The Module of Logarithmic p-Forms of a Locally Free Arrangement

For an essential, central hyperplane arrangement V kn + 1 we show that Ω1( ) (the module of logarithmic one forms with poles along ) gives rise to a locally free sheaf on Pn if and only if, for all X L with rank X < dim V, the module Ω1( X) is free. Motivated by a result of L. Solomon and H. Terao (1987, Adv. Math.64, 305–325), we give a formula for the Chern polynomial of a bundle on Pn in terms of the Hilbert series of m ZH0(Pn, i (m)). As a corollary, we prove that if the sheaf associated to Ω1( ) is locally free, then π( , t) is essentially the Chern polynomial. If Ω1( ) has projective dimension one and is locally free, we give a minimal free resolution for Ωp and show that ΛpΩ1( ) Ωp( ), generalizing results of L. Rose and H. Terao (1991, J. Algebra136, 376–400) on generic arrangements.