Homology of Local Systems on Real Line Arrangement Complements

TL;DR

The paper develops a Borel–Moore homology framework to compute the first homology of line arrangement complements with complex rank-one local systems, using the real figure to turn topological data into a combinatorial algorithm. It provides a computable upper bound $h_{1}\le\max(0,\#R_{0}-1)$ when monodromies are nontrivial, and then refines the analysis via exceptional divisors and bounded chambers to realize this bound in real, complexified arrangements. A key contribution is the explicit construction of generators from angles at resonant points and relations from bounded chambers, plus a detailed treatment of the sharp-pair case, yielding $h_{1}\le 1$ (and zero in certain constant-monodromy, even-order cases). These results link BM homology with combinatorial real-geometry data and provide a practical algorithm for evaluating $h_{1}$, contributing to understanding how much of the topology of arrangement complements is governed by combinatorics.

Abstract

We study the homology groups of the complement of a complexified real line arrangement with coefficients in complex rank-one local systems. Using Borel--Moore homology, we establish an algorithm computing their dimensions via the real figures of the arrangement. It enables us to give a new upper bound. We further consider the case where the arrangement contains a sharp pair and make partial progress on a conjecture proposed by Yoshinaga.

Figures (8)

• Figure 1: The angles at $p$
• Figure 2: Definition of $\alpha_{p}(\Delta)$
• Figure 3: The loop through $\left[x_{0}:y_{0}\right]$
• Figure 4: Coefficients $\lambda_{p}$
• Figure 5: $A_{3}$ arrangement

Theorems & Definitions (13)

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• proof : Proof of Theorem \ref{['thm: main']}