It is quite well understood how the Hodge structures of a compact Kähler manifold degenerate when the variety acquires singularities. The picture is however not so clear in the “non-abelian” version, modelled on the moduli of flat bundles of the manifold. In particular one would like to have a nice description of how these moduli spaces degenerate and which kind of monodromy appear around the singular fibres.
In this short survey, the author summarizes the current state of the art and the main difficulties on this question. He first explains how to extend the moduli spaces of flat bundles over a singular fibre (with base of dimension 1), either with moduli spaces of torsion-free sheaves with logarithmic connections, or some desingularizations of these. Then it is discussed how Hitchin’s hyper-Kähler metric (the non-abelian analogue of the Hodge metric) could be extended. Finally the case of higher-dimensional base is also considered.
For the entire collection see [Zbl 1437.11003].