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.


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