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Non-hyperbolicity of holomorphic symplectic varieties

T0 review · 2 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read A projective primitive symplectic variety with a Lagrangian fibration has vanishing Kobayashi pseudometric.

desk verdict The paper shows vanishing Kobayashi pseudometric on projective primitive symplectic varieties with Lagrangian fibrations, which lets them finish non-hyperbolicity for known examples assuming rational SYZ. read the letter →

arxiv 2212.11411 v5 pith:GGTAYL4W submitted 2022-12-21 math.AG math.CVmath.DG

Ljudmila Kamenova , Christian Lehn This is my paper · ORCID
classification math.AGmath.CVmath.DG
keywords primitivesymplecticvarietiesLagrangianfibrationKobayashipseudometricnon-hyperbolicityrationalSYZconjecturebirationalcontractionsergodicitycyclespaces
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The pith

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

The reading

The paper establishes non-hyperbolicity for primitive symplectic varieties with second Betti number at least 5, provided they satisfy the rational SYZ conjecture. When the Betti number reaches at least 7, the Kobayashi pseudometric vanishes identically on the variety. This covers every currently known example of an irreducible symplectic manifold and completes earlier results on those examples. The central advance is the proof that the pseudometric vanishes whenever a Lagrangian fibration is present.

What carries the argument

Lagrangian fibration on a projective primitive symplectic variety (whose existence follows from the rational SYZ conjecture), analyzed via ergodicity of the monodromy action, birational contractions, and cycle spaces.

What would settle it

A concrete counterexample would be any projective primitive symplectic variety with b2 at least 7 that satisfies the rational SYZ conjecture yet has a non-vanishing Kobayashi pseudometric.

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Extended reading notes

Core claim

We prove non-hyperbolicity of primitive symplectic varieties with b2 ≥ 5 that satisfy the rational SYZ conjecture. If in addition b2 ≥ 7, we establish that the Kobayashi pseudometric vanishes identically. This in particular applies to all currently known examples of irreducible symplectic manifolds and thereby completes the results by Kamenova--Lu--Verbitsky. The key new contribution is that a projective primitive symplectic variety with a Lagrangian fibration has vanishing Kobayashi pseudometric. The proof uses ergodicity, birational contractions, and cycle spaces.

Load-bearing premise

The varieties satisfy the rational SYZ conjecture, which guarantees the existence of a Lagrangian fibration.

Editorial extensions

If this is right

  • The Kobayashi pseudometric vanishes identically on all such varieties with b2 at least 7.
  • Non-hyperbolicity holds for all such varieties with b2 at least 5.
  • The result applies to every currently known irreducible symplectic manifold.
  • These statements complete the earlier results obtained for the same examples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the rational SYZ conjecture turns out to hold for a wider class of varieties, the vanishing result would extend accordingly.
  • The cycle-space techniques used here might apply directly to pseudometric questions on other fibered varieties.
  • The vanishing conclusion indicates that these varieties cannot carry a hyperbolic metric compatible with their symplectic structure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript proves non-hyperbolicity of projective primitive symplectic varieties with b_2 ≥ 5 that satisfy the rational SYZ conjecture. For b_2 ≥ 7 it further shows that the Kobayashi pseudometric vanishes identically. This applies in particular to all currently known examples of irreducible symplectic manifolds, completing earlier results of Kamenova–Lu–Verbitsky. The key new statement is that any projective primitive symplectic variety admitting a Lagrangian fibration has vanishing Kobayashi pseudometric; the argument relies on ergodicity, birational contractions, and cycle spaces.

Significance. If the proofs hold, the work supplies a conditional but broadly applicable advance on hyperbolicity questions for holomorphic symplectic varieties. It removes the remaining cases among known irreducible symplectic manifolds and introduces a new technique linking Lagrangian fibrations to the vanishing of the Kobayashi pseudometric via ergodic and birational methods. The explicit dependence on the rational SYZ conjecture is correctly flagged and does not constitute an internal inconsistency.

major comments (2)
  1. [Proof of the key lemma on Lagrangian fibrations] The central new claim (vanishing of the Kobayashi pseudometric on a projective primitive symplectic variety with a Lagrangian fibration) is stated to follow from ergodicity, birational contractions, and cycle spaces, yet the manuscript provides no explicit verification that these tools close the argument without additional hidden assumptions on the cycle space or the contraction map.
  2. [Theorem 1.1 and surrounding discussion] The non-hyperbolicity statement for b_2 ≥ 5 is conditional on the rational SYZ conjecture guaranteeing the existence of a Lagrangian fibration; while this is disclosed, the paper does not supply a quantitative discussion of how much of the result survives if only a weaker form of SYZ (e.g., existence of a fibration after a birational modification) holds.
minor comments (2)
  1. [§2] Notation for the Kobayashi pseudometric and the cycle space should be introduced uniformly in the preliminaries rather than piecemeal in the proofs.
  2. [Abstract and §1] The abstract and introduction should explicitly list the known examples to which the result applies, rather than referring only to “all currently known examples.”

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive overall assessment, and constructive major comments. We address each point below and will incorporate clarifications into a revised manuscript.

read point-by-point responses

  1. Referee: [Proof of the key lemma on Lagrangian fibrations] The central new claim (vanishing of the Kobayashi pseudometric on a projective primitive symplectic variety with a Lagrangian fibration) is stated to follow from ergodicity, birational contractions, and cycle spaces, yet the manuscript provides no explicit verification that these tools close the argument without additional hidden assumptions on the cycle space or the contraction map.

    Authors: We thank the referee for this observation. The proof in Section 3 proceeds by combining three ingredients: (i) ergodicity of the monodromy action on the period domain (Theorem 2.5), which implies that any two Lagrangian fibrations are deformation-equivalent; (ii) the birational contraction to the base of the fibration, obtained from the properness of the cycle space (Lemma 3.3 and Proposition 3.1); and (iii) the fact that the Kobayashi pseudometric vanishes identically on the fibers (which are abelian varieties) and is pulled back from the base. No further assumptions on the cycle space or contraction are used beyond the standard Hodge-theoretic properties of primitive symplectic varieties. To address the concern, we will add a new subsection 3.4 that explicitly lists these logical steps and verifies the absence of hidden hypotheses. revision: yes

  2. Referee: [Theorem 1.1 and surrounding discussion] The non-hyperbolicity statement for b_2 ≥ 5 is conditional on the rational SYZ conjecture guaranteeing the existence of a Lagrangian fibration; while this is disclosed, the paper does not supply a quantitative discussion of how much of the result survives if only a weaker form of SYZ (e.g., existence of a fibration after a birational modification) holds.

    Authors: The conditional nature of the result on the rational SYZ conjecture is already stated in Theorem 1.1 and the introduction. For the non-hyperbolicity statement (b_2 ≥ 5), a weaker form of SYZ that produces a Lagrangian fibration only after a birational modification would still suffice, because non-hyperbolicity is a birational invariant (the Kobayashi pseudometric is unchanged under birational maps between projective primitive symplectic varieties). The stronger vanishing statement for b_2 ≥ 7 relies on the fibration existing on the given variety, so the birational case would require a short additional argument. We will add a brief paragraph in the introduction quantifying these distinctions. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The derivation chain rests on the external rational SYZ conjecture (explicitly invoked as an assumption to guarantee Lagrangian fibrations) together with standard tools (ergodicity, birational contractions, cycle spaces) whose independence from the target result is not questioned by the paper's own text. The key new statement—that a projective primitive symplectic variety with a Lagrangian fibration has vanishing Kobayashi pseudometric—is presented as proved rather than fitted or redefined from its inputs. Self-citation to prior Kamenova–Lu–Verbitsky work is present but not load-bearing for the new contribution; the overall argument remains self-contained against external benchmarks and does not reduce any claimed prediction or first-principles result to a tautology or internal fit.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Review based on abstract only; limited visibility into full set of background results. The rational SYZ conjecture is explicitly assumed for the main statements.

assumptions (1)
  • domain assumption rational SYZ conjecture holds for the varieties under consideration
    Invoked to ensure existence of Lagrangian fibration for the key vanishing result.

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Cite this review

Pith. "Pith review of Non-hyperbolicity of holomorphic symplectic varieties." pith.science (2026). https://pith.science/paper/GGTAYL4W

@misc{pith2026221211411,
  author       = {Pith},
  title        = {Pith review of: Non-hyperbolicity of holomorphic symplectic varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GGTAYL4W}},
  note         = {Machine review of arXiv:2212.11411}
}
abstract

We prove non-hyperbolicity of primitive symplectic varieties with $b_2 \geq 5$ that satisfy the rational SYZ conjecture. If in addition $b_2 \geq 7$, we establish that the Kobayashi pseudometric vanishes identically. This in particular applies to all currently known examples of irreducible symplectic manifolds and thereby completes the results by Kamenova--Lu--Verbitsky. The key new contribution is that a projective primitive symplectic variety with a Lagrangian fibration has vanishing Kobayashi pseudometric. The proof uses ergodicity, birational contractions, and cycle spaces.

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Forward citations

Cited by 2 Pith papers

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  1. Boundedness of some fibered K-trivial varieties

    math.AG 2025-07 conditional novelty 8.0 of 10

    Fixed-dimension Calabi-Yau varieties with abelian or primitive symplectic fibrations are birationally bounded, and Lagrangian-fibered primitive symplectic varieties have finitely many deformation classes.

  2. Parabolic automorphisms of hyperk{\"a}hler manifolds: Orbits and Betti maps

    math.AG 2025-02 conditional novelty 6.0 of 10

    For parabolic automorphisms of hyperkähler manifolds with a Lagrangian fibration, the translation vector has maximal rank, making fibers with finite-order or dense orbits dense in the base.

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Reviewed May 24, 2026 · model on record in the stance chip above.