The p adic simpson correspondence and higgs isocrystals
In algebraic geometry, the Nonabelian Hodge correspondence or Corlette-Simpson correspondence (named after Kevin Corlette and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundamental group.
Contents
Takeshi tsuji notes on the local p adic simpson correspondence
Unitary representations
For a compact Riemann surface X of genus at least 2, Narasimhan and Seshadri established a bijection between irreducible unitary representations of
Arbitrary representations
These results were extended to arbitrary rank two complex representations of fundamental groups of curves by Hitchin and Donaldson and then in arbitrary dimension by Corlette and Simpson, who established an equivalence of categories between the finite-dimensional complex representations of the fundamental group and the semi-stable Higgs bundles whose Chern class is zero.