Bounded sets of sheaves on compact Kaehler manifolds

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.