Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.23976 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909559485890560 |
|---|---|
| author | Liu, Yongqiang Yoshinaga, Masahiko |
| author_facet | Liu, Yongqiang Yoshinaga, Masahiko |
| contents | Let $\mathcal{L}$ be a rank one local system with field coefficient on the complement $M(\mathcal{A})$ of an essential complex hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^\ell$. Dimca-Papadima and Randell independently showed that $M(\mathcal{A})$ is homotopy equivalent to a minimal CW-complex. It implies that $\dim H^k(M(\mathcal{A}),\mathcal{L}) \leq b_k(M(\mathcal{A}))$. In this paper, we show that if $\mathcal{A}$ is real, then the inequality holds as equality for some $0\leq k\leq \ell$ if and only if $\mathcal{L}$ is the constant sheaf. The proof is using the descriptions of local system cohomology of $M(\mathcal{A})$ in terms of chambers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal Betti number for local system cohomology of hyperplane arrangement complements Liu, Yongqiang Yoshinaga, Masahiko Algebraic Geometry Let $\mathcal{L}$ be a rank one local system with field coefficient on the complement $M(\mathcal{A})$ of an essential complex hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^\ell$. Dimca-Papadima and Randell independently showed that $M(\mathcal{A})$ is homotopy equivalent to a minimal CW-complex. It implies that $\dim H^k(M(\mathcal{A}),\mathcal{L}) \leq b_k(M(\mathcal{A}))$. In this paper, we show that if $\mathcal{A}$ is real, then the inequality holds as equality for some $0\leq k\leq \ell$ if and only if $\mathcal{L}$ is the constant sheaf. The proof is using the descriptions of local system cohomology of $M(\mathcal{A})$ in terms of chambers. |
| title | Maximal Betti number for local system cohomology of hyperplane arrangement complements |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2503.23976 |