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 × k matrices with (1, 1)-form entries in Cⁿ 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 ≥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 × 2 matrix with (1, 1)-form entries, if all entries are C-linear combinations of the diagonal entries, then its determinant also satisfies these theorems. Second, on a complex torus of dimension ≤5, the determinant of a Griffiths positive 2 × 2 matrix with diagonalized entries satisfies these theorems.