Batyrev's Stringy Hodge Number Conjecture
For a projective variety X with at worst Gorenstein canonical singularities whose stringy E-function E_st(X; u, v) is a polynomial, all stringy Hodge numbers h^{p,q}_st(X) are non-negative. (Batyrev 1998, Conjecture 3.10.)
- Result
- Disproved
- Field
- Algebraic Geometry
- Posed by
- Victor Batyrev
- Year posed
- 1998
- Years open
- 28y
- Solved
- 2026-07-21
- Model
- ChatGPT
- Vendor
- OpenAI
- Collaborators
- Matthew Satriano, Jeremy Usatine
- Verification
- Pending peer review
- Notability
- No dedicated article
What the AI did
Satriano and Usatine found the counterexample with the assistance of ChatGPT: X = M_0 x P^1, where M_0 is the coarse moduli space of rank-2 semistable bundles with trivial determinant over a genus-3 curve. X is a 7-dimensional projective variety with Gorenstein terminal singularities whose stringy E-function is a polynomial, yet its stringy Hodge number h^{2,5}_st(X) = -1 is negative.
Verification
arXiv preprint 2607.19184 (21 Jul 2026) by Matthew Satriano and Jeremy Usatine. The proof is short and fully explicit: the stringy E-function is written out and its u^2 v^5 coefficient gives h^{2,5}_st = -1, so it is hand-verifiable. A domain-expert preprint, not yet peer-reviewed. Distinct from the unrelated Batyrev-Manin conjecture on rational points.