Abstract

Every holomorphic vector bundle    on Riemann sphere  splits into the direct sum of line bundles and the total Chern number of this vector bundle  is equal to sum of Chern numbers of line bundles. The integer-valued vector with components Chern number of line bundles is called splitting type of holomorphic vector bundle and is analytic invariant of complex vector bundles.  
There exists a one-to-one correspondence between the H\”older continues matrix function and the holomorphic vector bundles described above, wherein the splitting type of vector bundles coincides with partial indices of matrix functions. It is known that every holomorphic vector bundle equipped with meromorphic (in general) connection  with logarithmic singularities at finite set of marked points and corresponding meromorphic 1-from  have first order poles in marked points and removable singularity at infinity.  
The Fucshian system of equations induced from this 1-form gives the monodromy representation of the fundamental group of Riemann sphere without marked points. The monodromy representation induces trivial holomorphic vector bundles  with connection. The extension of the pair (\texttt{bundle, connection}) on the Riemann sphere is not unique and defines a family of holomorphically nontrivial vector bundles.    
In the talk we present about the following statements:     
1. From the solvability condition (in the sense Galois differential theory) of the Fuchsian    system  follows formula for computation of partial indices of piecewise constant matrix function.     
2. All extensions of  vector bundle on noncompact Riemann surface correspond to    rational matrix functions  algorithmically computable by monodromy matrices of Fucshian system.

This work was supported, in part, by the Shota Rustaveli National Science Foundation under Grant No 17-96.