Title:

Free divisors and their deformations

A reduced divisor D = V (f) Cn is free if the sheaf Der(logD) := f 2 DerCn (f) 2 (f)OCng of logarithmic vector fields is a locally free OCnmodule. It is linear if, furthermore, Der(logD) is globally generated by a basis consisting of vector fields all of whose coefficients, with respect to the standard basis @=@x1;...; @=@xn of the space DerCn of vector fields on Cn, are linear functions. In principle, linear free divisors, like other kinds of singularities, might be expected to appear in nontrivial parameterised families. As part of this thesis, however, we prove that for reductive linear free divisors, there are no formally nontrivial families, where a linear free divisor is reductive if its associated Lie algebra is reductive, thus reductive linear free divisors are formally rigid. To prove this and to understand better the class of free divisors, we introduce a rigorous deformation theory for germs of free and linear free divisors. A (linearly) admissible deformation is a deformation in which we deform a germ of a (linear) free divisor (D; 0) c (Cn; 0) in such a way that each fiber of the deformation is still a (linear) free divisor and that the singular locus of (D; 0) is deformed atly. Moreover, we explain how to use the de Rham logarithmic complex to compute the space of first order infinitesimal admissible deformations and the Lie algebra cohomology complex to compute the space of first order infinitesimal linearly admissible deformations.
