The Douady space of a complex surface

Thumbnail Image

Date

Authors

Journal Title

Journal ISSN

Volume Title

Publisher

Abstract

Description

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: 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-Stromme 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 K-Theory for which, in the case of algebraic surfaces, Bezrukavnikov-Ginzburg have proposed a different approach.
Latex; 23 pages

Keywords

Algebraic Geometry

Citation

URI

http://salesiana.dossiersoluciones.com/handle/123456789/77765

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/9811159
 http://arxiv.org/abs/math/9811159

Collections

arXiv