Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries

TL;DR

This paper proves that determinants of Griffiths positive matrices with (1,1)-form entries satisfy Hodge-Riemann type theorems in certain cases, extending classical results to more general matrix classes.

Contribution

It establishes the Hodge-Riemann property for determinants of Griffiths positive matrices with (1,1)-form entries in specific dimensions and conditions, answering a question by Dinh-Nguyen.

Findings

Proves HRR, HLT, LD for 2x2 matrices in 2 and 3 dimensions.

Shows determinants of diagonalized matrices satisfy these theorems in higher dimensions.

Provides applications to matrices with linear combinations of diagonal entries and on complex tori.

Abstract

The classical Hard Lefschetz theorem (HLT), Hodge-Riemann bilinear relation theorem (HRR) and Lefschetz decomposition theorem (LD) are stated for a power of a K\"ahler class on a compact K\"ahler manifold. These theorems are not true for an arbitrary class, even if it contains a smooth strictly positive representative. Dinh-Nguy\^en proved the mixed HLT, HRR and LD for a product of arbitrary K\"ahler classes. Instead of products, they asked whether determinants of Griffiths positive $k\times k$ matrices with $(1,1)$-form entries in $\bc^n$ satisfies these theorems in the linear case. This paper answered their question positively when $k=2$ and $n=2,3$. Moreover, assume that the matrix only has diagonalized entries, for $k=2$ and $n\geqslant 4$, the determinant satisfies HLT for bidegrees $(n-2,0)$, $(n-3,1)$, $(1,n-3)$ and $(0,n-2)$. In particular, for $k=2$ and $n=4,5$ with this extra assumption, the determinant satisfies HRR, HLT and LD. Two applications: First, a Griffiths positive $2\times 2$ matrix with $(1,1)$-form entries, if all entries are $\mathbb{C}$-linear combinations of the diagonal entries, then its determinant also satisfies these theorems. Second, on a complex torus of dimension $\leqslant 5$, the determinant of a Griffiths positive $2\times 2$ matrix with diagonalized entries satisfies these theorems.