发表机构
East China Normal University（华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造显式Nakano正矩阵，证明其行列式在多数双次数下不满足Hodge-Riemann性质，并在同时可对角化条件下给出部分肯定与否定结果。
AI 中文摘要
Dinh和Nguyên提出了一个问题：具有$(1,1)$-形式分量的Griffiths正矩阵的行列式是否属于Hodge--Riemann锥。对于每个$n\geqslant4$和$2\leqslant k\leqslant n-2$，我们给出了在$\C^n$上常值$(1,1)$-形式的显式Nakano正$k\times k$矩阵，其行列式在双次数$(1,n-k-1)$处具有奇异的Lefschetz映射。这在整个范围内给出了否定答案。边界情形$n=k\geqslant4$和$n=k+1\geqslant5$仍然开放。接下来，我们在同时可对角化（SD）条件下研究该问题。在此条件下，我们证明了在满足$p+q=n-k$且$\min(p,q)\leqslant1$的每个双次数$(p,q)$处具有Hodge--Riemann性质。因此，当$1\leqslant k=n-2$或$n-3$时，SD给出了肯定答案。然而，对于每个$n\geqslant6$和$2\leqslant k\leqslant n-4$，我们给出了SD例子，其双次数$(2,n-k-2)$处的Lefschetz映射是奇异的，表明SD单独并不蕴含所有双次数下的Hodge--Riemann性质。肯定结果使用了Ross、Süß和Wannerer发展的对偶Lorentz多项式理论，特别是他们的广义Alexandrov--Fenchel不等式及其等式刻画。精确的Python验证程序伴随这些构造。
英文摘要
Dinh and Nguyên asked whether the determinant of a Griffiths positive matrix of $(1,1)$-forms belongs to the Hodge--Riemann cone. For every $n\geqslant4$ and $2\leqslant k\leqslant n-2$, 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 $\min(p,q)\leqslant1$. The proof uses the generalized Alexandrov--Fenchel inequality of Ross, Süß, and Wannerer. For every $n\geqslant6$ and $2\leqslant k\leqslant n-4$, 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 $n=k\geqslant4$ and $n=k+1\geqslant5$ 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 $2\times2$ counterexample on $\mathbb{C}^4$ can be both Nakano positive and dual Nakano positive.
Comments18 pages. 2 python files for computational verification. This version brings 2 open questions in the end
