"What Is a Complex Matroid?"

Following an ‘Ansatz’ of Björner & Ziegler [BZ], we give an axiomatic development of finite sign vector systems that we call complex matroids. This includes, as special cases, the sign vector systems that encode complex arrangements according to [BZ], and the complexified oriented matroids, whose complements were considered by Gel’fand & Rybnikov [GeR]. Our framework makes it possible to study complex hyperplane arrangements as entirely combinatorial objects. By comparing with the structure of 2-matroids, which model the more general 2-arrangements introduced by Goresky & MacPherson [GoM], one can isolate the essential combinatorial meaning of a “complex structure”. Our development features a topological representation theorem for 2-matroids and complex matroids, and the computation of the cohomology of the complement of a 2arrangement, including its multiplicative structure in the complex case. Duality is established in the cases of complexified oriented matroids, and for realizable complex matroids. Complexified oriented matroids are shown to be matroids with coefficients in the sense of Dress & Wenzel [Dr1], but this fails in general.