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

TL;DR

This paper characterizes when the module of logarithmic 1-forms of a hyperplane arrangement forms a locally free sheaf on projective space, linking it to the freeness of modules on subarrangements and relating the Poincaré polynomial to the Chern polynomial.

Contribution

It establishes a criterion for local freeness of the module of logarithmic forms based on subarrangement freeness and connects the Poincaré polynomial with the Chern polynomial, extending previous results.

Findings

Locally free sheaf condition characterized by subarrangement freeness.

Poincaré polynomial is essentially the Chern polynomial in this setting.

Provides a minimal free resolution for mega^p when projective dimension is one.

Abstract

For an essential, central hyperplane arrangement A in V=k^{n+1}, we show that \Omega^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 \Omega^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 \Omega^1(A)has projective dimension one and is locally free, we give a minimal free resolution for \Omega^p, and show that \wedge^p(\Omega^1(A))\iso\Omega^p(A), generalizing results of Rose and Terao on generic arrangements.