On the relative logarithmic connections and relative residue formula

TL;DR

This paper studies relative logarithmic connections on holomorphic vector bundles over complex families, providing conditions for their existence, defining relative residues, and relating them to Chern classes.

Contribution

It introduces a sufficient condition for the existence of relative logarithmic connections and defines the concept of relative residue, linking it to Chern classes.

Findings

Established a criterion for relative logarithmic connection existence.

Defined the notion of relative residue for such connections.

Expressed relative Chern classes via relative residues.

Abstract

We investigate the relative logarithmic connections on a holomorphic vector bundle over a complex analytic family. We give a sufficient condition for the existence of a relative logarithmic connection on a holomorphic vector bundle singular over a relative simple normal crossing divisor. We define the relative residue of relative logarithmic connection and express relative Chern classes of a holomorphic vector bundle in terms of relative residues.