Autour de l'irrégularité des connexions méromorphes.

Abstract : The first two parts of this thesis take place in the realm of analogies between the irregularity phenomenon for meromorphic connections and wild ramification for l-adic sheaves. We develope the analogue for meromorphic connections of Abbes and Saito's construction, first in the case of a trait, and then in higher dimension. In the first part, we prove an explicit formula relating the invariants produced by Abbes and Saito's construction applied to a differential module M to the most polar parts of the differential forms occuring in the Levelt-Turrittin decomposition of M. In the second part, we generalize to higher dimension the observation coming from the first part that on an algebraically closed field, the modules produced by Abbes and Saito's construction are finite sums of exponential modules associated to linear forms. In the last part of this thesis, we prove that the stable point locus of a meromorphic connection with poles along a smooth divisor is a subset of the intersection of the loci where the irregularity sheaves of M and End M are local systems. Finally, we discuss a strategy to attack the converse inclusion, and we prove that if it is true in dimension 2, then it is true in any dimension. This relies on André's criterion for stable points.
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Submitted on : Friday, November 1, 2013 - 3:07:28 PM
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Jean-Baptiste Teyssier. Autour de l'irrégularité des connexions méromorphes.. Géométrie algébrique [math.AG]. Ecole Polytechnique X, 2013. Français. ⟨pastel-00879175⟩

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