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Batyrev’s nonnegativity conjecture for stringy Hodge numbers

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Let XX be a projective algebraic variety with at worst Gorenstein canonical singularities. Assume that its stringy EE-function Est(X;u,v)E_{\rm st}(X;u,v) is a polynomial, and define the stringy Hodge numbers by writing

Est(X;u,v)=∑p,qap,qupvq,hstp,q(X)=(−1)p+qap,q.E_{\rm st}(X;u,v)=\sum_{p,q}a_{p,q}u^pv^q,\qquad h^{p,q}_{\rm st}(X)=(-1)^{p+q}a_{p,q}.

Nonnegativity conjecture. All stringy Hodge numbers hstp,q(X)h^{p,q}_{\rm st}(X) are nonnegative.

Stringy Hodge numbers extend the usual Hodge numbers to singular varieties for which the stringy EE-function is polynomial. Their nonnegativity is expected to reflect the positivity of ordinary Hodge numbers, but no resolution is supplied here.

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Batyrev’s nonnegativity conjecture for stringy Hodge numbers

    Must every projective variety with Gorenstein canonical singularities and polynomial stringy EE-function have nonnegative stringy Hodge numbers?

  2. Batyrev's conjecture on the non-negativity of stringy Hodge numbers

    Let XX be a projective algebraic variety with at worst Gorenstein canonical singularities. Assume that the stringy EE-function Est⁡(X;u,v)E_{\operatorname{st}}(X;u,v) is a polynomial. Batyrev's conjecture. All stringy Hodge numbers hst⁡p,q(X)h^{p,q}_{\operatorname{st}}(X) are non-negative. Batyrev introduced this conjecture in the context of mirror symmetry for singular Calabi--Yau varieties and motivic integration. It has remained a fundamental open problem, although the paper reports a counter-example to the conjecture.

    source: Matthew Satriano and Jeremy Usatine, “A counter-example to Batyrev's conjecture on the non-negativity of stringy Hodge numbers”, arXiv:2607.19184 (2026).

References

Primary source

Victor V. Batyrev, “Stringy Hodge numbers of varieties with Gorenstein canonical singularities”, arXiv:alg-geom/9711008 (1998).

geometry-topologyhodge-theorymotivic-integrationsingularitiesstringy-invariants
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Progress summary

Refreshed
Claimed solved

A 2026 paper claims the conjecture has a sharp answer: it holds through four dimensions and fails from five dimensions onward, but this has not been independently verified.

Batyrev’s conjecture asks whether stringy Hodge numbers are always nonnegative for the stated class of singular projective varieties. The original formulation is Batyrev’s Conjecture 3.10.

Known results

  • Batyrev proved nonnegativity when p+q≤4p+q\le 4 and the stringy EE-function is polynomial.
  • Batyrev’s conjecture was proved for projective threefolds with Gorenstein canonical singularities (2006).
  • Varieties admitting a crepant resolution satisfy the conjecture.
  • Nonnegativity is known for the determinantal varieties D^r,rk\hat{D}^{k}_{r,r} (2025).

August–September 2026 claimed threshold

Huang and Satriano’s August paper claims nonnegativity in dimensions at most 44 and counterexamples in every dimension at least 55, with hst2,3(Xn)=−1h_{\rm st}^{2,3}(X_n)=-1 for a family of dimension n+4n+4. A September follow-up claims a unified family. The earlier seven-dimensional example was discovered with assistance from OpenAI’s ChatGPT.

Current status (as of September 2026): A claimed sharp threshold settles the conjecture affirmatively through dimension 44 and negatively from dimension 55 onward, but the claims remain unverified.

Sources

Solutions 0

No solutions have been posted yet.