\(\mathbb{Z}\)-local system cohomology of hyperplane arrangements and a Cohen-Dimca-Orlik type theorem. (English) Zbl 1528.14067
This article generalizes a theorem by D. C. Cohen et al. [Ann. Inst. Fourier 53, No. 6, 1883–1896 (2003; Zbl 1054.32016)] for certain generic \(\mathbb{C}\)-local systems to \(\mathbb{Z}\)-local systems.
Consider a complexified arrangement \(\mathcal{A}= \{H_1, \dots, H_n\}\) of hyperplanes in \(\mathbb{C}^n\) (with real equations) and a \(\mathbb{Z}\)-local system \(\mathcal{L}\) on the complement \(M(\mathcal{A}) = \mathbb{C}^n \setminus \bigcup_{i=1}^n H_i\). Assume that the local system \(\mathcal{L}\) satisfies a certain CDO condition, defined in [loc. cit.].
{Theorem}. In this setting \[ H^k(M(\mathcal{A}); \mathcal{L}) = \begin{cases} \mathbb{Z}^{\beta_l} \oplus \mathbb{Z}_2^{\beta_{l-1}} & k=l \\ \mathbb{Z}_2^{\beta_{k-1}} & 1 \leq k \leq l-1 \\ 0 & \text{otherwise} \end{cases} \] where \(\beta_k= \lvert \sum_{i=0}^k (-1)^i b_i(M(\mathcal{A})) \rvert\).
The proof is based on the minimality of the complement \(M(\mathcal{A})\) and on a careful algebraic-combinatorial inspection of an explicit complex.
Consider a complexified arrangement \(\mathcal{A}= \{H_1, \dots, H_n\}\) of hyperplanes in \(\mathbb{C}^n\) (with real equations) and a \(\mathbb{Z}\)-local system \(\mathcal{L}\) on the complement \(M(\mathcal{A}) = \mathbb{C}^n \setminus \bigcup_{i=1}^n H_i\). Assume that the local system \(\mathcal{L}\) satisfies a certain CDO condition, defined in [loc. cit.].
{Theorem}. In this setting \[ H^k(M(\mathcal{A}); \mathcal{L}) = \begin{cases} \mathbb{Z}^{\beta_l} \oplus \mathbb{Z}_2^{\beta_{l-1}} & k=l \\ \mathbb{Z}_2^{\beta_{k-1}} & 1 \leq k \leq l-1 \\ 0 & \text{otherwise} \end{cases} \] where \(\beta_k= \lvert \sum_{i=0}^k (-1)^i b_i(M(\mathcal{A})) \rvert\).
The proof is based on the minimality of the complement \(M(\mathcal{A})\) and on a careful algebraic-combinatorial inspection of an explicit complex.
Reviewer: Roberto Pagaria (Bologna)
MSC:
| 14N20 | Configurations and arrangements of linear subspaces |
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
| 55N25 | Homology with local coefficients, equivariant cohomology |
Citations:
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