I will describe a program to describe Hitchin components as the moduli space of some new geometric structure on the surface. This geometric structure generalizes the complex structure. Its construction uses the punctual Hilbert scheme of the plane. It should give a unified description of Hitchin components without fixed complex structure on the surface. I also present a generalization to character varieties for non split real groups in the spirit of G-Higgs bundles.[-]

We consider a closed oriented surface of genus at least 2. For any positive integer k, an essential closed curve on the surface with k self-intersections is called a k-curve. A pair of curves on the surface are said to be k-equivalent, if they have the same intersection numbers with each k-curve. In this talk, I will discuss the general picture of a pair of k-equivalent curves and the relation between k-equivalence relations for different k's.
This is a joint-work with Hugo Parlier[-]