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Représentations d'algèbres de Lie dans des groupes de cohomologie à support

Abstract : We consider certain cohomology groups with support for sheaves over algebraic varieties. We study particularly, for line bundles over varieties where a reductive group $G$ operates, the cohomology with support in some subvarieties which are invariant by the action of a Borel subgroup of $G$. Thus, we obtain representations of the Lie algebra of $G$. We analyse them by giving filtrations whose successive quotients are ``generalized Verma modules''. By using the Grothendieck-Cousin complex, this yields back the Borel-Weil-Bott theorem about flag varieties and also determine all the cohomology groups of line bundles on $G \times G-$equivariant compactifications of $G$ (including the wonderful compactifications). This generalizes the well-known description of cohomology groups of line bundles on complete toric varieties.
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https://tel.archives-ouvertes.fr/tel-00002269
Contributor : Arlette Guttin-Lombard <>
Submitted on : Tuesday, January 14, 2003 - 3:29:09 PM
Last modification on : Wednesday, November 4, 2020 - 1:58:08 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 7:20:38 PM

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  • HAL Id : tel-00002269, version 1

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Alexis Tchoudjem. Représentations d'algèbres de Lie dans des groupes de cohomologie à support. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2002. Français. ⟨tel-00002269⟩

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