  We prove that a standard realization of the direct image complex via the
so-called Douady-Barlet morphism associated with a smooth complex analytic
surface admits a natural decomposition in the form of an injective
quasi-isomorphism of complexes. This is a more precise form of a special case
of the decomposition theorems of Beilinson-Bernstein-Deligne-Gabber and M.
Saito.
  The proof hinges on the special case of the bi-disk in the complex affine
plane where we make explicit use of a construction of Nakajima's and of the
corresponding representation-theoretic interpretation foreseen by Vafa-Witten.
Some consequences of the decomposition theorem: G\"ottsche Formula holds for
complex surfaces; interpretation of the rational cohomologies of Douady spaces
as a kind of Fock space; new proofs of results of Brian\c{c}on and
Ellingsrud-Stromme on punctual Hilbert schemes; computation of the mixed Hodge
structure of the Douady spaces in the K\"ahler case. We also derive a natural
connection with Equivariant K-Theory for which, in the case of algebraic
surfaces, Bezrukavnikov-Ginzburg have proposed a different approach.
