flat connections – Francis Bischoff

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 $X$ be a smooth complex manifold, and let $Y \subseteq X$ be a divisor (i.e. a codimension 1 subvariety). Then given a branch of the logarithm $\tau$ (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 $Y$ to a holomorphic vector bundle on $X$ with a flat logarithmic connection having poles along $Y$ :

Theorem. Let $(F,\nabla)$ be a holomorphic vector bundle with flat connection on $X \setminus Y$ . There exists a unique extension $(G,\tilde{\nabla})$ to all of $X$ such that $\tilde{\nabla}$ has logarithmic singularities along $Y \subseteq X$ and the eigenvalues of the residue of $\tilde{\nabla}$ lie in the image of $\tau$ .