Natural operations on holomorphic forms

TL;DR

This paper characterizes all natural differential operations on holomorphic forms on complex manifolds, showing they are limited to linear combinations, exterior products, and differentials, using a Galoisian categorical framework.

Contribution

It establishes a classification of natural differential operations on holomorphic forms via a Galoisian categorical approach, revealing their fundamental limitations.

Findings

Only linear combinations, exterior products, and differentials are natural operations on holomorphic forms.

Develops a Galoisian categorical framework for natural holomorphic bundles.

Provides a complete classification of natural differential 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 equivalence {\it \`a la Galois}.