nLab Hodge theorem

Idea

The Hodge theorem asserts, in particular, that for a compact Kähler manifold, the canonical $(p,q)$-grading of its differential forms descends to its de Rham cohomology/ordinary cohomology. The resulting structure is called a Hodge structure, and is indeed the archetypical example of such.

Definitions

Definition

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

The Hodge inner product

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

is given by

$$\langle \alpha , \beta\rangle = \int_X \alpha \wedge \star \beta \,.$$

Write $d^*$ for the formal adjoint of the de Rham differential under this inner product. Then

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

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

$$\Delta \omega = 0$$

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

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

For Riemannian manifolds

Observation

Harmonic forms are precisely those in the kernel of $d + d^*$, which are precisely those in the joint kernel of $d$ and $d^*$.

By the fact that the bilinear form $\langle -,-\rangle$ is non-degenerate.

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

Theorem

The canonical map

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

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 $\Delta$)

$$G : \Omega^\bullet(X) \to \Omega^\bullet(X)$$

such that

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

• $G(\mathcal{H}^\bullet(X)) = 0$;

• and

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

  where $\pi_{\mathcal{H}}$ 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).

For Kähler manifolds

(…)

Related concepts

• Hodge-Maxwell theorem

• Hodge theory, Hodge structure

• nonabelian Hodge theorem

• Poincaré lemma

• Stokes theorem

• de Rham theorem

• Dolbeault theorem

• Hodge conjecture

References

The theorem is due to

• William Hodge, (1947)

Textbook accounts include

• Claire Voisin, section 5 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

• Chris Peters, Jozef Steenbrink, section 1.1. of Mixed Hodge Structures, Ergebisse der Mathematik (2008) (pdf)

Lecture notes include

• Xi Yin, Notes on the Hodge Theorem (pdf)

• Jonathan Evans, Hodge theorem (pdf)

See also

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