Relative logarithmic cohomology and Nambu structures of maximal degree

Abstract

We present local classification results for isolated singularities of functions with respect to a Nambu structure (multi-vector field) of maximal degree, in a neighbourhood of a smooth point of its degeneracy hypersurface. The results depend on a logarithmic version of the Brieskorn-Sebastiani theorem, which guarantees the finiteness and freeness of the corresponding deformation module. This relates the functional moduli of the classification problem with the integrals of logarithmic forms along the vanishing cycles of the complement of the Milnor fibres of the restriction of the function on the degeneracy hypersurface of the Nambu structure, inside the Milnor fibres of the function itself.