On the formal structure of logarithmic vector fields

Abstract

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.

13 pages

Cited by in corpus (8)

• The Jacobian module, the Milnor fiber, and the $D$-module generated by $f^s$
• Hypersurface singularities with monomial Jacobian ideal
• Local cohomology of logarithmic forms
• Logarithmic comparison theorem versus Gauss-Manin system for isolated singularities
• Quasihomogeneity of isolated singularities and logarithmic cohomology
• Lifting free divisors
• Maximal multihomogeneity of algebraic hypersurface singularities
• On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors