A Miyaoka-Yau inequality for hyperplane arrangements in $\mathbb{CP}^n$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918394254589952 |
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| author | de Borbon, Martin Panov, Dmitri |
| author_facet | de Borbon, Martin Panov, Dmitri |
| contents | Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{CP}^n$. We define a quadratic form $Q$ on $\mathbb{R}^{\mathcal{H}}$ that is entirely determined by the intersection poset of $\mathcal{H}$. Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if $\mathbf{a} \in \mathbb{R}^{\mathcal{H}}$ is such that the weighted arrangement $(\mathcal{H}, \mathbf{a})$ is stable, then $Q(\mathbf{a}) \leq 0$.
As an application, we consider the symmetric case where all the weights are equal. The inequality $Q(a, \ldots, a) \leq 0$ gives a lower bound for the total sum of multiplicities of codimension $2$ intersection subspaces of $\mathcal{H}$. The lower bound is attained when every $H \in \mathcal{H}$ intersects all the other members of $\mathcal{H} \setminus \{H\}$ along $(1-2/(n+1))|\mathcal{H}| + 1$ codimension $2$ subspaces; extending from $n=2$ to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09573 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Miyaoka-Yau inequality for hyperplane arrangements in $\mathbb{CP}^n$ de Borbon, Martin Panov, Dmitri Algebraic Geometry Combinatorics Differential Geometry Symplectic Geometry Primary: 14N20, 14J60. Secondary: 32S22, 52C35 Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{CP}^n$. We define a quadratic form $Q$ on $\mathbb{R}^{\mathcal{H}}$ that is entirely determined by the intersection poset of $\mathcal{H}$. Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if $\mathbf{a} \in \mathbb{R}^{\mathcal{H}}$ is such that the weighted arrangement $(\mathcal{H}, \mathbf{a})$ is stable, then $Q(\mathbf{a}) \leq 0$. As an application, we consider the symmetric case where all the weights are equal. The inequality $Q(a, \ldots, a) \leq 0$ gives a lower bound for the total sum of multiplicities of codimension $2$ intersection subspaces of $\mathcal{H}$. The lower bound is attained when every $H \in \mathcal{H}$ intersects all the other members of $\mathcal{H} \setminus \{H\}$ along $(1-2/(n+1))|\mathcal{H}| + 1$ codimension $2$ subspaces; extending from $n=2$ to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane. |
| title | A Miyaoka-Yau inequality for hyperplane arrangements in $\mathbb{CP}^n$ |
| topic | Algebraic Geometry Combinatorics Differential Geometry Symplectic Geometry Primary: 14N20, 14J60. Secondary: 32S22, 52C35 |
| url | https://arxiv.org/abs/2411.09573 |