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

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 Kähler class on a compact Kähler manifold. These theorems are not true for an arbitrary class, even if it contains a smooth strictly positive representative.

Dinh-Nguyên proved the mixed HLT, HRR and LD for a product of arbitrary Kähler classes. Instead of products, they asked whether determinants of Griffiths positive $k\times k$ matrices with $(1,1)$-form entries in $\mathbb {C}^n$ satisfy 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.