Localization and residue of the Bott class. (Q1426426)

Secondary characteristic classes of foliations (as the Godbillon-Vey class) play a significant role as basic invariants of the geometry of foliations. It is a natural question to ask on which part of the foliated manifold some characteristic class exists. The author considers the above problem for transversely holomorphic foliations, in particular for codimension one. Roughly speaking, consider the complex normal vector bundle of a transversely holomorphic foliation of codimension \(q\), assume that a global trivializing \(q\)-form \(\omega\) exists for the normal bundle. From the Frobenius-Nierenberg theorem there is a \(1\)-form \(\eta\) such that \(d \omega = 2 \pi \sqrt{-1} \eta \wedge \omega\). The cohomology class \(\eta \wedge (d \eta)^q\) is called the Bott class of the foliation. The imaginary part of the Bott class is the most fundamental class of a transversely holomorphic foliation, it is very close to the Godbillon-Vey class, but it is not of maximal transverse degree. The author performs the (non trivial) construction of a localizable part of the Bott class, and relates it to the Godbillon measure of the foliation in the sense of Heitsch and Hurder. For the case of complex codimension one, a version of the residue is introduced and it is shown that the Bott class is localized near the Julia set of the foliation in the sense of Ghys, Gómez-Mont and Saludes. The rigidity of some of these invariants under deformation of the transverse holomorphic structure is studied. Several useful examples are given.