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Batyrev's Stringy Hodge Number Conjecture

Algebra · posed by Victor Batyrev, 1998 · disproved

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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.)

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Matthew Satriano and Jeremy Usatine, using GPT

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

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    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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