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Niveau: Supérieur, Doctorat, Bac+8

The A1–homotopy type of Atiyah–Hitchin schemes I: the geometry of complex points CHRISTOPHE CAZANAVE Given a smooth algebraic variety Y , we construct a family of new algebraic varieties RnY indexed by a positive integer n , which we baptize the Atiyah– Hitchin schemes of Y . This paper is the first of a series devoted to the study of the A1 –homotopy type (in the sense of Morel and Voevodsky) of these schemes. The interest of the Atiyah–Hitchin schemes is that we conjecture that, as n tends to infinity, the sequence of spaces RnY converges, in a precise sense, to ?P 1?P1 Y , the free P1 –loop space generated by Y . This first paper focuses on the geometry of the schemes RnY : the slogan is that RnY is a scheme-theoretic “completion” of the unordered configuration space of n distinct points in A1 with labels in Y . This makes RnY analogous—although in general different—to the May–Milgram model. 55P35, 55R80, 14F42; 57N80 1 Introduction This is the first of a series of papers devoted to the introduction and the study of some in- teresting families of algebraic varieties which we baptize the Atiyah–Hitchin schemes. There is a family of Atiyah–Hitchin schemes attached to any given algebraic variety Y ; they are indexed by a positive integer n and we denote them RnY . The fundamental example at the source of the definition, corresponding to the case where Y is the affine line minus the origin, is given by the schemes of pointed degree n rational functions (Fn)n>0 (see example

The A1–homotopy type of Atiyah–Hitchin schemes I: the geometry of complex points CHRISTOPHE CAZANAVE Given a smooth algebraic variety Y , we construct a family of new algebraic varieties RnY indexed by a positive integer n , which we baptize the Atiyah– Hitchin schemes of Y . This paper is the first of a series devoted to the study of the A1 –homotopy type (in the sense of Morel and Voevodsky) of these schemes. The interest of the Atiyah–Hitchin schemes is that we conjecture that, as n tends to infinity, the sequence of spaces RnY converges, in a precise sense, to ?P 1?P1 Y , the free P1 –loop space generated by Y . This first paper focuses on the geometry of the schemes RnY : the slogan is that RnY is a scheme-theoretic “completion” of the unordered configuration space of n distinct points in A1 with labels in Y . This makes RnY analogous—although in general different—to the May–Milgram model. 55P35, 55R80, 14F42; 57N80 1 Introduction This is the first of a series of papers devoted to the introduction and the study of some in- teresting families of algebraic varieties which we baptize the Atiyah–Hitchin schemes. There is a family of Atiyah–Hitchin schemes attached to any given algebraic variety Y ; they are indexed by a positive integer n and we denote them RnY . The fundamental example at the source of the definition, corresponding to the case where Y is the affine line minus the origin, is given by the schemes of pointed degree n rational functions (Fn)n>0 (see example

- functions has
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- called whitney
- main result
- rny
- topological space
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- bundles over
- functions

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Publié par | profil-zyak-2012 |

Nombre de lectures | 16 |

Langue | English |

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