Linear Free Divisors

A free divisor D in C is linear if its module of logarithmic vector fields has a basis of global vector fields of degree 0. It is then defined by a homogeneous polynomial of degree n and its complement is an open orbit of an algebraic subgroup GD in Gln(C). The best known example is the normal crossing divisor. Many other such divisors arise, for instance, from quiver representations. We give a large number of examples, and classify all linear free divisors for n ≤ 4. One says that the Logarithmic Comparison Theorem (LCT) holds for a reduced divisor D in a complex manifold X if the de Rham morphism Ω• X (logD) → Rj∗ CU , j : U = X rD →֒ X, is a quasi-isomorphism. This condition implies that the global de Rham morphism also induces an isomorphism H(Γ(C,Ω(logD)) → H(C rD;C) We refer to this second property as the Global Logarithmic Comparison Theorem GLCT. We characterize GLCT for linear free divisors by the property that the complex cohomology of GD coincides with the Lie algebra cohomology with complex coefficients of the Lie algebra gD of G. From this, we conclude that GLCT holds for a linear free divisor D if gD is reductive. In particular this is the case when D is the discriminant in a quiver representation space. Subject Classification 32S20, 14F40, 20G10, 17B66.