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On the formal structure of logarithmic vector fields. (English) Zbl 1096.32016

Summary: 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. Calderón-Moreno et al. conjectured this implication in all dimensions and proved it in dimension two. We prove a theorem that 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.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
32S20 Global theory of complex singularities; cohomological properties
14F40 de Rham cohomology and algebraic geometry
17B66 Lie algebras of vector fields and related (super) algebras
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