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 $n⩾4$ and $2⩽k⩽n-2$, we present counterexamples constructed by AI: Nakano positive $k\times k$ matrices of constant $(1,1)$-forms on ${\mathbb{C}}^n$ 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 $\min (p,q)⩽1$. The proof uses the generalized Alexandrov--Fenchel inequality of Ross, Süß, and Wannerer. For every $n⩾6$ and $2⩽k⩽n-4$, 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 $n=k⩾4$ and $n=k+1⩾5$ 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 $2\times 2$ counterexample on ${\mathbb{C}}^4$ can be both Nakano positive and dual Nakano positive.

arXiv comment18 pages. 2 python files for computational verification. This version brings 2 open questions in the end