# Théorie homotopique des schémas d'Atiyah et Hitchin

Abstract : This work introduces Atiyah-Hitchin schemes. They are a family, indexed by a positive integer m, of algebraic varieties $R_m(Y)$ attached to a fixed algebraic variety Y . We study the motivic homotopy properties of these “spaces” in the sense of Morel and Voevodsky. The schemes $F_m$ of pointed rational functions of degree m form a fundamental example. From the topological viewpoint, Segal and F. Cohen et al. proved that the topological space $F_m (C)$ approximates the loop space $Ω² S³$ . We formulate a precise series of conjectures aiming to generalize these results to the algebraic framework. The slogan is that the scheme $R_m (Y)$ should approximate the motivic loop space $Ω^{P¹} Σ^{P¹} Y$. We establish several results in this direction, among which: 1) First, we determine the monoid of naive algebraic connected components of the schemes of rational functions $F_m$ over a base field. The method is simple and elementary. We recover, up to a completion, the group of motivic homotopy classes of endomorphisms of the projective line $P¹$ , as computed by Morel. 2) We construct an algebraic morphism linking $R_m(Y)$ to $Ω^{P¹}Σ^{P¹} Y$. 3) When the algebraic variety Y is defined over C, we give an explicit description of the homotopy type of the topological space $R_m (Y)(C)$ as a functor in $Y(C)$. Moreover, we show that the space $(R_m Y )(C)$ stably splits with the same summands as in the Snaith splitting of $Ω² Σ² Y (C)$.
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Contributor : Christophe Cazanave <>
Submitted on : Sunday, March 14, 2010 - 4:15:51 PM
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• HAL Id : tel-00463680, version 1

### Citation

Christophe Cazanave. Théorie homotopique des schémas d'Atiyah et Hitchin. Mathématiques [math]. Université Paris-Nord - Paris XIII, 2009. Français. ⟨tel-00463680⟩

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