Kähler Identities

The Kähler identities, also called the Hodge identities, are a collection of identities which hold on a Kähler manifold. 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 . The following operators also act on differential forms on a Kähler manifold:

			(1)
			(2)
			(3)

where is the almost complex structure, extended to differential forms by applying to each argument. On k-forms, . The symbol denotes the interior product. Then

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

In addition,

			(10)
			(11)
			(12)
			(13)

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

(14)

and

(15)

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

(16)

where is projection onto the -Dolbeault cohomology, they satisfy

			(17)
			(18)
			(19)

In other words, these operators provide a Lie algebra 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.