Deligne’s construction for extending connections.

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 .

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