On the topology and combinatorics of decomposable arrangements
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909810999427072 |
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| author | Suciu, Alexander I. |
| author_facet | Suciu, Alexander I. |
| contents | A complex hyperplane arrangement $\mathcal{A}$ is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group $G$ decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the $I$-adic completion of the Alexander invariant $B(G)$ also decomposes as a direct sum of "local" invariants and the Chen ranks of $G$ are the sums of the local contributions. Moreover, if $B(G)$ is separated, then the degree 1 cohomology jump loci of the complement of $\mathcal{A}$ have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04784 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the topology and combinatorics of decomposable arrangements Suciu, Alexander I. Group Theory Algebraic Geometry Geometric Topology 52C35, 16W70, 17B70, 20F14, 20F40, 32S55, 57M07 A complex hyperplane arrangement $\mathcal{A}$ is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group $G$ decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the $I$-adic completion of the Alexander invariant $B(G)$ also decomposes as a direct sum of "local" invariants and the Chen ranks of $G$ are the sums of the local contributions. Moreover, if $B(G)$ is separated, then the degree 1 cohomology jump loci of the complement of $\mathcal{A}$ have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1. |
| title | On the topology and combinatorics of decomposable arrangements |
| topic | Group Theory Algebraic Geometry Geometric Topology 52C35, 16W70, 17B70, 20F14, 20F40, 32S55, 57M07 |
| url | https://arxiv.org/abs/2404.04784 |