# On the relative Connections

## TL;DR

This paper studies relative connections on sheaves of modules, providing conditions for their existence and showing that certain Chern classes vanish under specific geometric conditions.

## Contribution

It introduces a sufficient condition for the existence of relative holomorphic connections and proves the vanishing of relative Chern classes for compact Kähler fibers.

## Key findings

- Existence condition for relative holomorphic connections.
- Vanishing of relative Chern classes on compact Kähler fibers.
- Application to complex analytic families.

## Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and K\"ahler.