On the Non-abelian Hodge Correspondence for Higher-dimensional Quasiprojective Varieties

On a projective variety, Simpson showed that there is a homeomorphism between the moduli space of semisimple flat bundles and that of polystable Higgs bundles with vanishing Chern classes. Recently, Bakker, Brunebarbe and Tsimerman proved a version of this homeomorphism on a log smooth curve; for a log smooth variety of higher dimension, they got a continuous bijection. In this talk, I will give a sketch of their approach and give an argument that we do get a homeomorphism for arbitrary dimension.