Complex Arrangements: Algebra, Geometry, Topology

A hyperplane arrangement A is a finite collection of hyperplanes in some fixed (typically real or complex) vector space V. For simplicity, in this overview we work over the complex numbers C. There is a host of beautiful mathematics associated to the complement X = V A. Perhaps the first interesting result in the area was Arnol’d’s computation [2] of the cohomology ring of the complement of the pure braid arrangement; this was shortly followed by work of Brieskorn [3] computing the cohomology ring H(X) of X in terms of differential forms. Subsequently, Orlik and Solomon [21] gave a presentation of the ring H(X) as the quotient of an exterior algebra by an ideal determined by the intersection lattice L(A) of the arrangement; the rank one elements of L(A) are the hyperplanes, and a rank i element is a set of hyperplanes meeting in codimension i. A far more delicate invariant of X is the fundamental group; unlike the cohomology ring, π1(X) is not determined by L(A). In [15], Hirzebruch wrote ”The topology of the complement of an arrangement of lines in P 2 is very interesting, the investigation of the fundamental group very difficult.” For any group G, the lower central series is a chain of normal subgroups defined by G1 = G, G2 = [G,G], and Gk+1 = [G,Gk]). Because X is formal (that is, its rational homotopy type is determined by the rational cohomology ring), the ranks of the successive quotients φk = Gk/Gk+1 (the LCS ranks) are combinatorially determined for π1(X); so some information about the fundamental group does depend on the combinatorics of L(A). Combining results of Priddy, Sullivan, Kohno, and the Poincaré-Birkhoff-Witt theorem, one can show that the LCS ranks are determined by the graded pieces of the diagonal Yoneda Ext-algebra ExtH(X))i(C,C)i; this leads to beautiful connections to Koszul algebras and duality, first observed by Shelton-Yuzvinsky [28].