The Douady Space of a Complex Surface

Abstract

We prove that a standard realization of the direct image complex via the so-called Douady–Barlet (Hilbert-Chow in the algebraic case) 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 of which the proof we present is independent; in addition it is elementary and transparent and does not use either perverse sheaves, the methods of positive characteristic, nor Saito's theory of Mixed Hodge Modules. 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 and Witten. Some consequences of the decomposition theorem: the Göttsche Formula holds for complex surfaces; interpretation of the rational cohomologies of Douady spaces as a kind of Fock space; new proofs of results of Briançon and Ellingsrud and Stromme on punctual Hilbert schemes; and computation of the mixed Hodge structure of the Douady spaces in the Kähler case. We also derive a natural connection with Equivariant K-Theory for which, in the case of algebraic surfaces, Bezrukavnikov and Ginzburg have proposed a different approach.