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

讲座时间 Datetime: 2023年8月14日，星期一， 15:00-17:00

地点 Venue: 海琴二号A418

报告人 Speaker: 陈张弛 博士后

单位 Affiliation: 中科院晨兴数学中心

主持人Host：邵国宽 副教授

报告摘要 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. 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).

报告人简介：

陈张弛，中科院晨兴数学中心博士后，15年本科毕业于清华大学，21年博士毕业于巴黎萨克雷大学（巴黎第十一大学）。主要研究多复变函数论、全纯叶状结构、微分不变量理论等。目前已在J. Geom. Anal.，Ergodic Theory Dynam. Systems.，Linear Algebra Appl.等期刊发表或接收7篇论文。曾担任法国青年数学联赛决赛评委。主持博士后面上基金一项。