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

Abstract

For an essential, central hyperplane arrangement A in V=k^{n+1}, we show that Ω^1(A) (the module of logarithmic one forms with poles along A) gives rise to a locally free sheaf on P^n if and only if for all X in L_A with rank X<dim V, the module Ω^1(A_X) is free. Our main result is that in this case the Poicare polynomial of A is essentially the Chern polynomial. The proof is based on a result of Solomon and Terao and on a formula we give for the Chern polynomial of a bundle E on P^n in terms of the Hilbert series of \oplus_m H^0(\wedge^iE(m)). If Ω^1(A)has projective dimension one and is locally free, we give a minimal free resolution for Ω^p, and show that \wedge^p(Ω^1(A))\isoΩ^p(A), generalizing results of Rose and Terao on generic arrangements.