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If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with

  • $J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed in detail in the rather dated Gadea, Cruceanu, Fortuny - A Survey on Paracomplex Geometry (1996)

and

  • $J\in\operatorname{End}(TM)$ with $J^2 =0$ (also called almost dual number structure) have been studied in detail (within a larger class) by Shirokov, Shurygin, Vishnevskii and others.

Let $M$ now denote a $4n$-dimensional complex manifold and let $J\in\operatorname{End}_{\mathbb{C}}(T^{1,0}M)$ be an almost dual number structure on the holomorphic tangent bundle.

Have complex manifolds with such additional (almost-) dual number structures $(M,J)$ already been studied and if so, what are some references?

It was mentioned to me that (almost) dual number structures are important to physicists for some reason, but I am unsure if this was referring to the real or to the complex case or both. Also, he couldn't recall a precise reference, and my search has been unfruitful as I am quite unfamiliar with the mathematical physics venue.

Any pointers would be appreciated!

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