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On peut construire facilement des exemples de connexions plates de rang sur comme tirés en arrière de connexions sur . On donne un exemple de connexion qui ne peut être obtenue de cette manière. Cet exemple est construit à partir d’une solution algébrique de l’équation de Painlevé VI. On en déduit un feuilletage modulaire. La preuve de ce fait repose sur la classification des feuilletages sur les surfaces projectives par leurs dimensions de Kodaira, fruit du travail de Brunella, McQuillan et...
We consider tracefree meromorphic rank 2 connections over compact Riemann surfaces of arbitrary genus. By deforming the curve, the position of the poles and the connection, we construct the global universal isomonodromic deformation of such a connection. Our construction, which is specific to the tracefree rank 2 case, does not need any Stokes analysis for irregular singularities. It is thereby more elementary than the construction in arbitrary rank due to B. Malgrange and I. Krichever and it includes...
We investigate the interplay between invariant varieties of vector fields and the
inflection locus of linear systems with respect to the vector field. Among the
consequences of such investigation we obtain a computational criterion for the existence
of rational first integrals of a given degree, bounds for the number of first integrals
on families of vector fields, and a generalization of Darboux's criteria. We also provide
a new proof of Gomez--Mont's result on foliations...
We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeros of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. We give...
This paper is devoted to studying the growth and oscillation of solutions and their derivatives of higher order non-homogeneous linear differential equations with finite order meromorphic coefficients. Illustrative examples are also treated.
We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation
, (1)
where , P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z), (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.
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