Breakthrough Tracker record
A preprint claims the period–index conjecture is false in dimension three and higher
Alexander Perry constructs period-two Brauer classes of index 2^d on smooth projective d-folds, violating the proposed period–index bound in every dimension at least three. Version 2 extends the construction to positive characteristic and adds a corrected small-prime formulation with a proved Hodge-theoretic counterpart.
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- Stable ID
math-period-index-conjecture-counterexample-2026- Revision
math-period-index-conjecture-counterexample-2026.v2- Field
- Mathematics · Algebraic geometry, Brauer groups and Hodge theory
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed counterexample
- Last checked
AI role
The author says a flawed example produced in an interaction with ChatGPT supplied features of the eventual construction, and further discussion led to the theorem. The author wrote the paper and accepts responsibility for correctness.
Record details
- Problem or result
- The period–index conjecture for Brauer classes on algebraic varieties
- Authors
- Alexander Perry
- Institutions
- University of Michigan
- Result date
- Preprint submitted August 4 and revised September 3, 2026
Why it matters
Period and index measure different obstructions carried by a Brauer class. A valid Hodge-theoretic counterexample would overturn the proposed general bound in every dimension at least three while leaving the established surface case intact.
Limits
The expanded 41-page version 2 remains a non-peer-reviewed preprint with no located independent correctness assessment. Its positive-characteristic scope excludes characteristic two and depends on specified transcendence-degree hypotheses; the established surface theorem remains intact.
Sources
Correction and revision history
- 2026-08-08 — Added after primary-source review as a provisional claim; the scope, evidence level and material limitations are stated explicitly.
- 2026-09-04 — Version 2 expanded the construction to positive characteristic, corrected the proposed small-prime formulation and added a Hodge-theoretic counterpart; provisional status retained.
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