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

Relation to motives

Discussions of Hodge theory in relation to motives can be found in:

Nonabelian Hodge theory

General references on nonabelian Hodge theory include: