The Hodge filtration on nonabelian cohomology

Whereas usual Hodge theory concerns mainly the usual or abelian cohomology of an algebraic variety—or eventually the rational homotopy theory or nilpotent completion of π1 which are in some sense obtained by extensions—nonabelian Hodge theory concerns the cohomology of a variety with nonabelian coefficients. Because of the basic fact that homotopy groups in higher dimensions are abelian, and since cohomology theories can generally be interpreted as spaces of maps into classifying (or Eilenberg-MacLane) spaces, nonabelian cohomology occurs essentially only in degree 1. There are certainly some degree 2 aspects which are as of yet totally untouched; and the same goes for the degree 1 case with twisted coefficient systems. (See however [70] for a direction of development combining the nonabelian coefficients in degree 1 with abelian coefficients in higher degrees). If we leave these aside, we are left with the case of H(X,G) for G a nonabelian group. It is most natural to interpret this cohomology as a groupoid, or, when G is a group-scheme, to interpret H(X,G) as a stack. It is the stack of flat principal G-bundles on X. Recall from the usual abelian case that in order to obtain a Hodge structure, we must consider cohomology with complex coefficients. The analogue in the nonabelian case is that we must take as coefficient group a group-scheme G over the complex numbers (and in fact it should be affine too). This then is the domain of application of the work that has been done in nonabelian Hodge theory: the study of properties and additional structure on the moduli stack M(X,G) := H(X,G) which are the analogues in an appropriate sense of the main structures or properties of abelian cohomology. By its nature, the first nonabelian cohomology is an invariant of the fundamental group π1(X). The study of nonabelian Hodge theory may thus be thought of as the study of fundamental groups of algebraic varieties or compact Kähler manifolds. It is important to note, specially in light of Toledo’s examples of π1(X) not residually finite, that the study of π1(X) via its nonabelian cohomology, i.e. via the spaces of homomorphisms π1(X) → G, will only “see” a certain part of π1(X) and in particular will not at all see the intersection of subgroups of finite index. It is an interesting question to try to understand what Hodge-theoretic methods could say about this more mysterious part of the fundamental group. We start in §2 by reviewing Corlette’s nonabelian Hodge theorem [10] (cf also [18] and [17]) which is actually a generalization of the theorem of Eells and Sampson [19]. This