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.v4- 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
- Classification source v2 revised October 1, 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 remain unrefereed. The classification source is now v2 and still claims non-negativity through dimension four and failure from dimension five, but it shares an author with the original counterexample and is not independent validation.
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.
- 2026-10-02 — Reviewed classification v2; the dimension threshold is unchanged and remains non-independent of the original counterexample.
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