On the formal structure of logarithmic vector fields

摘要: In this article, we prove that a free divisor in three dimensional complex manifold must be Euler homogeneous strong sense if the cohomology of its complement is hypercohomology logarithmic differential forms. F.J. Calderon-Moreno et al. conjectured implication all dimensions and proved it dimension two. We theorem which describes special minimal system generators for module formal vector fields. This structure closely related to decomposition field by Kyoji Saito used proof above result. Another consequence truncated Lie algebras fields up are solvable. give an example may fail higher dimensions.