The [Kähler identities](http://mathworld.wolfram.com/KaehlerIdentities.html) (sometimes known as the Hodge identities) are an important collection of relationships between operators on the exterior algebra of a Kähler manifold. These relationships generalise to hermitian manifolds, sections of hermitian holomorphic vector bundles, and many other situations. 


I know that the notion of a Kähler metric was introduced by Kähler himself in 1933 and that the Kähler identities were first generalised to hermitian manifolds by Demailly in 1985 (although he mentions that the ideas were present in a paper by Griffiths in 1966). 



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> Can anyone fill in the historical gap and tell me where or when the Kähler identities first appeared?
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Where do the Kähler Identities first appear?
As you hinted yourself, they were indeed discovered by W.V.D. Hodge. They appear explicitly in his 1941 book "The Theory and Applications of Harmonic Integrals", which you can find [here,](http://en.bookfi.org/book/582372) see e.g. the lemmas on pages 172 and following. They were discovered by him some times in the 30s, see e.g. [this 1935 paper of his.](http://dx.doi.org/10.1112/plms/s2-39.1.249)


Hodge's notation is a bit different from the modern one. The first place that I know where they appear in the modern form is A.Weil's book "Introduction a l'étude des variétés kählériennes" (1957), available [here,](http://en.bookfi.org/book/1156268) Theoreme 1, p.42. The modern notation was probably introduced by Weil, since Kodaira in 1952 called the operator $\Lambda$ the "Hodge-Weil operator". 


