Derivations on algebras of holomorphic functions

Abstract

Suppose U is a domain in ℂⁿ, not necessarily pseudoconvex, and D is a derivation on the algebra %plane1D;4AA;(U) of holomorphic functions on U, i.e. D : %plane1D;4AA;(U)→%plane1D;4AA;(U) is additive and satisfies Dfg=fDg+gDf for all ƒ,g ε %plane1D;4AA;(U). It is shown that there are h₁,h_nε%plane1D;4AA;(U) such that Df = Σ_i=1ⁿh_i ∂ƒ/∂_Zi for all ƒ ε %plane1D;4AA;(U). The same techniques are then applied to show that, for a Stein manifold Ω, the natural map from the space of global holomorphic sections of the holomorphic tangent bundle of Ω to the space of derivations on %plane1D;4AA;(Ω) is a bijection.