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.
Mark Green, Claire Voisin, Jacob Murre, Algebraic cycles and Hodge theory Lecture Notes in Mathematics, 1594 (1993)
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: -geometry”, eds. Ron Y. Donagi and Katrin Wendland, 87-174
Claus Hertling, Christian Sevenheck, Twistor structures, -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
Discussions of Hodge theory in relation to motives can be found in:
General references on nonabelian Hodge theory include:
Alberto García Raboso, Steven Rayan, Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles, and complex variations of Hodge structure, Fields Inst. Monogr. 34 (2015), 131–171 (arXiv) (Springer)