In this post I will discuss a construction, due to Deligne, for extending a flat vector bundle to a flat logarithmic vector bundle. The setting of the construction is as follows: let be a smooth complex manifold, and let be a divisor (i.e. a codimension 1 subvariety). Then given a branch of the logarithm (it is in fact sufficient to choose a ‘set theoretical’ logarithm) Deligne’s construction gives a canonical way of extending a flat holomorphic vector bundle on the complement of to a holomorphic vector bundle on with a flat logarithmic connection having poles along :
Theorem. Let be a holomorphic vector bundle with flat connection on . There exists a unique extension to all of such that has logarithmic singularities along and the eigenvalues of the residue of lie in the image of .