Godbillon-Vey class of transversely holomorphic foliations
This monograph is devoted to the study of secondary characteristic classes of transversely holomorphic foliations. The most important classes of this kind are the Bott class and the Godbillon-Vey class; the two are related by the fact that the Godbillon-Vey class can be decomposed as the product of a power of a Chern class and another secondary class, that can be identified with the imaginary part of the Bott class. This work contains two main new results. First of all, the author gives a novel construction of locally homogeneous transversely holomorphic foliations of any codimension with non-trivial Godbillon-Vey classes. Then the author proves the rigidity of Godbillon-Vey classes, under both actual and infinitesimal deformations, in the category of transversely holomorphic foliations. The organization of the monograph is the following. Chapter 1 introduces secondary characteristic classes; Chapter 2 reviews the relationships between real and complex secondary classes. Chapter 3 contains the construction of transversely holomorphic foliations with non-trivial Godbillon-Vey class; Chapter 4 is devoted to the proof of the rigidity theorem. Chapter 5 discusses relationships with other constructions due to Heitsch, Fuks, Lodder and Kotschick. Finally, Chapter 6 presents a (weak) analogue for transversely holomorphic foliations of Duminy's theorem, that relates (in the real case) Godbillon-Vey classes with dynamical properties of the foliations.
scientific article; zbMATH DE number 5724301