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

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. [ABSTRACT FROM AUTHOR]