Speaker: Prof. Zhangchi Chen (East China Normal University)

Title: Nakano positivity may not imply Hodge-Riemann property

Time: 10:00-11:30  September 15, 2026 (Tuesday)

Inviter: Prof. Baohua Fu (MCM, CAS)

Place: MCM410

Abstract: Let M be a (k,k) matrix with (1,1)-form entries over C^n, with k<=n. Then Omega:=det M is a (k,k)-form over C^n. Dinh-Nguyen asked whether Griffiths positivity of M implies hard-Lefschetz or Hodge-Riemann properties of Omega. The question is trivial (Yes) for k=0,1,n-1,n. When M is diagonal, Dinh-Nguyen proved Yes. For general M, under assumption that entries of M can be simultaneously diagonalized (SD condition), Chen proved Yes when n=4,5 and k=2. Recently, AI constructed explicit counterexamples to DN's question when n>=4 and 2<=k<=n-2 unconditionally, or when n>=6 and 2<=k<=n-4 under SD condition. When n>=6 and k=n-3, n-2 under SD condition, the answer to DN's question is Yes, by using Lorentzian polynomials.

Thus DN's question is fully answered.