A Simpson Correspondence for Abelian Varieties in Positive Characteristic

Abstract

Let X/C be a smooth projective variety over the complex numbers. In the early 90's, Simpson established an equivalence between the category of local systems and that of the semi-stable Higgs bundles whose Chern class is zero. The correspondence between Higgs bundles and local systems can be viewed as a Hodge theorem for nonabelian cohomology. The theory is hence called the non-abelian Hodge theory, and sometimes is called the Simpson correspondence.

It has been a while to search for such a correspondence in positive characteristic. In this article, the following result is shown. Let X be an abelian variety over an algebraically closed field k of characteristic p > 0. Let X' be its Frobenius twist. Fix a natural number r, and consider the moduli stack (Higgs') of Higgs bundles on X' of rank r and the moduli stack (LocSys) of local systems on X of rank r. Then there are maps from (LocSys) and (Higgs') to a common base scheme B', which is an affine space over k. Moreover, there exists an étale surjective map U to B', such that after a base change to U, the two stacks (Higgs') and (LocSys) are isomorphic.

This result partially generalizes a previous result of Groechenig, where X is a curve, to higher dimensional varieties.