A unified family of counterexamples to Batyrev's non-negativity conjecture on stringy Hodge numbers
Abstract
Recently, Huang and Satriano determined the sharp dimension threshold for Batyrev's non-negativity conjecture on stringy Hodge numbers: the conjecture holds in dimensions at most four and fails in every dimension at least five. We show that their counterexamples fit into a single broad three-parameter family Ym,d:=V(∑i=0m si gi(x)+h(x))⊂ℙm+d+1, where each gi is homogeneous of degree d−1 and h is homogeneous of degree d, whose singular locus is a linear space Λ≅ℙm. Blowing up Λ gives a log resolution with one smooth exceptional divisor of discrepancy one. We compute the Hodge-Deligne polynomials of both the resolution and the exceptional divisor from their projective-space fibrations over ℙd, derive a master formula for Est(Ym,d×(ℙ1)n), and prove the structure formula hstp,q(Ym,d×(ℙ1)n)=Ap,q+Bp,q−Cp,q, where the three terms are non-negative. Thus the only possible source of negativity is −Cp,q, which is governed by the primitive middle cohomology of Zm,d=V(g0,…,gm)⊂ℙd. Off the diagonal, every negative entry equals −Cp,q, a binomially weighted sum of shifted primitive Hodge numbers of Zm,d. On the diagonal, the behaviour is governed by the parity of m+d. We also obtain a complete classification for 0≤ m≤ d−1, d≥3, and n≥1. The only non-counterexamples are (m,d)=(0,3) for every n≥1, and (m,d)=(2,3) for n≥4.
Related
- Counterexamples to the Monotone Completeness Conjecture
- Counterexamples to the ordinary Fried conjecture near hyperbolic metrics
- Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture
- Counterexamples to the Jacobian Conjecture from the viewpoint of lineability
- Counterexamples to a conjecture of Kamenetsky on OEIS A173419
BibTeX
@misc{dais2026unified,
title = {{A unified family of counterexamples to Batyrev's non-negativity conjecture on stringy Hodge numbers}},
author = {Dimitrios I. Dais},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06829},
eprint = {2609.06829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06829},
}
RIS
TY - UNPB
TI - A unified family of counterexamples to Batyrev's non-negativity conjecture on stringy Hodge numbers
AU - Dimitrios I. Dais
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.06829
UR - https://arxiv.org/abs/2609.06829
AB - Recently, Huang and Satriano determined the sharp dimension threshold for Batyrev's non-negativity conjecture on stringy Hodge numbers: the conjecture holds in dimensions at most four and fails in every dimension at least five. We show that their counterexamples fit into a single broad three-parameter family Y_m,d:=V(∑_i=0^m s_i g_i(x)+h(x))⊂ℙ^m+d+1, where each g_i is homogeneous of degree d−1 and h is homogeneous of degree d, whose singular locus is a linear space Λ≅ℙ^m. Blowing up Λ gives a log resolution with one smooth exceptional divisor of discrepancy one. We compute the Hodge-Deligne polynomials of both the resolution and the exceptional divisor from their projective-space fibrations over ℙ^d, derive a master formula for E_st(Y_m,d×(ℙ^1)^n), and prove the structure formula h_st^p,q(Y_m,d×(ℙ^1)^n)=A_p,q+B_p,q−C_p,q, where the three terms are non-negative. Thus the only possible source of negativity is −C_p,q, which is governed by the primitive middle cohomology of Z_m,d=V(g_0,…,g_m)⊂ℙ^d. Off the diagonal, every negative entry equals −C_p,q, a binomially weighted sum of shifted primitive Hodge numbers of Z_m,d. On the diagonal, the behaviour is governed by the parity of m+d. We also obtain a complete classification for 0≤ m≤ d−1, d≥3, and n≥1. The only non-counterexamples are (m,d)=(0,3) for every n≥1, and (m,d)=(2,3) for n≥4.
ER -