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 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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- November 1998
- DOI:
- 10.48550/arXiv.math/9811159
- arXiv:
- arXiv:math/9811159
- Bibcode:
- 1998math.....11159D
- Keywords:
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- Algebraic Geometry
- E-Print:
- Latex