Essential Projective Hyperplane Complements

• Essential projective hyperplane complements are open subsets of projective space obtained by removing a spanning arrangement of hyperplanes.
• They connect combinatorics, algebraic geometry, and topology, offering insights through matroid theory, CW decompositions, and incidence relations.
• Advanced tools like the Orlik–Solomon algebra and tropical compactifications help compute cohomological invariants and describe their topological structure.

An essential projective hyperplane complement is the open subset of projective space (or of a projective space bundle) obtained by removing an essential arrangement of hyperplanes—that is, a finite collection whose associated normal vectors span the whole space. Such complements encode rich interactions between combinatorics, algebraic geometry, topology, tropical and toric geometry, and incidence theory. The study of their structure, cohomology, and compactifications forms a central area within modern arrangement theory, with deep connections to matroid theory, tropical and toric geometry, and moduli spaces.

1. Definitions and Foundational Structures

Let $V$ be a $(d+1)$-dimensional vector space over a field $K$, with projective space $P(V) = \mathbb{P}^d_K$. An arrangement $A = \{H_0, \ldots, H_n\}$ of hyperplanes in $P(V)$ is called essential if $\bigcap_{i=0}^n \operatorname{Ker}(\ell_i) = \{0\}$ for defining forms $\ell_i \in V^*$. Equivalently, the $\ell_i$ are linearly independent up to scalars and have no common nontrivial factor. The complement

$\Omega = P(V) \setminus \bigcup_{i=0}^n H_i$

is then called an essential projective hyperplane complement.

Central to the structure is the associated matroid $M_A$ on $\{0, \ldots, n\}$; $A$ is connected if $M_A$ is connected (i.e., its intersection lattice is irreducible). Over $\mathbb{C}$, $\Omega$ is a smooth, affine, very affine variety of dimension $d$; it embeds intrinsically in $\mathbb{G}_m^n$ via $(\ell_1/\ell_0, \ldots, \ell_n/\ell_0)$ and is described as a linear (translated) subtorus cut out by independent affine-linear relations (Kurul et al., 2017, Elmaazouz et al., 20 Jan 2026).

The generalization to hyperplane sub-bundles $H$ inside projective space bundles $\pi: \mathbb{P}(E) \to \mathbb{P}^1$ involves replacing the ambient projective space by a relative analog. If $H = \mathbb{P}(F) \subset \mathbb{P}(E)$ for a rank $r-1$ subbundle $F$ of $E$, and $H$ is ample, then $\mathbb{P}(E) \setminus H$ is again an affine variety with properties determined purely by the self-intersection number $(H^r)$ (Dubouloz, 2011).

2. Combinatorics, Topology, and Cohomology

The topology of $\Omega$ is governed by the combinatorics of $A$. The sequence of Betti numbers is determined by the matroidal Whitney numbers of the first kind and the Orlik–Solomon algebra $A^*(A) = \Lambda(e_1, \ldots, e_d)/I$, where $I$ encodes dependence relations among $\{\ell_i\}$ (Budur, 2011, Denham, 2013). The cohomology Poincaré polynomial $P_\Omega(t)$, the (projective) characteristic polynomial $\chi_A(t)$, and properties such as log-concavity of the Betti numbers reflect deep combinatorial invariants (Budur, 2011, Denham, 2013):

• $P_\Omega(t) = \sum_{i=0}^n h_i t^i$, with $h_i = \dim_{\mathbb{C}} H^i(\Omega, \mathbb{C})$.
• $\chi_A(t) = \sum_{i=0}^n (-1)^i b_i t^{n-i}$, $b_i$ built from the $h_j$.

Higher-order resonance varieties $R^i(A)$, recording loci where the Orlik–Solomon differential $d_v$ has extra cohomology, are determinantal loci defined by the degeneracy of explicitly constructed vector bundle maps (Budur, 2011).

Minimal CW structures exist for $\Omega$: by Lefschetz pencil techniques and variation maps, $X = \mathbb{P}^n \setminus \bigcup_{i=1}^d H_i$ admits a minimal CW decomposition, with the number of $q$-cells precisely the Betti number $b_q$. The top Betti number $b_n$ is computable via sums of global polar invariants attached to the arrangement (Tibar, 2014).

3. Incidence Theory and Reconstruction from Complements

An essential structural result is that, under suitable assumptions on the sizes of lines (minimum number of points outside a fixed "horizon" set $H$), the pair $(\mathbb{P}(V), H)$ can be fully reconstructed from the incidence structure of $\mathbb{P}(V) \setminus H$. The main theorem asserts:

• If $H \subset \mathrm{PG}(V)$ with $0 < \operatorname{ind}(H) < \infty$ and all lines satisfy $|\ell| \geq 2\,\operatorname{ind}(H) + 2$, then $(\mathrm{PG}(V), H)$ is definable from the complement (Żynel et al., 2013).

The proof proceeds by defining an incidence-based parallelism among the remaining lines, identifying stars (cliques corresponding to improper points in $H$), and reconstructing improper lines via an incidence-theoretic "collinearity" relation.

This reconstruction method generalizes to Grassmannians: given a sufficient "thickness" condition and a mild partial-projective spanning hypothesis, the ambient Grassmann space with its horizon is recoverable from its complement. This approach shows that affine/projective completions, slit spaces, and affine Grassmannians are unified under the combinatorial incidence-theoretic paradigm (Żynel et al., 2013).

4. Tropical and Toric Geometry: Bergman Fan and Compactifications

Tropical geometry enters through tropicalizations and the Bergman fan associated to the matroid $M_A$ (Kurul et al., 2017, Denham, 2013). The tropicalization of $\Omega$ in its intrinsic torus embedding yields the tropical linear space of $M_A$: $$\operatorname{Trop}(M_A) = \{ w \in \mathbb{R}^{n+1}/\mathbb{R}(1,\ldots,1) : \forall \text{ circuits }C,\, \min\{w_i : i \in C\}\text{ attained at least twice}\}.$$ The Bergman fan $B(M_A)$, the coarsest fan structure on $\operatorname{Trop}(M_A)$, provides the indexing set for torus-orbit decompositions in natural compactifications.

Relevant compactifications include:

• Visible contour (Tevelev) compactification: The closure $\overline{j(\Omega)}$ in the toric variety associated to $B(M_A)$. For essential and connected $A$, every dominant endomorphism of $\Omega$ extends uniquely to its tropical compactification, and automorphisms extend to automorphisms (Kurul et al., 2017).
• Wonderful (De Concini–Procesi) compactification: Constructed by simultaneous blowups along all intersections of the arrangement as specified by a building set of flats (minimal or maximal), yielding a smooth projective variety with boundary a simple normal-crossings divisor indexed by nested sets (Denham, 2013). For many arrangements, the Bergman fan coincides with the nested-set fan, making the visible contour and (minimal) wonderful compactifications identical.

This tropical/toric perspective connects the topology and boundary strata of compactified hyperplane arrangement complements to matroid-theoretic and combinatorial data. Examples include classical moduli spaces (e.g., $\overline{M}_{0,n}$ as the minimal wonderful model of the braid arrangement) (Denham, 2013).

5. Homotopy-Theoretic and Pair-of-Pants Decomposition Results

The intrinsic toric embedding of essential complements leads to strong results on the topology of their co-amoebae. The angle map

$$\mathrm{ang}: (\mathbb{C}^\times)^n \to (S^1)^n, \quad (z_1,\ldots,z_n) \mapsto \left(\frac{z_1}{|z_1|},\ldots,\frac{z_n}{|z_n|}\right)$$

restricts to a map $\mathrm{ang}: \Omega \to \Theta$, where $\Theta$ is the co-amoeba (angle set). For any essential projective hyperplane complement, the angle map is a homotopy equivalence. The fibers are contractible, local trivializations exist over semi-algebraic toric cones, and a gluing argument shows the entire map is a homotopy equivalence (Elmaazouz et al., 20 Jan 2026).

This persists for finite Kummer covers: for any $m \geq 1$, the $m$th-root covering of $\Omega$ remains homotopy equivalent to its image under the angle map. Explicit analysis for arrangements such as $\mathbb{P}^1\setminus\{0,1,\infty\}$ recovers the known bouquet-of-circles topology and describes how Kummer covers correspond to unions of pair-of-pants, generalizing classical surface decompositions (Elmaazouz et al., 20 Jan 2026).

6. Classification and Moduli of Affine Complements in Bundles

The abstract isomorphism type of the complement of an ample hyperplane sub-bundle $H$ in a projective bundle $\mathbb{P}(E) \to \mathbb{P}^1$ depends only on the self-intersection number $(H^r)$. For such complements, there is no dependence on either the specific bundle $E$ or the choice of $H$ with fixed $(H^r)$. The structure is that of a nontrivial torsor under a vector bundle, and its moduli reduce to the single discrete invariant $(H^r)$. The automorphism group is generated by translations along the torsor and base automorphisms preserving the relevant cohomology classes (Dubouloz, 2011).

7. Broader Significance and Open Directions

Essential projective hyperplane complements unify several geometric and combinatorial phenomena. Key consequences include:

• Recovering the geometry and the "horizon" set from the incidence structure of the complement without additional structure (Żynel et al., 2013).
• Rigidity and extension properties of dominant self-maps and automorphisms to toric and wonderful compactifications (Kurul et al., 2017).
• Explicit topological and homological invariants computable from matroidal and polar data (Budur, 2011, Tibar, 2014).
• Generalizations to Grassmannians, affine Grassmannians, and moduli of configurations in higher dimensions, all within a combinatorial framework.

The interface of incidence geometry, tropical and toric methods, and matroid theory underlies many contemporary research trends, including the study of configuration spaces, moduli of curves, and the topology of non-compact but “almost toric” varieties. Essential projective hyperplane complements thus occupy a central organizing role in modern algebraic geometry, combinatorics, and topological studies of arrangements.