# Where do the Kähler Identities first appear?

The Kähler identities (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).

Can anyone fill in the historical gap and tell me where or when the Kähler identities first appeared?

-
(By the way, just a comment, not an answer: Kähler metrics seem to appear firstly in: J. A. Schouten and D. van Dantzig, Über unitäre Geometrie, Math. Ann. 103 (1930), no. 1, 319–346 ) –  daniele Dec 17 '12 at 8:54
In 'Eugenio Calabi and Kähler Metrics' by Bourguinon, he attributes the introduction of the notion of a Kähler metric to Kähler's 1933 paper, but makes a parenthetical remark listing the paper you mention, as well as a paper by Schouten from 1929, as "earlier attempts". I'm not sure why these two papers are only listed as attempts; personally, I have not had a look at either of them. –  Michael Albanese Dec 17 '12 at 9:39

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, Theoreme 1, p.42. The modern notation was probably introduced by Weil, since Kodaira in 1952 called the operator $\Lambda$ the "Hodge-Weil operator".