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The answers to this M.O. question give a history of the Kaehler identities. 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, or as Demailly refers 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?

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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.

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