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Hodge theorem

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Idea

The Hodge theorem asserts, in particular, that for a compact Kähler manifold, the canonical (p,q)(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)(X,g) be a compact oriented Riemannian manifold of dimension nn. Write Ω (X)\Omega^\bullet(X) for the de Rham complex of smooth differential forms on XX and :Ω (X)Ω n(X)\star : \Omega^\bullet(X) \to \Omega^{n-\bullet}(X) for the Hodge star operator.

The Hodge inner product

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

is given by

α,β= Xαβ. \langle \alpha , \beta\rangle = \int_X \alpha \wedge \star \beta \,.

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

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

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

Δω=0 \Delta \omega = 0

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

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

For Riemannian manifolds

Observation

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

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

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

Theorem

The canonical map

k(X)H 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)(X,g) has precisely one harmonic cocycle reprentative.

But more is true

Theorem

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

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

such that

  • GG commutes with dd and with d *d^*;

  • G( (X))=0G(\mathcal{H}^\bullet(X)) = 0;

  • and

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

    where π \pi_{\mathcal{H}} is the orthogonal projection on harmonic forms and the angular brackets denote the graded commutator [d,d *G]=[d,d *]G=ΔG[d, d^* G] = [d,d^*]G = \Delta G.

See for instance page 6 of (GreenVoisinMurre).

For Kähler manifolds

(…)

References

The theorem is due to

Textbook accounts include

Lecture notes include

See also

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

Revised on June 16, 2016 02:15:25 by Bartek (219.88.235.225)