The answers to this M.O. [question](https://mathoverflow.net/questions/116559/where-do-the-k%C3%A4hler-identities-first-appear) give a history of the [Kaehler identities](http://mathworld.wolfram.com/KaehlerIdentities.html). The identities can be extended to the vector bundle-valued setting, and play a central role in the proof of the Kodaira vanishing theorem. Where do these identities first appear? Moreover, is it more common to refer to these identities as the *Nakano identities* or the *Akizuki-Nakano identities*? 


A consequence of these identities is the [Bochner-Kodaira-Nakano identity](https://en.wikipedia.org/wiki/Bochner%E2%80%93Kodaira%E2%80%93Nakano_identity), or as Demailly [refers](https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/identite_KN.pdf) to it, the Bochner-Calabi-Kodaira-Nakano identity. Where does this first appear, and how are the contributions of Bochner, Calabi, Kodaira, and Nakano related?



Where do the (Akizuki)-Nakano Identities First Appear
* [Curvature and Betti
Numbers](https://www.jstor.org/stable/1969287), Salomon Bochner (1948).
* [On a differential-geometric
method in the theory of analytic stacks](http://www.jstor.org/stable/89226), Kunihiko Kodaira (1953).
* [On complex analytic vector bundles](https://projecteuclid.org/euclid.jmsj/1261414950), Shigeo Nakano (1955).
* [Note
on Kodaira-Spencer's Proof of Lefschetz Theorems](https://www.jstage.jst.go.jp/article/pjab1945/30/4/30_4_266/_pdf), Yasuo Akizuki
and Shigeo Nakano (1954).


The Bochner-Kodaira-Nakano identity expresses the antiholomorphic Laplace operator in terms of its conjugate operator plus extra terms involving the curvature of the manifold and the torsion of the metric. If the manifold is Kähler the torsion vanishes and one recovers the Akizuki-Nakano identity.


Calabi was Bochner's Ph.D. student and related results appear in his 1953 thesis, which is probably why Demailly refers to the Bochner-Calabi-Kodaira-Nakano identity.


