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. Cω-functions at the origin of IRN and
is an ideal with
. If we have R=R(ℂN) 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),Ωp,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.
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Reiffen, H.J. Fastholomorphe Algebren. Manuscripta Math 3, 271–287 (1970). https://doi.org/10.1007/BF01338660
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DOI: https://doi.org/10.1007/BF01338660