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Automorphismes réels d’un fibré et opérateurs de Cauchy–Riemann

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Soit \((N,c_N)\) un fibré vectoriel complexe muni d’une structure réelle au-dessus d’une courbe réelle \((\Sigma _g,c_\Sigma )\) de genre \(g\in \mathbb{N }\). Nous étudions le signe de l’action des automorphismes de \((N,c_N)\) relevant l’identité de \(\Sigma _g\) sur les orientations du fibré déterminant au-dessus de l’espace des opérateurs de Cauchy–Riemann réels de \((N,c_N)\). Ce signe s’obtient comme le produit de deux termes. Le premier calcule la signature des permutations induite par les automorphismes sur les structures \(Pin^\pm \) de la partie réelle de \((N,c_N)\). Le second provient de l’action des automorphismes du fibré sur les classes de bordisme de structures \(Spin\) réelles de \((\Sigma _g,c_\Sigma )\).

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Remerciements

Je remercie Jean-Yves Welschinger pour son soutien constant et ses conseils judicieux tout au long de ma thèse dont sont issus les résultats présentés ici.

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  1. Université de Genève, Mathématiques, villa Battelle, 7, route de Drize, 1227 , Carouge, Switzerland

    Rémi Crétois

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Crétois, R. Automorphismes réels d’un fibré et opérateurs de Cauchy–Riemann. Math. Z. 275, 453–497 (2013). https://doi.org/10.1007/s00209-013-1143-z

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