslash \{0\}$$\end{document}

and have isomorphic fibers over the base point 0.

Keywords

Holomorphic symplectic varieties / Geometric Shafarevich conjecture / Finiteness of families / Period map / Cone conjecture / 14J42 (Primary) / 14D10 / 14D23 / 32Q45 / 14D07

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Funding

Deutsche Forschungsgemeinschaft(491392403-TRR 358)

Université de Strasbourg

Agence Nationale de la Recherche(ANR-20-CE40-0023)

Centre National de la Recherche Scientifique(International Emerging Actions)

Shanghai Shuguang Program(21TQ00)

National Natural Science Foundation of China(12121001)

Japan Society for the Promotion of Science(JP22KJ1780)

Universität Bielefeld (3146)

RIGHTS & PERMISSIONS

The Author(s)