Bounded sets of sheaves on Kähler manifolds

Abstract

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