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Theses

Some problems in complex and almost-complex geometry

Abstract : In our thesis, we construct or adapt in other settings notions coming from algebraic geometry. We first concern ourselves with the theory of Chern classes for coherent sheaves. For projective manifolds, it is complete in the Chow rings via the existence of global locally free resolutions and it is a formal consequence of the theory for algebraic bundles. A result of Voisin shows that these resolutions do not always exist on general complex compact manifolds. We construct here Chern classes in rational Deligne cohomology for coherent analytic sheaves by induction on the dimension of the base manifold. To do so, we prescribe the Grothendieck-Riemann-Roch formula for immersions and we use dévissage methods. The classes we obtain are the only ones which verify the functoriality formula under pullback, the Whitney formula and the Grothendieck-Riemann-Roch formula for immersions; they then coincide with the topological classes and the Atiyah classes. Moreover, they satisfy the Grothendieck-Riemann-Roch theorem for projective morphisms. The second part of our work consists in studying the punctual Hilbert schemes of a symplectic or almost-complex fourfold. These manifolds have been built by Voisin and generalize the already known Hilbert schemes on projective surfaces. Using the relative integrable structures introduced in Voisin's construction, we can extend the classical theory to the symplectic or almost-complex setting. We compute the Betti numbers, define Nakajima operators, study the cohomology ring and the cobordism class of these Hilbert schemes, and we prove in this context a particular case of Ruan's crepant resolution conjecture.
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Contributor : Julien Grivaux <>
Submitted on : Saturday, February 27, 2010 - 3:14:22 PM
Last modification on : Wednesday, December 9, 2020 - 3:12:27 PM
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  • HAL Id : tel-00460334, version 1

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Julien Grivaux. Some problems in complex and almost-complex geometry. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2009. English. ⟨tel-00460334⟩

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