group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
Hodge theory is the study of properties of (complex, rational and integer) cohomology (and motives) of Kähler manifolds, induced by a Hodge filtration – given in the classical situation by harmonic differential forms – and the corresponding Hodge theorem.
Central aspects of the theory include Hodge decomposition, hard Lefschetz theorem, pure Hodge structures, mixed Hodge structures, variations of Hodge structure, Hodge to de Rham spectral sequence, periods… This basic setup is however by now vastly generalized.
The nomenclature “nonabelian”/“noncommutative” Hodge theory may be confusing - to clarify:
Nonabelian Hodge theory refers to the generalization of classical Hodge theory for commutative spaces from abelian coefficients to nonabelian coefficients.
Noncommutative Hodge theory refers to the generalization of classical Hodge theory from commutative geometry (e.g. algebraic varieties) to noncommutative geometry (e.g. dg-categories – see also the page on noncommutative algebraic geometry). The coefficients in noncommutative Hodge theory are abelian.
Ultimately, the goal is to develop a Hodge theory for noncommutative spaces with nonabelian coefficients.
The main historical contributors to Hodge theory include William Hodge, Phillip Griffiths, Alexander Grothendieck, Pierre Deligne. More recently a noncommutative generalization and refinement of Hodge theory is emerging in the work of M. Kontsevich and collaborators (D. Kaledin, L. Katzarkov, Tony Pantev…) with precursors in the works of Carlos Simpson, Mikio Sato, C. Hertling and others. Another very interesting and complex picture is developing for a number of years in works of Goncharov.
Bertin, Demailly, Luc Illusie, Chris Peters, Introduction to Hodge theory AMS (1996)
Claire Voisin, Hodge theory and the topology of complex Kähler and complex projective manifolds (survey, pdf)
Claire Voisin, Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3
Mark Green, Claire Voisin, Jacob Murre, Algebraic cycles and Hodge theory Lecture Notes in Mathematics, 1594 (1993)
Phillip Griffiths, Periods of integrals on algebraic manifolds I,II,III, Amer. J. Math. 90, 568–626, 808–865 (1968) ; Publ. Math. de l’IHÉS 38, 228–296 (1970) numdam
M. Saito, Mixed Hodge modules, Publ. R.I.M.S. Kyoto Univ. 26 (1990) pp. 221–333.
L. Katzarkov, M. Kontsevich, Tony Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/0806.0107 Proc. of Symposia in Pure Math. 78 (2008), “From Hodge theory to integrability and TQFT: $tt^*$-geometry”, eds. Ron Y. Donagi and Katrin Wendland, 87-174
D. Kaledin, Cartier isomorphism and Hodge theory in the non-commutative case, Arithmetic geometry, 537–562, Clay Math. Proc. 8, Amer. Math. Soc. 2009, arxiv/0708.1574
D. Kaledin, Tokyo lectures “Homological methods in non-commutative geometry”, pdf, TeX
Claus Hertling, Christian Sevenheck, Twistor structures, $tt^*$-geometry and singularity theory, arxiv/0807.2199
C. Hertling, C. Sabbah, Examples of non-commutative Hodge structure (v1 title: Fourier-Laplace transform of flat unitary connections and TERP structures), arxiv/0912.2754
A. B. Goncharov, Hodge correlators, arxiv/0803.0297, Hodge correlators II, arxiv/0807.4855
C. T. C. Wall, Periods of integrals and topology of algebraic varieties, jstor
Discussion of Hodge theory in relation to motives is in
Chris Peters, Tata Lectures on Motivic Aspects of Hodge Theory (pdf)
Chris Peters, Lectures on Motivic Aspects of Hodge Theory: The Hodge Characteristic., Summer School Istanbul 2014, pdf.