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arxiv: 2505.15295 · v2 · submitted 2025-05-21 · 🧮 math.AG

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

Lie Fu , Zhiyuan Li , Teppei Takamatsu , Haitao Zou This is my paper

Pith reviewed 2026-05-22 14:11 UTC · model grok-4.3

classification 🧮 math.AG
keywords primitive symplectic varietiesShafarevich conjecturehyper-Kähler manifoldslocally trivial familiesQ-factorial terminalfiniteness of familiespointed curvessemi-ampleness
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The pith

For a fixed pointed curve and fixed primitive symplectic variety X, locally trivial families of Q-factorial terminal fibers have only finitely many isomorphism classes of generic fibers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a finiteness result for families of primitive symplectic varieties over a fixed pointed curve. Given any curve B with marked point 0 and any fixed primitive symplectic variety X, only finitely many distinct generic fibers occur among locally trivial families of Q-factorial and terminal primitive symplectic varieties over B whose fiber at 0 is isomorphic to X. This is presented as a geometric form of the Shafarevich conjecture. When isotropic nef divisors are semi-ample, a condition known to hold for all hyper-Kähler manifolds of known deformation types, the projective families themselves are finite up to isomorphism.

Core claim

For any fixed pointed curve (B, 0) and fixed primitive symplectic variety X, among all locally trivial families of Q-factorial and terminal primitive symplectic varieties over B whose fiber over 0 is isomorphic to X, there are only finitely many isomorphism classes of generic fibers. Assuming semi-ampleness of isotropic nef divisors, there are only finitely many such projective families up to isomorphism.

What carries the argument

The pointed Shafarevich problem for locally trivial families of Q-factorial terminal primitive symplectic varieties, with the semi-ampleness assumption used to control projective cases.

Load-bearing premise

The families under consideration are locally trivial with Q-factorial and terminal fibers, and for projective families the isotropic nef divisors are semi-ample.

What would settle it

An explicit example of a pointed curve B with point 0 and a fixed X admitting infinitely many locally trivial families over B with fiber X at 0 but with pairwise non-isomorphic generic fibers.

read the original 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-K\"ahler 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-K\"ahler 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 $0$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves a pointed version of the Shafarevich conjecture for primitive symplectic varieties. For a fixed pointed curve (B,0) and fixed primitive symplectic variety X, it shows that there are only finitely many isomorphism classes of generic fibers among all locally trivial families of Q-factorial terminal primitive symplectic varieties over B with special fiber isomorphic to X. Under the additional hypothesis that isotropic nef divisors are semi-ample (known to hold for all hyper-Kähler manifolds of known deformation types), there are only finitely many such projective families up to isomorphism. The authors supply an explicit construction showing that the result is optimal: infinitely many pairwise non-isomorphic families exist that become isomorphic over the punctured curve B∖{0} while keeping the fiber over 0 fixed.

Significance. If the arguments are correct, the result gives a precise geometric finiteness statement for pointed families of symplectic varieties, paralleling classical Shafarevich-type theorems while respecting the distinction between locally trivial families and projective ones. The optimality construction is a genuine strength, as it demonstrates sharpness without relying on ad-hoc parameters or self-referential definitions. The work therefore supplies a concrete advance in the moduli theory of hyper-Kähler and symplectic varieties.

major comments (2)
  1. [§1] §1 (main theorem statement): the finiteness for generic fibers is stated for locally trivial families; the proof sketch in the introduction indicates that local triviality is used to control the deformation space, but it is not immediately clear whether the argument extends to non-locally-trivial families or whether a counter-example exists outside this class. This assumption appears load-bearing for the central claim.
  2. [§4] §4 (optimality construction): the families constructed are stated to consist of smooth hyper-Kähler varieties that remain Q-factorial and terminal; a brief verification that these properties are preserved in the deformation over the punctured curve would strengthen the claim that the construction truly shows optimality within the hypotheses of the main theorem.
minor comments (2)
  1. [Introduction] The semi-ampleness hypothesis is invoked only for the projective case; a short remark recalling why it holds for all known deformation types (e.g., by citing the relevant results on hyper-Kähler manifolds) would improve readability.
  2. [§2] Notation for primitive symplectic varieties and the distinction between Q-factorial terminal and smooth cases is introduced gradually; a consolidated definition paragraph early in the paper would help readers track the hypotheses.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§1] §1 (main theorem statement): the finiteness for generic fibers is stated for locally trivial families; the proof sketch in the introduction indicates that local triviality is used to control the deformation space, but it is not immediately clear whether the argument extends to non-locally-trivial families or whether a counter-example exists outside this class. This assumption appears load-bearing for the central claim.

    Authors: We thank the referee for highlighting this point. Our main result is deliberately stated and proved for locally trivial families, which is the natural and standard setting for the pointed geometric Shafarevich conjecture in the context of primitive symplectic varieties. Local triviality ensures that the family stays within a fixed deformation class of the special fiber X and allows control via the period map and monodromy representation. Without this assumption, the generic fibers need not remain deformation-equivalent to X, so the finiteness statement would not make sense in the same form. We do not claim the result for non-locally trivial families and do not provide a counter-example, as that lies outside the scope of the paper. We will add a clarifying sentence in the introduction and in §1 to explicitly note that local triviality is essential to the statement and the proof. revision: partial

  2. Referee: [§4] §4 (optimality construction): the families constructed are stated to consist of smooth hyper-Kähler varieties that remain Q-factorial and terminal; a brief verification that these properties are preserved in the deformation over the punctured curve would strengthen the claim that the construction truly shows optimality within the hypotheses of the main theorem.

    Authors: We appreciate this suggestion. The families in the optimality construction are obtained by deforming the symplectic form on a fixed underlying manifold in a manner that keeps the total space smooth. Since Q-factoriality and terminality are open conditions in the moduli space of hyper-Kähler varieties (see, e.g., results of Namikawa on deformations of symplectic varieties), these properties persist under small deformations and therefore hold over the punctured curve B∖{0}. We will insert a short paragraph in §4 with this verification, citing the relevant openness statements, to confirm that the constructed families lie within the class considered in the main theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard moduli and deformation techniques

full rationale

The paper proves finiteness of isomorphism classes of generic fibers (and of projective families under an additional semi-ampleness hypothesis) for locally trivial families of Q-factorial terminal primitive symplectic varieties with fixed special fiber X. The argument is presented as a theorem derived from deformation theory, moduli spaces of symplectic varieties, and standard algebraic geometry tools. The semi-ampleness assumption is explicitly noted to hold for all known hyper-Kähler deformation types without being used to define or fit the finiteness statement itself. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to prior unverified inputs are present. The optimality construction (infinitely many families isomorphic away from the marked point) is supplied explicitly and does not create circularity in the positive finiteness result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background facts from algebraic geometry concerning deformation theory of symplectic varieties, properties of terminal and Q-factorial singularities, and the behavior of isotropic nef divisors; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Locally trivial families of primitive symplectic varieties with Q-factorial terminal fibers admit well-behaved moduli spaces or deformation theory sufficient for finiteness arguments.
    Invoked implicitly when counting isomorphism classes of generic fibers over the fixed pointed curve.
  • domain assumption Semi-ampleness of isotropic nef divisors holds for hyper-Kähler manifolds of known deformation types.
    Explicitly stated as the extra hypothesis needed for the projective finiteness statement.

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Forward citations

Cited by 1 Pith paper

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  1. The Cone Conjecture for Primitive Symplectic Varieties over a Field of Characteristic Zero and an Application

    math.AG 2025-12 unverdicted novelty 6.0

    The Kawamata-Morrison cone conjecture holds for Q-factorial terminal projective primitive symplectic varieties with b2 > 5 over characteristic zero fields, with an application to relative movable and nef cone conjectu...

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