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?