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Structure of formal meromorphic connections in several variables and semicontinuity of the irregularity. (Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité.) (French) Zbl 1149.32017
Let \(f: Y \rightarrow X\) be a smooth morphism with connected fibers of complex-analytic varieties, let \(Z\subset Y\) be a hypersurface finite over \(X.\) Let \(\Omega^\bullet_{Y/X}(\star Z)\) be the relative de Rham complex of meromorphic differential forms with poles along the divisor \(Z,\) and let \({\mathcal N}\) be a locally free \({\mathcal O}_Y(\star Z)\)-module. Let us consider a relative connection on \(\mathcal N\) given by the operator \(\nabla:{\mathcal N} \rightarrow {\mathcal N}\otimes_{{\mathcal O}_Y(\star Z)} \Omega^1_{Y/X}(\star Z).\) Then the fiber \({\mathcal N}_{Y_x}\) on the preimage \(Y_x = f^{-1}(x),\, x\in X,\) is endowed with a meromorphic connection \(\nabla_{Y_x}\) having singularities contained in \(Z_x = Z\cap Y_x.\) Set \(i(\nabla, x) = \sum_{z\in Z_x}\text{ir}_z\nabla_{Y_x},\) where the irregularity of the differential operator \(\nabla_{Y_x}\) at \(z\in Z_x\) is denoted by \(\text{ir}_z;\) it is a basic invariant in investigations of asymptotic of meromorphic connections in a neighbourhood of singular points [Y. André and F. Baldassarri, De Rham cohomology of differential modules on algebraic varieties Basel: Birkhäuser (2001; Zbl 0995.14003)].
Under the assumptions that the morphism \(f\) has relative dimension 1 and the connection \(\nabla\) is integrable the author proves that \(i(\nabla,x)\) is a lower semicontinious function on \(X\) (B. Malgrange’s conjecture on the absence of confluence phenomenon for integrable meromorphic connections). The proof is mainly based on considerations from the microlocal theory in the style of [loc. cit.] involving the following topics: the rank of Poincaré-Katz of differential operators, the decomposition of Turritin-Levelt, the Newton polygon of a differential module, blowing ups and “tournant” points, stable and semi-stable points, etc. Thus, among other things the author establishes the semicontinuity of the rank of Poincaré-Katz. He also gives a pure algebraic proof of the well-known result due to P. Deligne [Équations différentielles à points singuliers réguliers. Berlin, etc.: Springer (1970; Zbl 0244.14004)] that the restriction of a regular algebraic integrable connection to any smooth curve is regular.
The author underlines that his proof was inspired by works of C. Sabbah [see Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2. Paris: Société Mathématique de France (2000; Zbl 0947.32005); Ann. Inst. Fourier 43, No. 5, 1619–1688 (1993; Zbl 0803.32005)] where relationships between the confluence problem and the existence of a good formal structure of integrable meromorphic connections in two variables are investigated in the context of microlocal analysis. He also remarks that his technique can be applied in the case of characteristic \(p>0,\) for example, in studies of \(p\)-adic versions of the monodromy theorem and related problems.

MSC:
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32C38 Sheaves of differential operators and their modules, \(D\)-modules
35A20 Analyticity in context of PDEs
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