A collection of identities which hold on a Kähler manifold, also called the Hodge identities. Let be a Kähler form, be the exterior derivative, where is the del bar operator, be the commutator of two differential operators, and denote the formal adjoint of A. The following operators also act on differential forms on a Kähler manifold:



where J is the almost complex structure,, and denotes the interior product. Then






In addition,




These identities have many implications. For instance, the two operators

and

(called Laplacians because they are elliptic operators) satisfy . At this point, assume that M is also a compact manifold. Along with Hodge's theorem, this equality of Laplacians proves the Hodge decomposition. The operators L and commute with these Laplacians. By Hodge's theorem, they act on cohomology, which is represented by harmonic forms. Moreover, defining

where is projection onto the (p, q)-Dolbeault cohomology, they satisfy



In other words, these operators provide a group representation of the special linear Lie algebra on the complex cohomology of a compact Kähler manifold. In effect, this is the content of the hard Lefschetz theorem.