Differential forms on free and almost free divisors (Q2766376)

The hypersurface \(D\) in the complex manifold \(X\) is a free divisor if the \({\mathcal O}_X\)-module \(\text{Der} (\log D)\) of germs of vector fields on \(X\) which are tangent to \(D\), is locally free. Examples of free divisors include smooth hypersurfaces, normal crossing divisors, reflection arangements and the discriminants of versal deformations of many classes of singularities.NEWLINENEWLINENEWLINEIn many contexts, one is given a divisor \(D_0\) which is free outside \(x_0\), and a family \((D,x_0)\to (\mathbb{C},0)\) with special fibre \((D_0, x_0)\), in which the general fibre is free. Typically, \(D_t\) has homology classes which vanish when \(t\) returns to \(0\). We use the framework for the study of such families provided by J. Damon's theory of almost free divisors. To describe the vanishing cohomology analytically, we introduce a variant of the usual Kähler forms. The \({\mathcal O}_D\)-modules of these forms, \(\check\Omega^k_D\), enjoy the same depth properties on free divisors as do the usual modules of Kähler forms on isolated hypersurface singularities (also the corresponding relative versions). Using these forms, we describe the Gauss-Manin connection on the vanishing cohomology, in close analogy with the classical description for the Milnor fibration of an isolated complete intersection singularity, due to Brieskorn and Greuel. In particular we prove coherence. This applies in particular to the family \(\{D(f_\lambda)\}_{\lambda\in\Lambda}\) of discriminants of a versal deformation \(\{f_\lambda\}_{\lambda \in\Lambda}\) of a singularity of mappings \(f_0:(\mathbb{C}^n,S) \to(\mathbb{C}^p,0)\), which carry the vanishing cohomology of the singularity. In this case, if \(D_0\) is the discriminant of \(f_0\), then the torsion submodule of \(\check \Omega^{p-1}_{D_0,0}\) has the same \(\mathbb{C}\)-dimension as the module \(N{\mathcal A}_ef_0\) of first-order deformations of \(f_0\).