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?

  • $\begingroup$ (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 ) $\endgroup$ – daniele Dec 17 '12 at 8:54
  • $\begingroup$ 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. $\endgroup$ – Michael Albanese Dec 17 '12 at 9:39

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, 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.

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".

  • $\begingroup$ May I ask, how did you find this out? $\endgroup$ – Michael Albanese Dec 17 '12 at 20:01
  • $\begingroup$ I remembered that Weil's book, which is arguably the first textbook written on Kahler geometry, so I looked there. He had the reference to Hodge, as part of his theory of "harmonic integrals". $\endgroup$ – YangMills Dec 18 '12 at 17:56
  • $\begingroup$ I've had a look at Hodge's paper, but I can't see where exactly the Kähler identities appear. $\endgroup$ – Michael Albanese Dec 19 '12 at 3:29
  • $\begingroup$ It's true, that paper does not really contain the Kahler identities (the 1941 book does), but for example equation (15) in the paper follows from one of the simpler Kahler identities. There is probably some other paper of Hodge that is a more suitable reference, but MR does not do a good job at indexing them. You can find some references in Hodge's book, page 225, but I haven't looked at them. $\endgroup$ – YangMills Dec 19 '12 at 6:13

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