# Bounded sets of sheaves on relative analytic spaces

## TL;DR

This paper extends boundedness results for coherent sheaves to relative analytic spaces, enabling new properness and semistability properties, including the existence of relative Harder-Narasimhan filtrations.

## Contribution

It generalizes boundedness and properness results for sheaves from smooth to relative analytic spaces, introducing new tools for semistability analysis.

## Key findings

- Proves properness of the relative Douady space.
- Establishes existence of relative Harder-Narasimhan filtrations.
- Extends boundedness results to non-smooth relative spaces.

## Abstract

We extend previous results on boundedness of sets of coherent sheaves on a compact K\"ahler manifold to the relative and not necessarily smooth case. This enlarged context allows us to prove properness properties of the relative Douady space as well as results related to semistability of sheaves such as the existence of relative Harder-Narasimhan filtrations.