THE INTEGRABILITY OF HOLONOMY COCYCLE ON FOLIATED COMPLEX SURFACE

Abstract

Viêt-Anh Nguyen proved the integrability of holonomy cocycles on a foliated compact complex surface under appropriate assumptions, in his recent paper titled Singular holomorphic foliations by curves I: integrability of holonomy cocycle in dimension 2 (Inventiones Mathematicae, 2018). The purpose of this thesis is to present key steps in the proof by Nguyen, fill in some arguments that are not fully explained in the original paper and look at the problem in higher dimension. The main object of interest is the holonomy cocycle of a foliated complex surface, which reflects the geometry and dynamics of such foliation. The main method of study is to examine the behavior of a path (contained inside a leaf of the foliation) as it approaches a singular point.