Abstract: The Hitchin map is a morphism from the moduli stack of Higgs bundles to the Hitchin base, e.g., the space of symmetric differentials over base manifold.  When the base manifold is a curve, the Hitchin map is surjective and it plays a very import role in the studying of the moduli space of Higgs bundles over curves. However, if the base manifold is of higher dimension, then  generally the Hitchin map is not  surjective anymore.  To study this map, Chen and Ngo introduced a closed subset of the Hitchin base, called the spectral base and conjectured that the Hitchin map is onto it.

In the first talk, I will introduce the basic definitions of Higgs bundles, the Hitchin map and the spectral base. Then I will show that the spectral base indeed vanishes for irreducible compact locally Hermitian symmetric spaces of rank at least two. This is based on my joint work with Siqi He and Ngaiming Mok.

In the second talk, I will introduce the spectral correspondence and then I will present our solution of Chen-Ngo’s conjecture for rank two Higgs bundles by using the so-called symmetric differentials of rank one introduced by Bogomolov and De Oliveira. This is based on my joint work with Siqi He.