Natural operations on holomorphic forms

Abstract

We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed manifold, making explicit its Galoisian structure by proving a categorical equivalenceà la Galois.MSC : 58A32, 32L05.

On Invariant Operations on a Manifold with a Linear Connection and an Orientation

“…The purpose of this section is twofold: On the one hand, we present the notion of natural operation (Definition 7); our definition strongly differs from the standard one (cf. [5]), although it is equivalent to it ( [18]). On the other hand, we prove a general result-Theorem 6-that relates these natural operations with certain smooth equivariant morphisms.…”

“…The present paper lays out complete proofs of the main results of this approach, whose novelties are a systematic use of the language of sheaves, ringed spaces, and a more elementary-yet equivalent (cf. [18])-notion of the natural bundle. In our opinion, the heart of the matter in this theory is the existence of an analogue of a Galois theorem (cf.…”

“…In our opinion, the heart of the matter in this theory is the existence of an analogue of a Galois theorem (cf. [18], Thm. 1.6), which allows the use of group theory to infer theorems in many areas of differential geometry, in many of which (such as Fedosov, contact, or Finsler geometry) this idea is still to be exploited.…”