L'irrégularité du complexe f+(Oeg)

Abstract : In the theory of D-modules, Gauss-Manin systems are defined by the direct image of the structure sheaf by a morphism. A major theorem says that these systems are regular. This thesis examines the irregularity of an analogue of the Gauss-Manin systems, it being the direct image of an elementary D-module associated to a polynomial g by a polynomial f, mainly in the case of two variables. We use two methods and compare them. This irregularity allows control of the immoderate growth of the integrales of an algebraic relative form on a locally constant collection of homology classes in the fibers of f with adapted closed supports. In our first method, we express the irregularity on c by the discriminant curve of f and g. The works of Lê Dung Trang and C. Weber on the resolution of polynomials at infinity help us to define a compactification of f and g. Using the theorem of Z. Mebkhout on the commutation between the direct image and the irregularity complex, we reduce the problem to calculating the Euler characteristics of the irregularity complexes by D-modules in two variables whose singular support is a normal crossing. According to a result of C. Sabbah, we can calculate these Euler characteristics using those of the Milnor fibres. Concerning the irregularity at infinity, we are led to take into account a special curve which comes from the dicritical divisors of a resolution at infinity for f and g. In our second method, we reduce the calculations to the case of two projections. Finally, we express the irregularity of the Gauss-Manin analogue as a function of characteristic cycles of the direct image complex of the structural sheaf by (f,g).
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Submitted on : Friday, September 3, 2004 - 9:54:30 AM
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  • HAL Id : tel-00006796, version 2

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Céline Roucairol. L'irrégularité du complexe f+(Oeg). Mathématiques [math]. Université d'Angers, 2004. Français. ⟨tel-00006796v2⟩

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