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I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting.

After a quick thought, I've gone through the standard reference (Kolar-Michor-Slovak: Natural operations in differential geometry, Springer, 1993), and through references quoting this, but found nothing about the holomorphic case.

So, my first question is:

  • Is there any reference about natural operations in complex geometry?

As far as I can see, a big part of the theory is purely "algebraic", so it extends without much pain (definition of natural bundle, the universal natural bundle of "references", its Galois theory...).

Nevertheless, there are some points that seem specific of the smooth case; for example:

  1. The Peetre-Slovak theorem on the finiteness of the order of local operators (pdf).
  2. The finiteness of the order of any natural bundle.
  3. The fact that the regularity condition in the definition of natural bundle can be deduced from the other two.

My question is:

  • Does any of the above facts hold in the holomorphic theory? In particular, is there a version of Peetre's theorem for local operators between holomorphic bundles?
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