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