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Hodge theory

Overview

Hodge theory is centered around fine structure seen at the level of (complex, rational and integer) cohomologies (and motives) of Kähler manifolds, based on the study of harmonic forms on them: 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.

For de Rham cohomology on a Riemannian manifold

Definitions

Definition

Let $(X,g)$ be a compact oriented Riemannian manifold of dimension $d$. Write ${\Omega }^{•}(X)$ for the de Rham complex of smooth differential forms on $X$ and $\star :{\Omega }^{•}(X)\to {\Omega }^{d-•}(X)$ for the Hodge star operator.

The Hodge inner product

$$⟨-,-⟩:{\Omega }^{•}(X)\otimes {\Omega }^{d-•}(X)\to ℝ$$\langle -,-\rangle : \Omega^\bullet(X) \otimes \Omega^{d-\bullet}(X) \to \mathbb{R}

is given by

$$⟨\alpha ,\beta ⟩={\int }_X\alpha \wedge \star \beta .$$\langle \alpha , \beta\rangle = \int_X \alpha \wedge \star \beta \,.

Write $d^{*}$ for the formal adjoint? of the de Rham differential under this iner product. Then

$$\Delta :=[d,d^{*}]:=d{d}^{*}+d^{*}d=(d+d^{*}{)}^2$$\Delta := [d,d^*] := d d^* + d^* d = (d + d^*)^2

is the Hodge Laplace operator ($d+d^{*}$ is the corresponding Dirac operator). A differential form $\omega$ in the kernel of $\Delta$

$$\Delta \omega =0$$\Delta \omega = 0

is called a harmonic form on $(X,g)$.

Write ${\mathscr{H}}^k(X)$ for the abelian group of harmonic $k$-forms on $X$.

Results

Observation

Harmonic forms are precisley those in the kernel of $d+d^{*}$, which are pecisely those in the joint kernel of $d$ and $d^{*}$.

By the fact that the bilinear form $⟨-,-⟩$ is non-degenerate.

Therefore we have a canonical map ${\mathscr{H}}^k(X)\to H_{\mathrm{dR}}^k(X)$ of harmonic forms into the de Rham cohomology of $X$.

Theorem

The canonical map

$${\mathscr{H}}^k(X)\to H_{\mathrm{dR}}^k(X)$$\mathcal{H}^k(X) \to H_{dR}^k(X)

is an isomorphism.

This means that every de Rham cohomology class on $(X,g)$ has precisely one harmonic cocycle reprentative.

But more is true

Theorem

For $(X,g)$ as above, there exists a unique degree-preserving operator (the Green operator? of the Laplace operator $\Delta$)

$$G:{\Omega }^{•}(X)\to {\Omega }^{•}(X)$$G : \Omega^\bullet(X) \to \Omega^\bullet(X)

such that

• $G$ commutes with $d$ and with $d^{*}$;

• $G({\mathscr{H}}^{•}(X))=0$;

• and

  $$\mathrm{Id}-{\pi }_{\mathscr{H}}=[d,d^{*}G],$$Id - \pi_{\mathcal{H}} = [d, d^* G] \,,

  where ${\pi }_{\mathscr{H}}$ is the orthogonal projection on harmonic forms and the angular brackets denote the graded commutator $[d,d^{*}G]=[d,d^{*}]G=\Delta G$.

See for instance page 6 of (GreenVoisinMurre).

Related entries

• harmonic form, Hodge star, nonabelian Hodge theory, Lefschetz hyperplane theorem

References

The main historical contributors to Hodge theory include W. Hodge, P. Griffiths, A. Grothendieck, P. 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, T. Pantev…) with precursors in the works of C. Simpson, M. Saito, C. Hertling and others. Another very interesting and complex picture is developing for a number of years in works of Goncharov.

• wikipedia: Hodge structure, Hodge conjecture

• A.I. Ovseevich, Hodge structure, Period mapping, J. Steenbrink, Variation of Hodge structure, Springer Enc. of Math.

• Bertin, Demailly, Illusie, Peters, Introduction to Hodge theory AMS (1996)

• Claire Voisin, Hodge theory and the topology of complex Kähler and complex projective manifolds (survey, pdf)

• C. Voisin, Hodge theory and complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77

• Mark Green, Claire Voisin, Jacob Murre, Algebraic cycles and Hodge theory Lecture Notes in Mathematics, 1594 (1993)

• P.A. Griffiths, J.E. Harris, Principles of algebraic geometry, 1978

• P. 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

• J. Steenbrink, S. Zucker, Variation of mixed Hodge structure I, Invent. Math. 80 (1985), 489-542.

• M. Saito, Mixed Hodge modules, Publ. R.I.M.S. Kyoto Univ. 26 (1990) pp. 221–333.

• L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/0806.0107

• 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, ${\mathrm{tt}}^{*}$-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