The spectral base of Hitchin maps | GRASP seminar

Abstract

Let X be a projective manifold. The Hitchin morphism is a map from the moduli stack of Higgsbundles over X to the Hitchin base, which sends a Higgs bundle to its characteristic polynomial. lfX is a curve, it is well-known that the Hitchin morphism is surjective and it plays an important rolein the study of the moduli space of Higgs bundles. However, if X has dimension at least two, theHitchin morphism in general is not suriective. Thus a closed subset of the Hitchin base. called thespectral base, is introduced by Tsao-Hsien Chen and Bao Chau Ngo and it is coniectured that theHitchin morphism is onto the spectral base. This conjecture is confirmed when X is a surface bythe works of Tsao-Hsien Chen and Bao Chau Ngo and Lei Song and Hao Sun. in this talk, l willpresent our solution to this conjecture for rank two Higgs bundles and also show the vanishing ofthe spectral base for Hermitian locally symmetric spaces with higher rank. This is joint work withSigi He and Ngaiming Mok.