  Summary: The Hodge conjecture asks whether rational Hodge classes on a smooth
projective manifolds are generated by the classes of algebraic subsets, or
equivalently by Chern classes of coherent sheaves. On a compact Kaehler
manifold, Hodge conjecture is known to be false if algebraic subsets are
replaced with analytic subsets. Here we show that it is even false that for a
Kaehler manifold, Hodge classes are generated by Chern classes of coherent
sheaves. We also show that finite free resolution do not in general exist for
coherent sheaves on compact Kaehler manifolds.
