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arxiv: 1906.11528 · v1 · pith:T5GQALXYnew · submitted 2019-06-27 · 🧮 math.AG

Twisted cotangent bundles of Hyperk\"ahler manifolds

Fabrizio Anella , Andreas H\"oring This is my paper

Pith reviewed 2026-05-25 15:01 UTC · model grok-4.3

classification 🧮 math.AG
keywords hyperkähler manifoldspseudoeffective vector bundlesBeauville-Bogomolov formtwisted cotangent bundlesK3 surfacesHilbert schemesample divisors
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The pith

The Beauville-Bogomolov form q(H) supplies a lower bound for the twisted cotangent bundle Ω_X ⊗ H to be pseudoeffective on a hyperkähler manifold X with ample divisor H.

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

This paper establishes a numerical criterion, expressed via the Beauville-Bogomolov quadratic form, that guarantees pseudoeffectiveness of the cotangent bundle twisted by an ample line bundle on any hyperkähler manifold. A reader would care because pseudoeffectiveness controls the asymptotic behavior of sections and the birational geometry of the manifold. The bound is made fully explicit when the manifold belongs to the deformation class of the Hilbert scheme of points on a K3 surface, where its sharpness is also examined.

Core claim

Let X be a hyperkähler manifold and H an ample divisor. The twisted cotangent bundle Ω_X ⊗ H is pseudoeffective provided that the Beauville-Bogomolov form q(H) exceeds a certain lower bound. When X is deformation equivalent to the Hilbert scheme of a K3 surface, this lower bound can be written explicitly and its optimality is studied.

What carries the argument

The Beauville-Bogomolov quadratic form q on the second cohomology of X, which supplies the numerical threshold controlling pseudoeffectiveness of the twisted cotangent bundle.

If this is right

  • For hyperkähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface the lower bound on q(H) becomes fully explicit.
  • The same bound applies uniformly to every hyperkähler manifold equipped with an ample divisor.
  • Optimality of the bound can be tested directly inside the K3 Hilbert-scheme deformation class.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same numerical control might extend to other positivity notions such as nefness for the twisted bundle.
  • Similar thresholds could be sought for the cotangent bundle twisted by non-ample classes.
  • The result suggests that many positivity questions on hyperkähler manifolds reduce to evaluations of the Beauville-Bogomolov form.

Load-bearing premise

The Beauville-Bogomolov quadratic form q is defined on the second cohomology of any hyperkähler manifold and can be compared against a fixed positivity threshold.

What would settle it

A hyperkähler manifold X with ample H such that q(H) meets or exceeds the stated lower bound yet Ω_X ⊗ H fails to be pseudoeffective.

read the original abstract

Let $X$ be a Hyperk\"ahler manifold, and let $H$ be an ample divisor on $X$. We give a lower bound in terms of the Beauville-Bogomolov form $q(H)$ for the twisted cotangent bundle $\Omega_X \otimes H$ to be pseudoeffective. If $X$ is deformation equivalent to the Hilbert scheme of a K3 surface the lower bound can be written down explicitly and we study its optimality.

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

0 major / 3 minor

Summary. The paper claims to establish a lower bound, expressed in terms of the Beauville-Bogomolov quadratic form q(H), on the value of q(H) that guarantees pseudoeffectiveness of the twisted cotangent bundle Ω_X ⊗ H, where X is a hyperkähler manifold and H is an ample divisor. When X is deformation equivalent to the Hilbert scheme of points on a K3 surface, the bound is written explicitly and its optimality is examined.

Significance. If the stated bound holds, the result supplies a concrete, computable criterion using the standard BBF form for a basic positivity question on hyperkähler manifolds. The explicit formula and optimality analysis in the K3^[n] deformation class constitute verifiable, falsifiable content that strengthens the contribution.

minor comments (3)
  1. [Abstract] The abstract announces the existence of a lower bound but does not record its explicit functional form; the introduction or §2 should state the inequality q(H) > f(dim X, n) or equivalent at the outset.
  2. [Introduction] Notation for the pseudoeffectiveness threshold is introduced without a dedicated definition; a short paragraph or displayed equation defining the constant appearing in the bound would improve readability.
  3. [§4] The optimality discussion for the K3^[n] case refers to specific numerical checks; a table listing the bound versus known examples of non-pseudoeffective bundles would make the sharpness claim easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript and the recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard BBF form without reduction to inputs

full rationale

The abstract states a lower bound on q(H) for pseudoeffectiveness of Ω_X ⊗ H, with an explicit form in the K3 Hilbert scheme case. No equations, lemmas, or proof steps are visible that equate the claimed bound to a fitted parameter, self-definition, or self-citation chain. The Beauville-Bogomolov form is an established external tool on H^2(X), and the statement restricts to the projective case consistently. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; no explicit free parameters, invented entities, or non-standard axioms are visible. Full paper would be required to list any hidden assumptions or fitted constants.

axioms (1)
  • domain assumption Standard properties of the Beauville-Bogomolov quadratic form on the second cohomology of hyperkähler manifolds
    Invoked to express the numerical lower bound in the abstract statement.

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