Unfoldings of complex analytic foliations with singularities.(English)Zbl 0591.32020

A codim q foliation F (of complete intersection type) on a complex manifold M is a locally free sub-$${\mathcal O}_ M$$-module of $$\Omega_ M$$ of rank q which is integrable in the sense that $$dF_ z\subset (\Omega_ M\wedge F)_ z$$ for all $$z\in M-S(F),$$ where S(F) is the singular set of the coherent sheaf $$\Omega_ F:=\Omega_ M/F$$. A local foliation is, by definition, the germ of a foliation at a distinguished point. An unfolding of F consists of a deformation $${\mathcal M}\to Q$$ of M with smooth parameter space Q and a codim q foliation F on $${\mathcal M}$$ such that the restriction of F to M is equal to F. There is a natural way (following standard arguments in deformation theory) to define an infinitesimal unfolding map $$\rho: T_{Q,0}\to Ext^ 1_{{\mathcal O}_ M}(\Omega_ F,{\mathcal O}_ M)$$ which is $${\mathbb{C}}$$-linear and, roughly, picks up the first order terms in $${\mathcal M}$$ and F in the given tangent direction. Via this infinitesimal map the set of equivalence classes of first order unfoldings of F can be identified with a subspace $$U(M,F)\subset Ext^ 1_{{\mathcal O}_ M}(\Omega_ F,{\mathcal O}_ M).$$ The aim of this paper is to study this subspace if F is a foliation on a compact manifold or a reduced local foliation. Actually, a detailed description is only given in the local case. In general, U(F) is smaller than $$Ext^ 1_{{\mathcal O}}(\Omega_ F,{\mathcal O})$$. To describe the difference, the author associates to elements of $$Ext^ 1_{{\mathcal O}}(\Omega_ F,{\mathcal O})$$ certain local cohomology classes which may be interpreted as primary obstructions to finding integrating factors for a system of generators of F. The author shows that the vanishing of these obstructions characterizes the subspace U(F). Furthermore, he gives an algorithm to find the first order unfoldings of a given local foliation.

MSC:

 32S30 Deformations of complex singularities; vanishing cycles 57R30 Foliations in differential topology; geometric theory 32Sxx Complex singularities