The Douady space of a complex surface
Abstract
We prove that a standard realization of the direct image complex via the socalled DouadyBarlet morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasiisomorphism of complexes. This is a more precise form of a special case of the decomposition theorems of BeilinsonBernsteinDeligneGabber and M. Saito. The proof hinges on the special case of the bidisk in the complex affine plane where we make explicit use of a construction of Nakajima's and of the corresponding representationtheoretic interpretation foreseen by VafaWitten. Some consequences of the decomposition theorem: 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 EllingsrudStromme on punctual Hilbert schemes; computation of the mixed Hodge structure of the Douady spaces in the Kähler case. We also derive a natural connection with Equivariant KTheory for which, in the case of algebraic surfaces, BezrukavnikovGinzburg have proposed a different approach.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 1998
 DOI:
 10.48550/arXiv.math/9811159
 arXiv:
 arXiv:math/9811159
 Bibcode:
 1998math.....11159D
 Keywords:

 Algebraic Geometry
 EPrint:
 Latex