The only initial datum in section 7.1 is a monodromy operator. For the corresponding flat vector bundle over the punctured plane ℂ* notions such as the elementary sections and the V-filtration are introduced. In section 7.2 Oℂ-free extensions over 0 with regular singularity at 0 of the sheaf of holomorphic sections of the vector bundle are discussed. Comparison with the V-filtration leads to the spectral numbers and certain filtrations. Sections 7.1 and 7.2 are elementary and classical.

The subject of section 7.3 is those extensions over 0 which not only have a regular singularity at 0, but also a logarithmic pole. There is a correspondence between such extensions and certain filtrations, which is not so well known. It has a generalization in section 8.2. It is used in section 7.4 for the solution of a Riemann–Hilbert–Birkhoff problem. This is based on ideas of M. Saito. It is central to the construction of Frobenius manifolds in section 11.1.

In section 7.5 a formula is given for the sum of the spectral numbers in a global situation, when one has on a compact Riemann surface a locally free sheaf and a flat connection with several singularities as above. It is useful in the case of ℙ1 for the Riemann–Hilbert–Birkhoff problem.