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

Zhangchi Chen

TL;DR

This work investigates whether determinants of Griffiths positive $k\times k$ matrices with $(1,1)$-form entries satisfy the Hard Lefschetz theorem, Hodge-Riemann bilinear relations, and Lefschetz decomposition in the linear setting. It proves a positive answer for $k=2$ when $n=2,3$, and, under the extra assumption of diagonalized entries, for $k=2$ and $n\ge 4$ in several key bidegrees, including $(n-2,0)$, $(n-3,1)$, $(1,n-3)$, and $(0,n-2)$; for $n=4,5$ these extend to full HRR/HLT/LD. The paper introduces a diagonalized-entry framework using layer matrices and a multi-layer hyperdeterminant to express $\det(M)$ as a positive $(k,k)$-form, enabling explicit Lefschetz isomorphisms in many cases, while noting a remaining difficulty at $n=6$ for the $(2,2)$-bidegree. Two applications are given: (i) on a compact Kähler manifold, when $\alpha_{1,2}$ lies in the span of the diagonal entries, $\det(M)$ satisfies HRR/HLT/LD; (ii) on complex tori of dimension $\le 5$, $\det(M)$ with diagonalized entries also satisfies these theorems. These results extend the reach of Hodge-Riemann-type statements to Griffiths-cone determinants and have connections to mixed volumes and torus geometry.

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. 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 $\bc^n$ satisfies 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.

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

TL;DR

This work investigates whether determinants of Griffiths positive matrices with -form entries satisfy the Hard Lefschetz theorem, Hodge-Riemann bilinear relations, and Lefschetz decomposition in the linear setting. It proves a positive answer for when , and, under the extra assumption of diagonalized entries, for and in several key bidegrees, including , , , and ; for these extend to full HRR/HLT/LD. The paper introduces a diagonalized-entry framework using layer matrices and a multi-layer hyperdeterminant to express as a positive -form, enabling explicit Lefschetz isomorphisms in many cases, while noting a remaining difficulty at for the -bidegree. Two applications are given: (i) on a compact Kähler manifold, when lies in the span of the diagonal entries, satisfies HRR/HLT/LD; (ii) on complex tori of dimension , with diagonalized entries also satisfies these theorems. These results extend the reach of Hodge-Riemann-type statements to Griffiths-cone determinants and have connections to mixed volumes and torus geometry.

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. 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 matrices with -form entries in satisfies these theorems in the linear case. This paper answered their question positively when and . Moreover, assume that the matrix only has diagonalized entries, for and , the determinant satisfies HLT for bidegrees , , and . In particular, for and with this extra assumption, the determinant satisfies HRR, HLT and LD. Two applications: First, a Griffiths positive matrix with -form entries, if all entries are -linear combinations of the diagonal entries, then its determinant also satisfies these theorems. Second, on a complex torus of dimension , the determinant of a Griffiths positive matrix with diagonalized entries satisfies these theorems.
Paper Structure (8 sections, 21 theorems, 71 equations, 1 figure)

This paper contains 8 sections, 21 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hodge-Riemann forms
  3. Proof for $k=2$ and $n=2,3$
  4. Case all entries of $M$ being diagonalized
  5. Case $k=2$, $n=4$, $M$ with diagonalized entries
  6. Case $k=2$, $n\geqslant 4$, $M$ with diagonalized entries
  7. Difficulty of the case $k=2$, $n=6$ and $M$ with diagonalized entries
  8. Applications on compact Kähler manifolds and on torus

Key Result

Theorem 1.1

The linear map is an isomorphism, where $\smile$ denotes the cup-product on the cohomology ring $\oplus H^*(X,\mathbb{C})$.

Figures (1)

  • Figure :

Theorems & Definitions (24)

  • Theorem 1.1: HLT
  • Theorem 1.2: HRR: Hodge-Riemann biliener relations
  • Theorem 1.3: LD: Lefschetz decomposition theorem
  • Corollary 1.4: Hodge index theorem
  • Theorem 1.5: mixed HLT, HRR, LD
  • Theorem 1.6: Linear mixed HLT, HRR, LD
  • Theorem 1.7: Dinh-Nguyên 2013
  • Corollary 1.8: Answer 1
  • Theorem 1.9: Answer 2
  • Corollary 1.10
  • ...and 14 more