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space for rank n representations of the fundamental group of the usual topological space X t~ A vector bundle with integrable connection is a pair (E, V) where E is a vector bundle and V:E-+ E | f~ is an operator satisfying the Leibniz rule and V2-- - 0. The de Rham moduli space MDg(X, n) is a coarse moduli space for rank n vector bundles with integrable connection on X. A Higgs bundle [Hill [Si5] is a pair (E, ~) where E is a vector bundle and q~ : E --> E | f/~ is a morphism of 0x-modules such that q0 ^ q0 = 0. There is a condition of semistability analogous to that for vector bundles, but only concerning subsheaves preserved by ~. The Dolbeault moduli space MDot(X , n) is a coarse moduli space for rank n semlstabte Higgs bundles with Chern classes vanishing in rational cohomology. In all three cases, the objects in question form an abelian category in which we can apply the Jordan-H61der theorem. Let gr(E) denote the direct sum of the subquotients in a Jordan-H61der series for E, and say that E 1 is Jordan equivalent to E~ if gr(E1) _-__ gr(E~). The points of the coarse moduli spaces parametrize Jordan equivalence classes of objects. The constructions of these moduli spaces are reviewed in w 5. The construction of Mu is a classical one from the theory of representations of discrete groups. The construction of MD~ follows from the construction of Part I, w 4, for the case where A "t~ = ~x is the full sheaf of rings of differential operators on X. We give two constructions of MDo ~. One is based on an interpretation of Higgs sheaves as coherent sheaves on T* X, and uses the construction of the moduli space of coherent sheaves constructed in Part I, w 1. The other consists of applying the general construction of Part I, w 4, to the case A"~ Sym'(TX). The three types of objects are related to each other. The Riemann-I-Iilbert correspondence between systems of ordinary differential equations and their monodromy repre

Moduli of representations of the fundamental group of a smooth projective variety. II