# Correspondance de McKay : variations en dimension trois

Abstract : Let $X$ be a quasiprojective scheme smooth over $\mathbb(C)$ and $G$ a finite group. In a first part, we study the $G$-Hilbert scheme of $X$, and construct the Hilbert-Chow morphism by linearization of the determinant. We are then interested in a family of dimension three non abelian singularities, wich admit two natural crepant resolutions. One of them is the result of a Jung process of desingularization of singularities, and the other is the equivariant Hilbert scheme. To compare those resolutions, we calculate the fibres over any singular point in both cases. We exhib a McKay correspondance phenomenon for the Hilbert scheme resolution and construct a new family of vector bundles on $E$ parametrized by the $W_+$-Hilbert scheme. In the last part, we are interested in the $G$-equivariant derived category of $X$. We compare the $G$-equivariant derived category of $X$ with the derived category of the quotient by giving a descent criterion. We also give an equivariant version of Be(\u\i)linson's equivalence of categories.
Keywords :
Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00006683
Contributor : Arlette Guttin-Lombard <>
Submitted on : Thursday, October 14, 2004 - 8:45:17 AM
Last modification on : Tuesday, May 11, 2021 - 11:36:03 AM
Long-term archiving on: : Wednesday, November 23, 2016 - 5:11:48 PM

### Identifiers

• HAL Id : tel-00006683, version 2

### Citation

Sophie Térouanne. Correspondance de McKay : variations en dimension trois. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2004. Français. ⟨tel-00006683v2⟩

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