Euclidean $\vee$-systems and real PK arrangements

Abstract

We establish a correspondence between two structures arising in the geometry of hyperplane arrangements: Euclidean $\vee$-systems and real polyhedral K\"ahler (PK) arrangements. We prove that every irreducible Euclidean $\vee$-system determines a real PK arrangement, and conversely that every real PK arrangement arises this way. As a consequence, we show that, up to equivalence, there are exactly three irreducible rank-three Euclidean $\vee$-systems whose vectors have equal length; their arrangements are the mirrors of the reflection groups of the regular tetrahedron, cube, and icosahedron. The correspondence also yields a description of the moduli space of Euclidean $\vee$-systems in a fixed projective class: it is homeomorphic to the relative interior of a polytope. We also give a direct proof that the hyperplane arrangement associated with a Euclidean $\vee$-system is simplicial. Among the currently known simplicial line arrangements, we identify precisely those that arise from $\vee$-systems. As a consequence, we prove that the Schreiber--Veselov catalog is complete for irreducible rank-three Euclidean $\vee$-systems with at most $27$ vectors.