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The Hyperkahler Period-Index Conjecture

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hyperkahler-period-index-conjectureGeometry & topologyposed by Daniel Huybrechts, 2019recorded: disproved

1 attempt · no person has looked

Statement

Huybrechts conjectured that for every Brauer class alpha on a hyperkahler variety X, the index divides the period raised to the power dim(X)/2, strengthening the usual period-index conjecture. Disproved on certain hyperkahler fourfolds, in both the K3^[2] and Kum^2 deformation types.

Context

The counterexamples come with divisibility bounds on the Hodge-theoretic index, at 2-torsion and 5-torsion, on very general hyperkahler fourfolds.

A conjecture by a leading figure strengthening the period-index conjecture, well known within hyperkahler geometry and Brauer-group theory.

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  • #1

    Attempt 1

    constructionUnspecified with Pieter Belmans, James Hotchkiss ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    Unspecified
    people
    Pieter Belmans, James Hotchkiss

    The AI disclosure names the role precisely while leaving the model unnamed: "The starting points for this paper were two different LLM-assisted constructions of counterexamples, obtained independently by the authors. The paper is the synthesis of these constructions, with LLMs used to help with the copyediting." Two authors reaching counterexamples independently with model help is a stronger signal than either would be alone.

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