Fastholomorphe Algebren

Abstract

In this paper we study residue class rings

in the sequel called algebras - where R is the ring of germs of complex-valued C^∞- resp. C^ω-functions at the origin of IR^N and

is an ideal with

. If we have R=R(ℂ^N) and

with an ideal

of germs of holomorphic functions, then we call A holomorphic. For holomorphic algebras it is possible to introduce the modules Ω(A),Ω^p,q(A) of differential forms of degree r and of bidegree (p,q) in a natural way. We have

. An almost holomorphic (almost complex) structure on an algebra A is a direct sum Ω¹(A)=Ω′+Ω″ with the following property: for the projections p′:Ω¹→Ω', p″:Ω¹→Ω″ we have$\overline {p'\alpha } = p''\bar \alpha$ for all αεΩ¹. Holomorphic algebras have an almost holomorphic structure. We collect some elementary properties of almost holomorphic algebras and then we prove a criterion for integrability in the analytic case, using Cartan's theory of germs of real-analytic sets.