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Systèmes locaux rigides et transformation de Fourier.

Abstract : In 1857, translating into modern language, Riemann showed that the hypergeometric equation can be reconstructed up to isomorphism, from the knowledge of its mono dromes points 0, 1 and ∞. In modern language, we say that the hypergeometric equation is rigid and that its local system is physically rigid. Katz, in his book Rigid Local Systems [11] gives a necessary and tiring for a local system L on P1 is physically hard (Theorem 1.1.2 on page 14). In Section 3.1 we extend this definition to the scope of DP1-modules using the notion of minimal extension, which is presented in Chapter 2. Katz shows, see. [11] Theorem 3.0.2 on page 91 that the Fourier transform, positive characteristic to maintain the rigidity index of irreducible perverse sheaves, provided that neither beam nor are Fourier transformed to be ad hoc support. On the other hand Katz also believes that the Fourier transform in the context of D-modules must maintain rigidity index, cf. [11] page 10. Using these conditions as a guide, we infer the statement of Theorem 3.2.1, cf. Section 3.2, and it shows in the case where the starting module is scheduled on P1. During the preparation of this thesis, S. H. Bloch Esnault showed this result in full generality in [2]. We propose here a different proof of departure when the module is regular singularities on P1. The demonstration is done by comparing the index of rigidity of a DP1-module cf. Theorem 3.1.1, and its Fourier transform, cf. Theorem 3.1.7. The expression of the stiffness index of DP1-starting module uses the knowledge of the monodromy on each of its singular points and the expression of the stiffness index of the Fourier transform uses the knowledge of monodromy at 0 and mono drome decomposition Turrittin to infinity. The concepts of Fourier transform and decomposition Turrittin are presented in Chapter 1. In his book differential equations with polynomial coefficients Malgrange shows in an analytical way, that these mono dromes are not independent, cf. [16] Theorem XII.2.9 page 203. In chapter 3 we demonstrated an algebraically also using the concept of pairs of vector spaces, a concept introduced in Chapter 2.
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  • HAL Id : pastel-00002259, version 1



Adelino Paiva. Systèmes locaux rigides et transformation de Fourier.. Mathématiques [math]. Ecole Polytechnique X, 2006. Français. ⟨pastel-00002259⟩



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