Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929204985069568 |
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| author | Chen, Zhangchi |
| author_facet | Chen, Zhangchi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11578 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries Chen, Zhangchi Complex Variables Algebraic Geometry 15A15, 32Q15, 58A14 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. |
| title | Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries |
| topic | Complex Variables Algebraic Geometry 15A15, 32Q15, 58A14 |
| url | https://arxiv.org/abs/2211.11578 |