nLab
Hodge theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Complex geometry

Contents

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.

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.

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

General

  • 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 *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 *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

Relation to motives

Discussion of Hodge theory in relation to motives is in

Revised on June 4, 2014 22:08:05 by Urs Schreiber (82.136.246.44)