The Douady space of a complex surface

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.