Nakano-positive determinants outside the Hodge-Riemann cone
Abstract
Dinh and Nguyên asked whether the determinant of a Griffiths positive matrix of (1,1)-forms belongs to the Hodge--Riemann cone. For every ngeqslant4 and 2leqslant kleqslant n−2, we present counterexamples constructed by AI: Nakano positive k× k matrices of constant (1,1)-forms on ℂn whose determinants have singular Lefschetz maps in bidegree $(1,n-k-1)$. Under the simultaneous diagonalizability (SD) condition, we prove the Hodge--Riemann property in every bidegree (p,q) with p+q=n−k and min(p,q)leqslant1. The proof uses the generalized Alexandrov--Fenchel inequality of Ross, Süß, and Wannerer. For every ngeqslant6 and 2leqslant kleqslant n−4, we also present SD examples whose determinants have singular Lefschetz maps in bidegree $(2,n-k-2)$. Their underlying idea comes from Ross--Toma's construction. The remaining cases n=kgeqslant4 and n=k+1geqslant5 are open for general Griffiths positive matrices. They reduce to the top-Chern case of Griffiths' positivity question, equivalently to Finski's double mixed discriminant problem. We also ask whether a 2×2 counterexample on ℂ4 can be both Nakano positive and dual Nakano positive.
Related
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BibTeX
@misc{chen2026nakanopositive,
title = {{Nakano-positive determinants outside the Hodge-Riemann cone}},
author = {Zhangchi Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07964},
eprint = {2609.07964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07964},
}
RIS
TY - UNPB
TI - Nakano-positive determinants outside the Hodge-Riemann cone
AU - Zhangchi Chen
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.07964
UR - https://arxiv.org/abs/2609.07964
AB - Dinh and Nguyên asked whether the determinant of a Griffiths positive matrix of (1,1)-forms belongs to the Hodge--Riemann cone. For every ngeqslant4 and 2leqslant kleqslant n−2, we present counterexamples constructed by AI: Nakano positive k× k matrices of constant (1,1)-forms on ℂ^n whose determinants have singular Lefschetz maps in bidegree $(1,n-k-1)$. Under the simultaneous diagonalizability (SD) condition, we prove the Hodge--Riemann property in every bidegree (p,q) with p+q=n−k and min(p,q)leqslant1. The proof uses the generalized Alexandrov--Fenchel inequality of Ross, Süß, and Wannerer. For every ngeqslant6 and 2leqslant kleqslant n−4, we also present SD examples whose determinants have singular Lefschetz maps in bidegree $(2,n-k-2)$. Their underlying idea comes from Ross--Toma's construction. The remaining cases n=kgeqslant4 and n=k+1geqslant5 are open for general Griffiths positive matrices. They reduce to the top-Chern case of Griffiths' positivity question, equivalently to Finski's double mixed discriminant problem. We also ask whether a 2×2 counterexample on ℂ^4 can be both Nakano positive and dual Nakano positive.
ER -