Logarithmic Connections and their Residues

Abstract

A central theme in geometry is the construction of characteristic classes of vector bundles from geometric data. This has its roots in Chern-Weil theory, where smooth connections and their curvature are used to construct representatives of these classes. An analogous theory exists in the complex analytic category, whose study was first initiated by Michael Atiyah in [1]. In his 1982 paper [16], Makoto Ohtsuki showed that characteristic classes can also be described using residues. Specifically, given a holomorphic vector bundle E over a compact complex manifold M and a logarithmic connection D on E with pole along a normally crossing divisor Z, he showed the existence of a formula relating the Chern classes of E with the residues of D. In this thesis, we present a mild generalisation of the notion of residue to holomorphic principal bundles over complex manifolds. We also show that Ohtsuki’s formula generalises, under reasonable assumptions on the structure group, to principal bundles over compact complex manifolds.