Toen, Katzarkov, Pantev: Schematic homotopy types and nonabelian Hodge theory. File Toen web publ nht.pdf. Constructs a “Hodge decomposition” (a certain action) on the schematic homotopy type of a smooth projective complex variety. This recovers many other Hodge invariants, short review of these.
C. Simpson. Algebraic aspects of higher nonabelian Hodge theory, math.Algebraic Geometry/9902067.
Chapter 7 of Amoros et al in Complex manifolds folder
References The abelian Hodge theorem and its consequences are nicely explained in the textbook of Griffiths and Harris, but also look at the beautiful papers of Deligne: [1] P. Deligne, Travaux de Griffiths, Seminar Bourbaki 376, Lecture Notes in Math 180, Springer Verlag 1970, 213–237 [2] P. Deligne, Theorie des Hodges II, III. Inst. Hautes tudes Sci. Publ. Math. No. 40 (1971), 5–57; Inst. Hautes tudes Sci. Publ. Math. No. 44 (1974), 5–77. and for the non-abelian, some references are: [3] C. Simpson, Nonabelian Hodge theory. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), [4] C Simpson, Higgs bundles and local systems. Inst. Hautes Etudes Sci. Publ. Math. No. 75 (1992), 5–95. [5] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I,II, Publ. IHES. [6] N. Hitchin, The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.
nLab page on Nonabelian Hodge theory