Holomorphic foliations and deformations of the Hopf foliation

Abstract

A deformation theory for transversally holomorphic is developed here and used to give an explicit description of the transver- sally holomorphic near the foliations on odd dimen- sional spheres. Introduction. In (1) and (2) we began the study of the deformation theory of holomorphic on a smooth compact manifold. Our aim was to construct a reasonably explicit parameterization of a neighborhood of a fixed holomoφhic foliation % in the space of all by generalizing Kuranishi's theorem on deformations of complex structures on compact complex manifolds. However, in (1) we assumed the existence of a smooth foliation ψ- transverse to the foliation %. The purpose of the present paper is to eliminate this rather artificial assumption. In (3) Gomez-Mont observed that the Kodaira-Spencer machine can be used to show the existence of such a parameterizati on by an analytic subset of a finite dimensional vector space. However, as is the case for the deforma- tion theory of complex structures, the proof is rather abstract and is not easily adapted to computations. To illustrate our results, we present here a classification of all holomoφhic near the foliation given by the Hopf fibration S2n+X -* CP n. We shall now give a more precise statement of our results. The reader is assumed to be somewhat familiar with the notations and results of (1); but we begin with a short review. Let % be a fixed holomorphic foliation of real codimensions 2q on the smooth, compact, oriented manifold Mn, i.e., % is given locally by smooth submersions into Cq which patch together via local biholomoφhisms of Cq. Let L C TM and Q — TM/L be the (real) tangent and normal bundles of % and fix once and for all a splitting TM — Lθ Q and a Riemannian metric on M respecting it. (In (1) this splitting was assumed to be induced by a transverse foliation. This is not necessarily the case here.) The complex structure map on Q induces a splitting of the complexified normal bundle in the standard way, Qc = <2(10) θ β (01) and there is a split exact sequence r P