The non-abelian Hodge locus

Given a family of smooth projective varieties, one can consider the relative de Rham moduli space, of flat vector bundles of rank n on the fibers. The flat vector bundles which underlie a Z-polarized variation of Hodge structure form the “non-abelian Hodge locus”. Simpson proved that this locus is closed and analytic, and he conjectured it is algebraic. Simpson's conjecture would imply a conjecture of Deligne that only finitely many representations of the fundamental group underlie a Z-PVHS on some fiber. I will discuss a proof of Deligne’s and Simpson’s conjectures, under the additional hypothesis that the Z-Zariski closure of monodromy is a cocompact arithmetic group. This is joint work with Salim Tayou.