Invariants of vector bundles induced from regular system of differential equations

Author: gega gulaghashvili
Keywords: vector, bundle, invariant
Annotation:

In the work regular system of differential equations on Riemann sphere are investigated. It is known that system of equation on sphere with meromorphic coefficients defines monodromy representation on fundamental group of non compact Riemann surface (Riemann sphere without singular points of system of equations) and induces trivial holomorphic vector on Riemann surface. In work considered extension of this bundle on entire Riemann sphere. The bundle obtained after the extention is nontrivial and has topological invariant Chern number, also splitting type, which completely defines analytic structures of this bundle. The splitting type of bundle is integer valued vector and sum his coordinates coincides of Chern number of this bundle. In First and second sections of the work is given necessary definition, constructions and propositions which used next sections. In the third section technique of construction of bundles by cocycles is given. In fifth section given construction of bundles with meromorphic connection on Riemann sphere, here the algorithm to calculate splitting type of vector bundle is given. This topic not well described in scientific literature, hence in work detailed analysis of this topic is given. In Finally general constructions from previous sections is used for hypergeometric equations and in case, when monodromy representation induced from hypergeometric equations is nonreduced is investigated. In this case is obtained completely solution of the problem, calculated splitting type of two dimensional vector bundle on Riemman sphere.

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რეგულარულ განტოლებათა სისტემისაგან ინდუცირებული ვექტორული ფიბრაციის ინვარიანტების შესახებ [ka]