nLab Hodge theory

Overview

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. Hodge theory also applies in combinatorics, for instance to matroids (Huh 22).

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.

Related entries

• p-adic Hodge theory

• harmonic differential form, Hodge star operator

• Hodge theorem

• Hodge structure

• Hodge conjecture

• nonabelian Hodge theory, Lefschetz hyperplane theorem

• motive

References

Contributors

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, Claus Hertling and others. Another very interesting and complex picture is developing for a number of years in works of Goncharov.

General

• 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) [doi:10.1017/CBO9780511615344]

• 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

• P. Griffiths, Hodge theory - from Abel to Deligne, talk, Inst. Adv. Studies 2016 yt; Some geometric applications of Hodge theory, Oct 2020, yt

• 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, pp. 49–74 in: “From tQFT to tt* and integrability”, Proceedings of Symposia in Pure Mathematics 78 (2007, Augsburg, Germany), AMS 2008; arxiv:0807.2199

• Claus Hertling, Claude 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 A course in Hodge theory : with emphasis on multiple integrals

• Hossein Movasati, A course in Hodge theory: with emphasis on multiple integrals, Somerville: International Press of Boston 2021.

Relation to motives

Discussions of Hodge theory in relation to motives can be found 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

Nonabelian Hodge theory

General references on nonabelian Hodge theory include:

• Ron Donagi, Tony Pantev, Lectures on the geometric Langlands conjecture and non-abelian Hodge theory, 2009 (pdf)

• 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)

In combinatorics

• June Huh, Combinatorics and Hodge theory, Proc. Int. Cong. Math. 1 (2022)

  [doi:10.4171/ICM2022/205, pdf, pdf]