Complexified Real Line Arrangement

• Complexified real line arrangements are finite collections of complex lines defined by real linear equations, bridging real and complex geometric methods.
• They reveal intricate relationships between combinatorics, topology, and geometry, impacting areas like fundamental groups and cohomology rings.
• The study involves operations such as pencil addition and parallel connections to construct examples, while addressing classification and realization challenges.

A complexified real line arrangement is a finite collection of complex lines in affine or projective space defined by linear equations with real coefficients. The study of their combinatorial, topological, and geometric properties illuminates central phenomena in arrangement theory, algebraic topology, and singularity theory. These arrangements are the complexification of real affine line arrangements in $\mathbb{R}^2$, and their complements in $\mathbb{C}^2$ (and associated spaces) exhibit intricate relationships between combinatorics, topology, and geometry. The class of complexified real arrangements provides a natural bridge between real and complex methods, with deep consequences for the topology of complements, classification of arrangements, and realization problems.

1. Definitions and Constructions

Let $\mathcal{A} = \{H_1,\dots,H_n\}$ be a finite set of affine real lines in $\mathbb{R}^2$, each defined by a real-linear form $\alpha_j(x,y) = a_j x + b_j y + c_j$, with $a_j, b_j, c_j \in \mathbb{R}$. The complexification, or complexified real line arrangement, consists of the lines $$H_j^\mathbb{C} = \{(x,y) \in \mathbb{C}^2 \mid \alpha_j(x,y) = 0\}$$ in $\mathbb{C}^2$.

The complement of the arrangement is $$M(\mathcal{A}) = \mathbb{C}^2 \setminus \bigcup_{j=1}^n H_j^\mathbb{C}$$ (Williams, 2015).

Such arrangements may also be defined in the projective setting: a real arrangement in $\mathbb{RP}^2$ given by real homogeneous forms yields a complexified arrangement in $\mathbb{CP}^2$ by extending scalars to $\mathbb{C}$. The intersection properties, such as the intersection lattice formed by non-empty intersections of subsets of lines, are combinatorially invariant under complexification.

The associated objects include:

• Multiple intersection points, with multiplicity $m(P,\mathcal{A})$ being the number of lines passing through $P$.
• The combinatorial data (intersection lattice, or matroid).
• The complement: $\mathbb{C}^2 \setminus \bigcup_j H_j^\mathbb{C}$ or the projective analog.

2. Arrangements, Construction Operations, and Examples

Operations on complexified real arrangements include:

• Addition of pencils: For any $H \in \mathcal{A}$, one can adjoin a pencil of $m \geq 3$ lines $C$ through a point $p \in \mathbb{C}^2$ with $C \cup \{H\}$, where $p$ is not a multiple point of $\mathcal{A}$ and the new lines intersect $A \setminus \{H\}$ only in double points. This produces a "2-generic section" of a parallel connection and realizes arrangements $\mathcal{B}_H = \mathcal{A} \cup C$ (Williams, 2015).
• Parallel connection and truncations: The "parallel connection" and "3-generic direct sum" operations allow for the construction of more complex arrangements, preserving combinatorics and (in specific circumstances) real structure (Williams, 2010). Tree-like arrangements—those whose Fan graph is a forest—are built from such operations.
• Combinatorial models: The Salvetti complex provides a cell complex whose $k$-cells correspond to flags of faces of the arrangement, and which models the homotopy type of the complement (Deshpande, 2012). For ranked oriented matroids, this construction extends to non-realizable or "pseudo-arrangements".

Explicit examples include augmented arrangements where non-diffeomorphic, but homotopy equivalent, complements are constructed by varying the pencil base-point in the construction above (Williams, 2015).

3. Topology of Complements and Homotopy Invariants

The topology of $M(\mathcal{A})$ is rich and highly sensitive to both the intersection lattice and the real geometric embedding.

Key results include:

• Homotopy invariance under generic pencil addition: For any two lines $H_1, H_2 \in \mathcal{A}$, the complements $M(\mathcal{B}_{H_1})$ and $M(\mathcal{B}_{H_2})$ are always homotopy equivalent, but—except in special cases—are not diffeomorphic (Williams, 2015).
• Fundamental group: The Arvola-Randell presentation encodes generators (meridians) around lines and relations from multiple points (cyclic commutators). The fundamental group is highly sensitive to the ordering of meridians at multiple points—two arrangements with the same intersection lattice may have non-isomorphic fundamental groups (Bartolo et al., 2017). Williams showed that if the fundamental group is a direct sum of free groups, then the arrangement can be replaced (up to diffeomorphism of complements) by a complexified-real arrangement with the same combinatorics (Williams, 2010).
• Cohomology ring: The Orlik–Solomon algebra associated to the intersection lattice governs $H^*(M(\mathcal{A}), \mathbb{Z})$.
• Sharp pairs and upper bounds: The existence of a sharp pair (two lines such that all other intersections are on one side) provides control over the cohomology of local systems, with various vanishing and upper bound results (Yoshinaga, 2013, Xie et al., 25 Dec 2025, Yoshinaga, 2013).

The minimal CW-complex for $M(\mathcal{A})$ is combinatorially defined via the real figure and chamber structure, and algorithms for local system cohomology and monodromy eigenspaces (via resonant bands) reduce to linear algebra on chamber decompositions (Yoshinaga, 2013, Yoshinaga, 2013).

4. Arrangement Realization and Classification Problems

Not every intersection lattice can be realized by a complexified real line arrangement—realizability over $\mathbb{R}$ is strictly more restrictive than over $\mathbb{C}$. The notion of supersolvable arrangements—those with a modular point—has significant implications:

• Supersolvable arrangements have strong bounds on the number of simple intersection points and relate to classical theorems like the Dirac–Motzkin conjecture. Over $\mathbb{R}$, every line not through the modular point must contain a simple intersection, yielding $t_2(\mathcal{A}) \geq n/2$ for a rank-3 arrangement of $n$ lines (Anzis et al., 2015).
• The complexification preserves incidence and intersection multiplicity data.
• The Hirzebruch property provides rigidity: in the real case, there are exactly four real arrangements with this property, each corresponding to a real reflection group (Panov, 2016).

Classification questions focus on whether the combinatorics (intersection lattice) determines the topological type and whether isomorphism of fundamental groups implies diffeomorphism of complements (Bartolo et al., 2017, Williams, 2015).

5. Homology of Cyclic Covers and Milnor Fibers

Given a complexified real arrangement, its Milnor fiber and finite cyclic covers of the complement are central objects:

• Cyclic covers and torsion: For a cyclic cover of order $d$, $M(\mathcal{A})_d \to M(\mathcal{A})$, the first integral homology $H_1(M(\mathcal{A})_d; \mathbb{Z})$ is torsion-free if at every multiple point of multiplicity $m_X>2$, $\gcd(m_X,d)=1$ (Liu et al., 2023, Williams, 2011). This generalizes prior results for the Milnor fiber ($d=n$) and includes the classical condition of Cohen–Dimca–Orlik.
• Rank formula: The explicit formula

$$\operatorname{rank} H_1(M(\mathcal{A})_d; \mathbb{Z}) = (n-1) + \sum_{X} (m_X-2)(\gcd(m_X,d)-1)$$

holds for equal weights assigned to all meridians; with coprimality conditions, the rank is $(n-1)$ and $H_1$ is free (Liu et al., 2023, Williams, 2011).

Obstructions to torsion-freeness appear precisely where multiple points have multiplicity sharing a common factor with the cover degree, as in arrangements constructed to test this phenomenon (Liu et al., 2023).

6. Topological and Geometric Realizations

Complexified real arrangements connect real, complex, and symplectic geometry:

• Complexification and symplectic realization: Any arrangement of real lines (or more generally, pseudolines) in $\mathbb{RP}^2$ can be complexified to a configuration of smooth symplectic $S^2$'s in $\mathbb{CP}^2$ with incidence and intersection data preserved (Ruberman et al., 2016). No obstruction exists to symplectic realization at the level of arrangement; rigid algebraic constraints (e.g. Pappus theorem) only appear for algebraic realizations.
• Kirby diagrams and handle structures: The complement $M(\mathcal{A})$ admits an explicit handle decomposition up to 2-handles—0-handle (4-ball), 1-handles for each line, 2-handles for each chamber—reflected in a combinatorially determined Kirby diagram (Sugawara et al., 2021).

The minimal CW structure, discrete Salvetti-type models, and combinatorial definitions (via oriented matroid theory) deliver tractable, combinatorially determined invariants for complements and their covers (Yoshinaga, 2013, Deshpande, 2012, Delucchi et al., 2013).

7. Combinatorial and Classification Phenomena

Distinguishing topological, algebraic, and combinatorial invariants is central:

Property	Determined by intersection lattice?	References
Homotopy type of complement	Not always	(Williams, 2015)
Diffeomorphism type	Not always	(Williams, 2015)
Fundamental group	Not always	(Bartolo et al., 2017)
Cohomology ring	Yes (Orlik–Solomon algebra)	(Williams, 2015)

Explicit Zariski pairs—arrangements with the same intersection lattice but non-homeomorphic complements—arise via modifications that preserve combinatorics but alter embedding (Guerville-Ballé et al., 2017). The "chamber weight" and other real-geometric invariants can distinguish such pairs even when the cohomological invariants coincide.

The ambient classification of arrangements, realization problems (including those for pseudo-arrangements), and the effect of real combinatorics on the topology of complex complements remain areas of active research. Complexified real line arrangements are thus a primary testing ground for the interplay between combinatorics, geometry, and topology in the broader context of configuration and arrangement theory.