Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries (Zhangchi Chen - Chinese Academy of Science, Beijing)

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. Explicit counterexamples of bidegree (2,2) classes in dimension 4 can be found in Timorin (1998) and Berndtsson-Sibony (2002). Dinh-Nguyên (2006, 2013) proved the mixed HLT, HRR, 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 C^n satisfies these theorems in the linear case. In a recent work I gave positive answer 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, Dinh-Nguyên's question has positive answer when k=2 and n=4,5 with this extra assumption. The proof uses a Heron's formula type factorization, observed by computer (Mathematica).