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Batyrev's Stringy Hodge Number Conjecture

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batyrev-stringy-hodge-numbersAlgebraposed by Victor Batyrev, 1998recorded: disproved

1 attempt · no person has looked

Statement

For a projective variety XX with at worst Gorenstein canonical singularities whose stringy EE-function Est(X;u,v)E_{\mathrm{st}}(X; u, v) is a polynomial, all stringy Hodge numbers hstp,q(X)h^{p,q}_{\mathrm{st}}(X) are non-negative. (Batyrev 1998, Conjecture 3.10.)

Context

Batyrev's stringy invariants are foundational in birational geometry and mirror symmetry.

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  • #1

    Attempt 1

    constructionGPT with Matthew Satriano, Jeremy Usatine ·
    AI involvement
    ai co developed
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    GPT
    people
    Matthew Satriano, Jeremy Usatine

    Satriano and Usatine found the counterexample with the assistance of GPT: X=M0×P1X = M_0 \times \mathbb{P}^1, where M0M_0 is the coarse moduli space of rank-2 semistable bundles with trivial determinant over a genus-3 curve. XX is a 7-dimensional projective variety with Gorenstein terminal singularities whose stringy EE-function is a polynomial, yet its stringy Hodge number hst2,5(X)=−1h^{2,5}_{\mathrm{st}}(X) = -1 is negative.

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