A counter-example to Batyrev's conjecture on the non-negativity of stringy Hodge numbers

Jeremy Usatine; Matthew Satriano

REVIEW 1 major objections 2 minor 1 cited by

A counter-example to Batyrev's conjecture on the non-negativity of stringy Hodge numbers

T0 review · 1 major / 2 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A 7-dimensional projective variety with Gorenstein terminal singularities has stringy Hodge number h^{2,5}_st = -1, refuting Batyrev's nonnegativity conjecture.

desk verdict A short, apparently correct counterexample to Batyrev's conjecture, resting on a published formula the authors do not re-derive—worth refereeing, with a request to show the verification. read the letter →

arxiv 2607.19184 v2 pith:RUDMVFIN submitted 2026-07-21 math.AG

Matthew Satriano , Jeremy Usatine This is my paper · ORCID

classification math.AG MSC 14J1714D2014E18

keywords stringyHodgenumbersBatyrevconjecturecounterexamplemoduliofrank-2bundlesE-functionGorensteinterminalsingularitiesmotivicintegration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Batyrev's conjecture asserts that stringy Hodge numbers, defined combinatorially from a log resolution, are nonnegative whenever the stringy E-function is a polynomial. This paper gives a counterexample: the product X = M0 × P^1, where M0 is the coarse moduli space of rank-2 semistable bundles with trivial determinant on a genus-3 curve, is a 7-dimensional projective variety with Gorenstein terminal singularities. The authors compute its stringy E-function via a known formula for M0 and the E-function of P^1, obtaining a polynomial whose u^2 v^5 coefficient is +1, so the signed coefficient h^{2,5}_st(X) equals -1. This invalidates the conjecture as stated and indicates that stringy Hodge numbers cannot be interpreted as dimensions of a standard cohomology theory.

What carries the argument

The machinery is the explicit rational function for the stringy E-function of M0, quoted from Kiem–Li, together with the product behaviour of stringy E-functions for products with smooth factors. Multiplying the rational expression P/(1+uv) by the Hodge–Deligne polynomial of P^1, namely 1+uv, cancels the denominator and leaves the polynomial P. The sign convention for stringy Hodge numbers then converts the positive u^2 v^5 coefficient of P into the negative number h^{2,5}_st = -1.

What would settle it

Recompute the stringy E-function of M0 independently via a log resolution or the stack-theoretic crepant resolution, and verify whether the numerator P has u^2 v^5 coefficient +1. If the coefficient differs, the negative Hodge number disappears.

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Extended reading notes

Core claim

The central discovery is that the variety X = M0 × P^1 satisfies all hypotheses of Batyrev's Conjecture 1.1 yet yields a negative stringy Hodge number. Here M0 is the coarse moduli space of rank-2 semistable bundles with trivial determinant over a smooth projective curve of genus 3; M0 is known to have Gorenstein terminal singularities, so X is Gorenstein terminal of dimension 7. Using the formula of Kiem and Li for the stringy E-function of M0, the stringy E-function of X simplifies to a polynomial P(u,v) because the E-function of P^1 is 1+uv. The coefficient of u^2 v^5 in P is +1, so Batyrev's sign convention gives h^{2,5}_st(X) = (-1)^{2+5} · 1 = -1. Since E_st(X;u,v) is polynomial, this

Load-bearing premise

The counterexample stands or falls on the quoted formula of Kiem and Li for the stringy E-function of M0; if that formula is incorrect, the coefficient of u^2 v^5 in P could change sign and the negative Hodge number would vanish.

Editorial extensions

If this is right

Batyrev's Conjecture 1.1 is false as stated, so the nonnegativity of stringy Hodge numbers cannot be used as a guiding principle for motivic integration.
No pure cohomology theory can reproduce all stringy Hodge numbers, since these are negative for X; a mixed cohomology theory remains consistent with the data.
The counterexample lies in a well-studied moduli space, so the failure of positivity is not an exotic pathology.
The stringy E-function of X is an explicit polynomial P, providing a concrete test object for future positivity statements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The method suggests a search program: take moduli spaces whose stringy E-functions are rational with a pole at 1+uv, multiply by P^1, and check coefficients; other negative Hodge numbers may be found.
The result strengthens the case that stringy Hodge numbers are inherently mixed; a mixed cohomology interpretation is consistent with this failure of pure cohomology.
One might test the robustness of the counterexample by checking whether small deformations or blow-ups of X preserve the negative coefficient; if the sign is stable, the phenomenon is geometric rather than coordinate-dependent.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a legitimate counterexample to Batyrev's 1998 conjecture, assuming the quoted formula from Kiem–Li is right. The math is a short computation, and I checked the coefficient extraction: the coefficient of u^2 v^5 in P is indeed 1, so h^{2,5}_st = -1. The main ingredients—terminal Gorenstein for M0, multiplicativity under products, and E(P^1) = 1+uv—are all standard and correctly cited. The contribution is the observation that the product with P^1 kills the denominator and leaves a polynomial with a negative signed coefficient. That is new, and it settles a 25-year-old question.

What the paper does well: it is honest and compact. It gives a complete proof modulo two external results (Kiem–Li's formula and the singularities of M0), and it does not overclaim. The discussion of why Schepers–Veys's example did not count is accurate. The stringy cohomology framing in the introduction is a bit self-promotional but not inaccurate.

Soft spots: the load-bearing E-function formula is quoted, not re-derived. The stress-test is right that a transcription error there would kill the example. That said, the formula is in a peer-reviewed paper and appears to be consistent with Kiem's independent computation of the same stringy E-function for genus 3, so the risk is small. Still, for a counterexample to a famous conjecture, it would be better practice to include a cross-check or at least state that they compared with Kie03. Second, the algebra from the KL04 expression to P is asserted without intermediate steps; I reproduced it, and it's fine, but a referee will want the simplification shown or attached. Third, the ChatGPT acknowledgment is irrelevant to the math, but it might strike some readers as odd; I would not hold it against the paper.

Who this is for: anyone working in motivic integration, stringy Hodge numbers, or moduli of bundles. It is a counterexample, so the immediate value is to close a conjecture; the long-term value is to redirect the search for a cohomological interpretation.

Recommendation: send to a good referee. The proof is short, but the result is important and the dependence on external formula justifies verification. I would accept after minor revision—mainly asking them to include the simplification and a note on cross-checking the KL04 formula.

Referee Report

1 major / 2 minor

Summary. The paper presents a counterexample to Batyrev's conjecture on the non-negativity of stringy Hodge numbers. The authors take a smooth projective genus-3 curve C, let M0 be the coarse moduli space of rank-2 semistable bundles with trivial determinant, and set X = M0 × P^1. They note that M0 is terminal and Gorenstein by [KL04, Cor 5.4] and [Kie03], so X is a 7-dimensional projective variety with Gorenstein terminal singularities. Using [KL04, Thm 6.1], they write the stringy E-function E_st(M0;u,v) as a rational function, simplify it to P(u,v)/(1+uv), and then, since E_st(P^1)=1+uv, they obtain E_st(X)=P(u,v), a polynomial. The coefficient of u^2 v^5 in P is 1, giving h^{2,5}_st(X)=(-1)^{2+5}·1=-1, contradicting Batyrev's conjecture.

Significance. If correct, this resolves a central conjecture in motivic integration in the negative. The construction is simple and the computation is explicit and checkable. The paper's main strength is that it reduces the counterexample to a single coefficient extraction from a known formula; the weakness is that the quoted formula is taken on faith, and the simplification to P is not shown. Given the importance of the claim, independent verification is desirable.

major comments (1)

[§2, Eq. (1)] The identity E_st(M0;u,v) = P(u,v)/(1+uv) is asserted without displaying the algebra that transforms the rational expression quoted from [KL04, Thm 6.1] into P/(1+uv). The entire counterexample hinges on this identity: the negative Hodge number is exactly the coefficient [u^2 v^5]P, and any error in the simplification would change the sign. Please provide the full calculation, or a reproducible computer-algebra verification, and ideally compare the formula with the independent computation in [Kie03].

minor comments (2)

[§2] The multiplicativity statement E_st(M0 × P^1) = E_st(M0)E_st(P^1), with E_st(P^1)=1+uv, is used without proof or citation. This is standard for log-terminal varieties, but a reference would help the reader.
[§2] The paper moves from the quoted rational expression to 'Thus, E_st(M0) = P/(1+uv)' with no intermediate steps. Even if the algebra is routine, showing the factorization would make the proof self-contained and easier to audit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the counter-example is a direct algebraic consequence of externally quoted E-function formulas for M0.

full rationale

The derivation in Section 2 combines three ingredients: (1) terminality and Gorensteinness of M0 quoted from [KL04, Cor. 5.4] and [Kie03, §2]; (2) the stringy E-function E_st(M0;u,v) quoted from [KL04, Thm. 6.1]; and (3) the standard identity E_st(P^1;u,v)=1+uv, followed by reading the coefficient of u^2 v^5. None of these steps defines the target quantity in terms of itself. The coefficient producing h^{2,5}_st(X)=-1 is read off the polynomial P(u,v), not fitted or imposed. The paper does not claim to re-derive the [KL04] formula, and reliance on an external, previously published computation is not circularity; an error there would be a correctness issue, not a circular one. The authors' own prior work [SU24a, SU24b, HSU26] appears only in the introduction as background and is not load-bearing in the proof of Theorem 1.2. Thus the central claim is self-contained conditional on the quoted external formula, with no circular step.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The stringy E-function coefficients come from a cited theorem, not from fitting. The axioms are either standard properties of motivic integration or previously-published facts about M0.

assumptions (6)

domain assumption M0 has terminal singularities
Cited [KL04, Corollary 5.4]; ensures X has at worst canonical singularities as required by Batyrev's conjecture.
domain assumption M0 is Gorenstein
Cited [Kie03, §2]; Gorenstein condition is part of the conjecture's hypotheses and needed for stringy E-function definition.
domain assumption Stringy E-function of M0 equals the rational function displayed in the proof
Quoted from [KL04, Theorem 6.1]; this is the central computational input, not re-derived.
standard math Stringy E-function is multiplicative under products
Used to write E^st(M0 × P^1) = E^st(M0) E^st(P^1); standard property of motivic integration.
standard math E^st(P^1; u, v) = 1 + uv
For a smooth variety, the stringy E-function is the Hodge-Deligne polynomial; for P^1 this is 1+uv.
standard math Stringy Hodge numbers are signed coefficients of the E-function
Batyrev's definition: h^{p,q}_st = (-1)^{p+q} times the coefficient of u^p v^q; gives -1 when the coefficient is 1.

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Cite this review

Pith. "Pith review of A counter-example to Batyrev's conjecture on the non-negativity of stringy Hodge numbers." pith.science (2026). https://pith.science/paper/RUDMVFIN

@misc{pith2026260719184,
  author       = {Pith},
  title        = {Pith review of: A counter-example to Batyrev's conjecture on the non-negativity of stringy Hodge numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RUDMVFIN}},
  note         = {Machine review of arXiv:2607.19184}
}

abstract

Batyrev's conjecture on the non-negativity of stringy Hodge numbers has been a fundamental open problem, guiding and motivating many beautiful mathematical results in motivic integration, mirror symmetry, and the McKay correspondence. Let $M_0$ be the coarse moduli space of rank 2 semistable bundles with trivial determinant over a fixed smooth projective genus 3 curve. Using a formula, obtained by Kiem and Kiem-Li, for the stringy $E$-function of $M_0$, we verify that $M_0 \times\mathbb{P}^1$ is a counter-example to Batyrev's conjecture.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The dimension threshold for Batyrev's non-negativity conjecture on stringy Hodge numbers
math.AG 2026-08 accept novelty 8.0 of 10

Batyrev's non-negativity conjecture on stringy Hodge numbers is true for Gorenstein canonical projective varieties in dimension at most 4 and false in all dimensions 5 and higher.

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