Independent rigid secondary classes for holomorphic foliations

Abstract

A foliation ~on a smooth manifold M is said to transversely holomorphic if ~ ' is locally defined by submersions into ~", and the associated transition functions are biholomorphic. If M is also a complex manifold and the local submersions are holomorphic, then F is said to be holomorphic. The purpose of this paper is to study the characteristic classes of transversely holomorphic foliations. The universal chern classes for transversely holomorphic foliations of codimension n are shown to be independent up to degree 2n, above which they must vanish. Using this result, we then construct for all even complex codimensions the first examples of transversely holomorphic foliations for which a set of rigid secondary classes is non-trivial. Applications of the independence results we obtain are given to the study of BF2 r and to the study of the space of foliations on an open manifold. Heitsch showed that the secondary classes of a foliation divide into two categories [93: the variable classes are those whose value can change under a deformation of the foliation; rigid classes are those invariant under deformation. The examples of Baum-Bott [w 2] established that all of the variable classes of degree 2n+ 1 in codimension n are independently variable in Hz"+I(F~r The Baum-Bott examples are extended in [Theorem 5; 11], so that all of the variable classes in H*(W,,)| which are not products are independently variable in H*(FF~). The examples of Rasmussen [18] show that some of the decomposable classes can also vary. The natural question raised by these examples is whether any rigid classes are non-trivial in H*(FF,,~). We give a positive answer, using the dual homotopy techniques of [10] and a detailed study of the topology of the map v: BF, e-* BU,. The rest of the paper is organized as follows. Section 1 gives the general non-triviality results for the chern classes and rigid secondary classes. Background material about secondary classes and dual homotopy invariants is presented in w 2. Section 3 gives specific results about codimension 2, and w 4 is devoted to the space of foliations. The proofs of Theorems 1.3 and 1.4 are deferred until w 5.