Papers
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  • "Tannaka duality for noetherian projective schemes", in preparation.

    • Abstract: Let X be a noetherian projective scheme over an algebraically closed field k and M the stack of vector bundles on X. We show that X can be recovered as a fine moduli spaces of the isomorphism classes associated functor of the stack of vector bundles on M that verify some direct sum and tensor conditions. This is the analogue of the Tannaka duality for topological compact groups.

  • "Déformation du fibré de Poincaré", Rev. Roum. Math. P.A. vol. 48, no.4, 2003 : postscript

    • Deformations of the Poincaré bundle.
    Abstract: We give a description, in terms of analytic groups cohomology, of the deformations of the Poincaré bundle over a moduli space of vector bundles on an analytic variety X. Knowing that X itself can be described in terms of groups cohomology, we obtain a cohomological analogue of the Kodaira-Spencer map, when X is considered to be the parameter space of the Poincaré bundle.

  • "Sur la stabilité du fibré de Picard", C.R. Acad. Sci. Paris, Ser. I334 (2002), 885-888 : postscript

    • On the stability of the Picard bundle.
    Abstract: We study the stability of the Picard bundle on the moduli space of stable vector bundles of rank r and determinant &xi, over an irreducible, smooth, algebraic curve of genus g > 1. Under existence conditions of (1, 1)-stable vector bundles, this means that (g, deg&xi mod(r)) should not be equal to (2, ± 1), we prove that the Picard bundle is (0, -1/r)-semi-stable.

  • "Propriétés de bidualité des espaces de modules en termes de cohomologie de groupes", PhD thesis, J. Fourier Institute (Grenoble 1), France, October 2001 : postscript

    • Bi-duality properties of the moduli spaces of vector bundles in terms of group cohomology.
    Abstract: The subject of this thesis is the study of duality properties of the moduli spaces of stable (or simple) vector bundles over a complex, compact variety. The thesis breaks into two halves:
    The first one is algebraic, and treats the case of moduli spaces of stable vector bundles on a projective curve. The starting point is a theorem due to Narasimhan and Ramanan, which asserts that one can recover the curve, like an universal deformation space of certain vector bundles over the moduli space. This may be viewed as a bi-duality property for the moduli spaces on projective curves. In order to have good duality properties, the restrictions of the Poincaré bundle on the moduli space should be stable. Using the Hecke cycles we prove that this is indeed the case, and we apply the same techniques to study the stability of the adjoint of the Poincaré bundle, and of the Picard bundle.
    The second part of this thesis is analytical, and it treats the deformations of the Poincaré bundle in terms of groups cohomology. The starting point is a result of S. Kosarew which allows us to identify locally a complex variety with the cohomology of some non-abelian group. Using the analytical description of the moduli space of stable bundles over a complex variety, we also express the deformations of the Poincaré bundle in terms of non-abelian cohomology. Then we obtain a map between two cohomological sets whose differential corresponds to the Kodaira-Spencer infinitesimal deformation map, and we establish a connection with the classical point of view. Using this method we get a filtered group whose successive quotients are additive groups. Finally, we investigate the cohomology of additive groups and the deformations of filtered groups. We prove that, under certain conditions, a filtered group and its associated graduated group are deformation equivalent.
    Mathematical Subject Classification : 14D20, 14H60, 32G08, 32G13, 20G10.