On a characteristic class associated with deformations of foliations

TL;DR

This paper investigates a characteristic class related to foliation deformations, providing conditions for its triviality and examples where it is non-trivial, especially in transversely holomorphic cases.

Contribution

It establishes conditions for the triviality of the FLK class and constructs examples demonstrating its non-triviality in certain foliations.

Findings

Conditions for FLK class triviality are identified.

FLK class is often non-trivial in transversely holomorphic foliations.

An explicit example and a general construction are provided.

Abstract

A characteristic class for deformations of foliations called the Fuks-Lodder-Kotschick class (FLK class for short) is studied. It seems unknown if there is a real foliation with non-trivial FLK class. In this article, we show some conditions to assure the triviality of the FLK class. On the other hand, we show that the FLK class is easily to be non-trivial for transversely holomorphic foliations. We present an example and give a construction which generalizes it.