Batyrev's Stringy Hodge Number Conjecture

What the AI did

Satriano and Usatine found the counterexample with the assistance of GPT: $X = M_0 \times \mathbb{P}^1$, 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 $h^{2,5}_{\mathrm{st}}(X) = -1$ 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 $h^{2,5}_{\mathrm{st}} = -1$, so it is hand-verifiable. A domain-expert preprint, not yet peer-reviewed. Distinct from the unrelated Batyrev-Manin conjecture on rational points.

Source

• PaperarXiv:2607.19184