Breakthrough Tracker record
A counterexample to Batyrev’s non-negativity conjecture for stringy Hodge numbers
Satriano and Usatine give a projective variety with a negative stringy Hodge number, refuting Batyrev’s general non-negativity conjecture. A separate Huang–Satriano preprint now claims the exact dimensional threshold: non-negativity through dimension four and counterexamples in every dimension from five onward.
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- Stable ID
math-batyrev-stringy-hodge-counterexample-2026- Revision
math-batyrev-stringy-hodge-counterexample-2026.v3- Field
- Mathematics · Algebraic geometry and motivic integration
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional counterexample
- Last checked
AI role
The authors state that the counterexample was discovered with assistance from OpenAI’s ChatGPT. They present and take responsibility for the proof.
Record details
- Problem or result
- Batyrev’s conjecture that polynomial stringy E-functions of projective varieties with Gorenstein canonical singularities have non-negative stringy Hodge numbers
- Authors
- Matthew Satriano and Jeremy Usatine
- Institutions
- University of Waterloo and Florida State University
- Result date
- Original counterexample submitted July 21; threshold preprint submitted August 8, 2026
Why it matters
The counterexample rules out non-negativity in full generality; the newer result, if confirmed, draws the boundary sharply by preserving the conjecture through dimension four and disproving it from dimension five onward.
Limits
Both controlling results are recent non-peer-reviewed preprints. The threshold manuscript includes an author of the original counterexample and is not independent validation; no external specialist correctness assessment was located.
Sources
- Primary: Matthew Satriano and Jeremy Usatine, A counter-example to Batyrev’s conjecture on the non-negativity of stringy Hodge numbers
- Primary: Huang and Satriano, dimension threshold for Batyrev’s conjecture
Correction and revision history
- 2026-07-23 — Added as a provisional first-version counterexample after checking the four-page proof, its cited prerequisites and the authors’ AI disclosure.
- 2026-08-08 — Reviewed v2 and replaced the obsolete first-version caveat; no material change to the claimed counterexample or provisional status was found.
- 2026-08-22 — Added the separate dimension-threshold claim: non-negativity through dimension four and counterexamples from dimension five; provisional status retained.
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