ar X iv : m at h / 06 07 04 5 v 3 [ m at h . A G ] 3 0 Ju l 2 00 7 LINEAR FREE DIVISORS

A hypersurface D in C n is a linear free divisor if the module of logarithmic vector fields along D has a basis of global linear vector fields. It is then defined by a homogeneous polynomial of degree n and its complement is an open orbit of an algebraic subgroup G D of Gln(C) whose Lie algebra g D can be identified with that of linear logarithmic vector fields along D. We classify all linear free divisors for n ≤ 4. We introduce and study the global logarithmic comparison theorem (GLCT), which is the property of D that the global de Rham morphism induces an iso-It is a global/weak version of the logarithmic comparison theorem (LCT), a logarithmic analog of Grothendieck's comparison theorem. We give a general characterization of GLCT for linear free divisors in terms of Lie algebra and complex group cohomology and conclude that GLCT holds whenever g D is a reductive Lie algebra but that the converse implication is false. Linear free divisors arise naturally as discriminants in quiver representation spaces. We show that all those in a representation space of a real Schur root fulfill GLCT. As a by-product our approach yields a simplified proof of a theorem of V. Kac on the number of components of such discriminants. While our classification shows that linear freeness is a strong condition, we describe a method to generate numerous examples through generalized quiver groups. Guided by properties related to LCT we investigate local quasihomogeneity, strong Euler homogeneity, and Koszul freeness for linear free divisors. We show that for n ≤ 4 all linear free divisors are locally quasihomogeneous and hence LCT and GLCT hold. By counterexample we show that Koszul freeness and hence local quasihomogeneity fails for general linear free divisors and even for quiver discriminants.