group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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.
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
is given by
Write $d^*$ for the formal adjoint of the de Rham differential under this inner product. Then
is the Hodge Laplace operator ($d + d^*$ is the corresponding Dirac operator). A differential form $\omega$ in the kernel of $\Delta$
is called a harmonic form on $(X,g)$.
Write $\mathcal{H}^k(X)$ for the abelian group of harmonic $k$-forms on $X$.
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 $X$.
This means that every de Rham cohomology class on $(X,g)$ has precisely one harmonic cocycle reprentative.
But more is true
For $(X,g)$ as above, there exists a unique degree-preserving operator (the Green operator of the Laplace operator $\Delta$)
such that
$G$ commutes with $d$ and with $d^*$;
$G(\mathcal{H}^\bullet(X)) = 0$;
and
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).
(…)
The theorem is due to
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