Bounded sets of sheaves on compact Kaehler manifolds

TL;DR

This paper establishes boundedness criteria for quotients of coherent sheaves with fixed Chern classes on compact Kähler manifolds, leading to compactness results for components of the Douady space.

Contribution

It adapts Grothendieck's boundedness criterion to the Kähler setting and proves boundedness of sheaf quotients with fixed Chern classes.

Findings

Boundedness of quotients with fixed Chern classes on Kähler manifolds

Compactness of connected components of the Douady space

Extension of boundedness criteria to Kähler geometry

Abstract

We show that any set of quotients with fixed Chern classes of a given coherent sheaf on a compact Kaehler manifold is bounded in a sense which we define. The result is proved by adapting Grothendieck's boundedness criterium expressed via the Hilbert polynomial to the Kaehler set-up. As a consequence we obtain the compactness of the connected components of the Douady space of a compact Kaehler manifold.