Section 8.1 is a reminder of logarithmic vector fields and differential forms and some other classical facts. In sections 8.2–8.4 extensions with logarithmic poles along a divisor D ⊂ M of the sheaf of holomorphic sections of a flat vector bundle on M – D are discussed. In the case of a smooth divisor D in section 3.2, there are three important tools for working with such extensions: the correspondence to certain filtrations, the classical residue endomorphism along D, and the (less familiar) residual connections along D, whose definitions require the choice of a transversal coordinate.

Extensions to singular divisors D are treated in section 8.3 in greater generality than in the literature. If an (automatically locally free) extension to Dreg with logarithmic pole is given, then there exists a unique maximal coherent extension to D. It is locally free only under special circumstances. The case of a normal crossing divisor is discussed in section 8.3, the Gauß–Manin connections for singularities provide other very instructive examples (Theorem 10.3, Theorem 10.7 (b)). In Section 8.4 only some remarks on regular singularities are made.

Logarithmic vector fields and differential forms

For the reader's convenience we put together some definitions and results from [SK4][De1][Ser], which will be useful in sections 8.2 and 8.3.